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Volume 8, Issue 1

# On a class of fully nonlinear parabolic equations

Stanislav Antontsev
• Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk, Russia; and CMAF-CIO, University of Lisbon, Portugal
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/ Sergey Shmarev
• Corresponding author
• Departamento de Matemáticas, Universidad de Oviedo, c/Calvo Sotelo s/n, Oviedo 33007, Spain
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Published Online: 2016-11-23 | DOI: https://doi.org/10.1515/anona-2016-0055

## Abstract

We study the homogeneous Dirichlet problem for the fully nonlinear equation

with the parameters $m>1$, $\sigma >1$ and $d\ge 0$. At the points where $\mathrm{\Delta }u=0$, the equation degenerates if $m>2$, or becomes singular if $m\in \left(1,2\right)$. We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as $t\to \mathrm{\infty }$. Sufficient conditions for exponential or power decay of ${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}$ are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

MSC 2010: 35K55; 35K65 35K67

## 1 Introduction

The paper is devoted to studying the Dirichlet problem for the fully nonlinear parabolic equation

(1.1)

with a nonnegative coefficient d and exponents $m>1$, $\sigma >1$.

The fully nonlinear equation (1.1), without lower-order terms, falls into the class of equations ${u}_{t}=\beta \left(\mathrm{\Delta }u\right)$, which includes the dual porous medium equation

${u}_{t}={|\mathrm{\Delta }u|}^{m-2}\mathrm{\Delta }u,m>1,$(1.2)

and the Barenblatt equation

$F\left({u}_{t}\right)=\mathrm{\Delta }u.$(1.3)

Equation (1.3), with

was introduced in [6] as a model of flows of elastic fluids in elasto-plastic porous media. We refer here to [9, 11, 14, 15, 16] for a discussion on the issues of solvability, asymptotic behavior and some specific properties of solutions to equations (1.2)–(1.3). Equations of the type $F\left({\partial }_{t}u\right)+Au=g$ with maximal monotone operators F and A, as well as the stochastic versions of these equations, are studied in [7, 8]. These works contain a detailed review of the relevant literature and the results on solvability of boundary-value problems for generalizations of the Barenblatt equation (1.3).

The Dirichlet and Cauchy problems for the equation

${u}_{t}=\mathrm{\Phi }\left(u\right){|\mathrm{\Delta }u|}^{m-1}\mathrm{\Delta }u+\mathrm{\Psi }\left(u\right)$(1.4)

were studied in [17, 26], see also [18]. It is shown in [17] that the Cauchy problem admits continuous weak solutions, provided that $u\left(x,0\right)\in {C}^{2+\alpha }\left({ℝ}^{n}\right)$, $u\left(x,0\right)\ge 0$, and the following structural assumptions are fulfilled:

$m\ge 1,\mathrm{\Phi }\left(s\right)\ge 0,\mathrm{\Phi }\left(s\right)\in {C}^{0}\left[0,\mathrm{\infty }\right),\mathrm{\Psi }\left(s\right)\in {C}^{1}\left[0,\mathrm{\infty }\right),\mathrm{\Psi }\left(0\right)=0.$

In [26], local in time existence of a classical solution to the Dirichlet problem for (1.4) is proved under the assumptions that $\mathrm{\Phi }\left(s\right)>0$, Φ and Ψ are Lipschitz-continuous and bounded, and ${u}_{0}\in {C}^{2+\alpha }\left(\overline{\mathrm{\Omega }}\right)$.

At the points where $\mathrm{\Delta }u=0$ equation (1.4) degenerates if $m>2$, or becomes singular if $m\in \left(1,2\right)$. Li et al. [19] consider the Dirichlet problem for the singular equation

${u}_{t}=g\left(x\right){|\mathrm{\Delta }u|}^{m-2}\mathrm{\Delta }u+k\left({u}_{0}-u\right)$(1.5)

with a given smooth function $g\left(x\right)>0$, $k=\text{const}>0$ and $m\in \left(1,2\right)$. It is proved that this problem has an integral solution $u\in C\left(0,T;{C}^{\alpha }\left(\overline{\mathrm{\Omega }}\right)\right)$ in the sense of [22, Chapter 9]. When used for image restoration [19], equation (1.5) prevents the “staircase” effect in the denoising process.

Conditions of existence and nonexistence of nonnegative global solutions of the Cauchy problem for the degenerate equation

${\left({u}^{q}\right)}_{t}={|\mathrm{\Delta }u|}^{m-1}\mathrm{\Delta }u+{u}^{p}$(1.6)

were studied in [13] for $q=1$, $m>1$, $p>1$, and in [18] for the range of parameters $m\ge 1$, $q\ge 1$, $p>0$. A local in time strong (maximal) solution of the Cauchy problem for equation (1.6) is obtained in [18] as the limit of a sequence of solutions of the regularized problems posed in expanding cylinders. The initial datum ${u}_{0}$ is assumed continuous and nonnegative if $p\ge q\ge 1$, while in the case $0, the condition $0<ϵ\le {u}_{0}$ in ${ℝ}^{n}$ is required. Because of the non-divergent form of equation (1.6), the study of conditions for nonexistence of global solutions required a new approach, which was proposed in [13] for the case $q=1$ and extended in [18] to the case $q>1$. In these papers, analogues of the critical Fujita exponents are derived, which indicate the ranges of the exponents of nonlinearity where the solutions of the Cauchy problem for (1.6) blow-up in a finite time or exist globally in time.

To the best of our knowledge, by now there are no results on the asymptotic behavior of solutions of problem (1.1) as $t\to \mathrm{\infty }$ nor the possibility of extinction of solutions in a finite time. We deal with the strong solutions of problem (1.1) understood in the following sense.

#### Definition 1.

A function $u\left(x,t\right)$ is called strong solution of problem (1.1) if the following hold:

• (i)

$u\in C\left(\left[0,T\right];{H}_{0}^{1}\left(\mathrm{\Omega }\right)\right)\cap {L}^{\sigma }\left({Q}_{T}\right)\cap {L}^{m}\left(0,T;{W}^{2,m}\left(\mathrm{\Omega }\right)\right)$, ${u}_{t}\in {L}^{{m}^{\prime }}\left({Q}_{T}\right)$, ${|u|}^{\sigma -1}\in {L}^{{m}^{\prime }}\left({Q}_{T}\right)$, ${m}^{\prime }=\frac{m}{m-1}$,

• (ii)

for every test-function $\varphi \in {L}^{m}\left({Q}_{T}\right)$,

${\int }_{{Q}_{T}}\left({u}_{t}-{|\mathrm{\Delta }u|}^{m-2}\mathrm{\Delta }u+d{|u|}^{\sigma -2}u-f\right)\varphi 𝑑z=0,$(1.7)

• (iii)

for every $\varphi \left(x\right)\in {L}^{m}\left(\mathrm{\Omega }\right)$, ${\left(u\left(t\right)-{u}_{0},\varphi \right)}_{2,\mathrm{\Omega }}\to 0$ as $t\to {0}^{+}$.

In Section 2 we derive sufficient conditions of global in time existence of strong solutions of problem (1.1). The solution is constructed as the limit of a sequence of Galerkin’s approximations. This is in fact an energy solution, in the sense that the solution itself can be taken for the test-function in identity (1.7). We do not distinguish the singular and degenerate cases ($m\in \left(1,2\right)$ or $m>2$). It turns out that for $n=1,2$ a strong solution exists for any $m>1$ and $\sigma \in \left(1,\mathrm{\infty }\right)$, while in the case $n\ge 3$, the exponent $\sigma >1$ has to satisfy the inequality $\sigma <\mathrm{max}\left\{m,\frac{2n}{n-2}\right\}$. The existence theorem is proved under the natural assumptions ${u}_{0}\in {H}_{0}^{1}\left(\mathrm{\Omega }\right)$ and $f\in {L}^{{m}^{\prime }}\left({Q}_{T}\right)$.

In Section 3 we derive conditions of uniqueness of strong solutions. These conditions are different for various ranges of the exponents m, σ and the space dimension n.

Section 4 is devoted to study the asymptotic behavior of solutions as $t\to \mathrm{\infty }$, and the possibility of extinction of solutions in a finite time. For energy solutions, this analysis is reduced to the study of the behavior of the energy function $y\left(t\right)={\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}$ as $t\to \mathrm{\infty }$. The energy $y\left(t\right)$ satisfies the ordinary differential inequality, linear or nonlinear in dependence on the assumptions on the parameters of nonlinearity, and may be nonhomogeneous if $f\ne 0$. It is shown, in particular, that for certain ranges of the exponents m and σ every strong solution of problem (1.1) vanishes at a finite moment, and this moment can be estimated through the data. The bulk of the material of this section is the derivation of differential inequalities for the energy function under various assumptions on the parameters of nonlinearity. We follow here the ideas of monographs [3] and [1] and adapt them to the study of fully nonlinear equations in non divergent form. As an illustration, let us summarize here the results that correspond to the model case when $n=2$, $d>0$, $f\equiv 0$ and ${u}_{0}\in {H}_{0}^{1}\left(\mathrm{\Omega }\right)$.

• (a)

For every $m>1$ and $\sigma >1$ problem (1.1) has at least one strong solution.

• (b)

The solution is unique if either $m\in \left(1,2\right]$ and $\sigma \ge 2$, or $m>2$ and $3\le \sigma \le m+2$.

• (c)

If $m\ge 2$ and $\sigma \in \left(1,2\right)$, then every strong solution:

• vanishes in a finite time if $m+\sigma <4$,

• decreases exponentially if $m+\sigma =4$,

• has the power decay

with L being a constant independent of u.

• (d)

If $m\in \left(1,2\right)$, then every solution vanishes in a finite time.

It is worth noting that all results can be extended to more general equations with variable coefficients,

${u}_{t}=a\left(x,t\right){|\mathrm{\Delta }u|}^{m-2}\mathrm{\Delta }u-d\left(x,t\right){|u|}^{\sigma -2}u+f,$

under suitable regularity assumptions and the conditions $0<{a}_{0}\le a<\mathrm{\infty }$ and $0\le d<\mathrm{\infty }$. We do not discuss this issue in order to avoid unnecessary technical complications.

The asymptotic behavior of solutions of nonlinear parabolic equations in divergence form has been studied by many authors. The most precise results have been obtained for the equation

${u}_{t}=\mathrm{div}\left({|u|}^{m-1}\nabla u\right)+\mu u,m>0,\mu \ge 0,$(1.8)

and the doubly nonlinear equation

${u}_{t}=\mathrm{div}\left({|u|}^{m-1}{|\nabla u|}^{p-2}\nabla u\right),p>1,m+p>3.$(1.9)

It is proved in [10] that for $\mu =0$ and $m\in \left(0,1\right)$ (the fast diffusion equation) every solution of the homogeneous Dirichlet problem for equation (1.8) vanishes at a finite moment ${T}^{\ast }$. Moreover, as $t\to {T}^{\ast }$, the solution converges to a separable solution of the form

$v\left(x,t\right)=K{\left({T}^{\ast }-t\right)}^{\frac{m}{1-m}}w\left(x\right),K=\text{const}>0.$

The case $m>1$ (the slow diffusion equation) and $\mu \ge 0$ is studied in [4]. It is shown that in this case every solution converges as $t\to \mathrm{\infty }$ to a separable solution of the same problem. The rate of convergence is estimated in both cases.

The asymptotic behavior of solutions of the homogeneous Dirichlet problem for equation (1.9) is studied in [20]. It turns that the solutions of equation (1.9) converge, as $t\to \mathrm{\infty }$, to a nonnegative separable solution $v=\theta \left(t\right)w\left(x\right)$ of the same equation. The rate of convergence is different in the cases $m+p>3$ and $m+p=3$. There exist positive constants C and β such that for $t>1$,

$|u\left(x,t\right)|\le C{t}^{\frac{1}{3-m-p}}w\left(x\right)$$|u\left(x,t\right)|\le C{\mathrm{e}}^{-\beta t}w\left(x\right)$

The cited results are established for the equations that admit separable solutions. It seems plausible that similar results could be established for the equations of the type (1.1) that admit separable solutions, but for the moment this question is left open.

For a review of the previous work we refer to papers [20] and [21]. The latter contains results on the asymptotic behavior of solutions for several classes of second-order parabolic equations in divergence form, which are obtained by means of the energy estimates.

The influence of the lower-order terms on the asymptotic behavior of solutions is studied in [2] for a class of anisotropic parabolic equations with variable exponents of nonlinearity which includes, as a special case, the equation

${u}_{t}=\sum _{i=1}^{n}{D}_{i}\left(a\left(x,t,u\right){|{D}_{i}u|}^{p-2}{D}_{i}u\right)+c\left(x,t\right){|u|}^{\sigma -2}u,$(1.10)

with constant exponents $p>1$, $\sigma >1$, and the coefficients $0<{a}_{0}\le a\left(x,t,u\right)\le {a}_{1}$, $0\le {c}_{0}\le c\left(x,t\right)\le {c}_{1}$. Set

It is shown in [2, Theorems 6.1, 6.2, 8.1] that every weak solution of the Dirichlet problem for equation (1.10):

• vanishes in a finite time if either ${c}_{0}>0$ and $\nu <1$, or ${c}_{0}=0$ and $1<\frac{1}{\nu }\le 1+\frac{2}{n}$,

• decreases exponentially if $\nu =1$, i.e., ${\parallel u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le L{\mathrm{e}}^{-Ct}$, $L,C=\text{const}>0$,

• has the power decay ${\parallel u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le C{\left(1+t\right)}^{-\frac{\nu }{\nu -1}}$ if $\nu >1$.

A revision of the proofs given in [2] shows that the same properties hold for the solutions of the isotropic equation

${u}_{t}=\mathrm{div}\left(a\left(x,t,u\right){|\nabla u|}^{p-2}\nabla u\right)+c\left(x,t\right){|u|}^{\sigma -2}u.$(1.11)

Observe that equations (1.1) and (1.11) coincide when $p=m=2$, and the coefficients a and c are constant. If this happens, the conditions on σ that guarantee one or another type of asymptotic behavior of solutions also coincide.

A comparison of the results of [10] and [20] on the asymptotic behavior of solutions of the porous medium equation

${u}_{t}=\mathrm{div}\left({|u|}^{m-1}\nabla u\right),$(1.12)

and the results for the dual porous medium equation (1.2) (see Theorem 4 below) shows that their solutions share the asymptotic properties. The solutions of (1.12) vanish in a finite time if $m\in \left(0,1\right)$, decrease exponentially if $m=1$, or decrease like ${t}^{\frac{1}{1-m}}$ if $m>1$, while the solutions of (1.2) display the same behavior in the ranges of the exponent $m\in \left(1,2\right)$, $m=2$, or $m>2$.

Throughout the text, we denote ${\parallel u\parallel }_{p,\mathrm{\Omega }}={\parallel u\parallel }_{{L}^{p}\left(\mathrm{\Omega }\right)}$ and ${\parallel u\left(t\right)\parallel }_{p,\mathrm{\Omega }}={\parallel u\left(\cdot ,t\right)\parallel }_{p,\mathrm{\Omega }}$ for the functions depending on $\left(x,t\right)\in \mathrm{\Omega }×\left(0,T\right)$. ${W}^{k,p}\left(\mathrm{\Omega }\right)$ is the Sobolev space with the norm ${\parallel u\parallel }_{{W}^{k,p}\left(\mathrm{\Omega }\right)}$, with

$\parallel u{\parallel }_{{W}^{k,p}\left(\mathrm{\Omega }\right)}^{p}=\sum _{0\le |\alpha |\le k}\parallel {D}^{\alpha }u{\parallel }_{p,\mathrm{\Omega }}^{p}.$

The Hilbert space ${H}_{0}^{k}\left(\mathrm{\Omega }\right)={W}_{0}^{k,2}\left(\mathrm{\Omega }\right)$ is the closure of the set ${C}_{c}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ (smooth functions with compact support in Ω) with respect to the norm of ${W}^{k,2}\left(\mathrm{\Omega }\right)$. Notice that ${\parallel \mathrm{\Delta }u\parallel }_{m,\mathrm{\Omega }}$ is an equivalent norm of the space ${W}^{2,m}\left(\mathrm{\Omega }\right)\cap {H}_{0}^{1}\left(\mathrm{\Omega }\right)$. Indeed, on the one hand we have $\parallel \mathrm{\Delta }u{\parallel }_{m,\mathrm{\Omega }}^{m}\le {\sum }_{i=1}^{n}\parallel {D}_{i}^{2}u{\parallel }_{m,\mathrm{\Omega }}^{m}\le n\parallel u{\parallel }_{{W}^{2,m}\left(\mathrm{\Omega }\right)}^{m}$, while on the other, $u\in {W}^{2,m}\left(\mathrm{\Omega }\right)\cap {H}_{0}^{1}\left(\mathrm{\Omega }\right)$ is a solution of the Poisson equation with the right-hand side set to $\mathrm{\Delta }u$ and with zero trace on $\partial \mathrm{\Omega }$. Thus, it satisfies the estimate

${\parallel u\parallel }_{{W}^{2,m}\left(\mathrm{\Omega }\right)}\le C{\parallel \mathrm{\Delta }u\parallel }_{m,\mathrm{\Omega }},$(1.13)

see, e.g., [23, 24].

In the rest of the text C stands for a constant which can be explicitly calculated but whose exact value is unimportant and may change from line to line. We repeatedly use the Young inequality: for every $a,b\ge 0$ and $ϵ>0$, $p>1$,

$ab\le ϵ{a}^{p}+{C}_{ϵ}{b}^{q},{C}_{ϵ}=\frac{\left(p-1\right){ϵ}^{-\frac{1}{p-1}}}{{p}^{q}},q=\frac{p}{p-1}.$

## 2 Existence of strong solutions

#### Theorem 1.

Let $\mathrm{\partial }\mathit{}\mathrm{\Omega }\mathrm{\in }{C}^{\mathrm{2}}$. Assume that

(2.1)

Then, for every ${u}_{\mathrm{0}}\mathrm{\in }{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$, $f\mathrm{\in }{L}^{{m}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}{Q}_{T}\mathrm{\right)}$, problem (1.1) has at least one strong solution which satisfies the energy equality

(2.2)

## 2.1 Galerkin’s approximations

Let $s\ge 1$ be a natural number. By $\left\{{\psi }_{i}\right\}$, we denote the system of eigenfunctions of the problem

Without loss of generality, we assume that $\left\{{\psi }_{i}\right\}$ forms an orthonormal basis of ${L}^{2}\left(\mathrm{\Omega }\right)$. A solution of problem (1.1) will be constructed as the limit of the sequence $\left\{{u}^{\left(k\right)}\right\}$, where ${u}^{\left(k\right)}={\sum }_{i=1}^{k}{c}_{i}\left(t\right){\psi }_{i}\left(x\right)$. The coefficients ${c}_{i}\left(t\right)$ are defined from the system of ordinary differential equations

$\left\{\begin{array}{cc}& {c}_{i}^{\prime }\left(t\right)={\int }_{\mathrm{\Omega }}{\psi }_{i}\left(x\right)|\mathrm{\Delta }{u}^{\left(k\right)}{|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}dx-d{\int }_{\mathrm{\Omega }}|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}{\psi }_{i}\left(x\right)dx+{\int }_{\mathrm{\Omega }}{\psi }_{i}\left(x\right)fdx,\hfill \\ & {c}_{i}\left(0\right)={\left({u}_{0},{\psi }_{i}\right)}_{2,\mathrm{\Omega }},i=1,2,\mathrm{\dots },k.\hfill \end{array}$(2.3)

For every finite k the right-hand side in (2.3) is a Hölder-continuous function of ${c}_{1},{c}_{2},\mathrm{\dots },{c}_{k}$, and by Peano’s theorem there exists at least one solution, $𝐜\left(t\right)=\left({c}_{1},\mathrm{\dots },{c}_{k}\right)$, defined on an interval $\left[0,{T}_{k}\right]$. To prove that each of $𝐜\left(t\right)$ can be continued to the maximal existence interval $\left[0,T\right]$, it is sufficient to derive independent of k a priori estimates on ${u}^{\left(k\right)}$.

## 2.2 A priori estimates

#### Lemma 2.

Let $m\mathrm{>}\mathrm{1}$, $\sigma \mathrm{>}\mathrm{1}$ and ${u}_{\mathrm{0}}\mathrm{\in }{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$, $f\mathrm{\in }{L}^{{m}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}{Q}_{T}\mathrm{\right)}$. Then the functions ${u}^{\mathrm{\left(}k\mathrm{\right)}}$ satisfy the following uniform estimate:

$\frac{1}{2}\underset{\left(0,T\right)}{\mathrm{ess}\mathrm{sup}}\parallel \nabla {u}^{\left(k\right)}\left(t\right){\parallel }_{2,\mathrm{\Omega }}^{2}+\frac{1}{2}\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,{Q}_{T}}^{m}+d\left(\sigma -1\right){\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{\sigma -2}|\nabla {u}^{\left(k\right)}{|}^{2}dz\le \parallel \nabla {u}_{0}^{\left(k\right)}{\parallel }_{2,\mathrm{\Omega }}^{2}+C\parallel f{\parallel }_{{m}^{\prime },{Q}_{T}}^{{m}^{\prime }},$(2.4)

with the constant C depending only on m and n.

#### Proof.

It is easy to check that for every sufficiently smooth function u,

$-{\int }_{\mathrm{\Omega }}|u{|}^{\sigma -2}u\mathrm{\Delta }udx=-{\int }_{\mathrm{\Omega }}{\left({u}^{2}\right)}^{\frac{\sigma -2}{2}}u\mathrm{\Delta }udx$$=\frac{\sigma -2}{2}{\int }_{\mathrm{\Omega }}{\left({u}^{2}\right)}^{\frac{\sigma -4}{2}}2{u}^{2}|\nabla u{|}^{2}dx+{\int }_{\mathrm{\Omega }}|u{|}^{\sigma -2}|\nabla u{|}^{2}dx$$=\left(\sigma -1\right){\int }_{\mathrm{\Omega }}|u{|}^{\sigma -2}|\nabla u{|}^{2}dx.$(2.5)

Multiplying each one of equations (2.3) by ${\lambda }_{i}{c}_{i}\left(t\right)$, summing up from 1 to k, integrating the result over an interval $\left(0,t\right)\subset \left(0,{T}_{k}\right)$, and then applying (2.5), we obtain

$-\frac{1}{2}{\int }_{0}^{t}\frac{d}{dt}\left(\parallel \nabla {u}^{\left(k\right)}\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}^{2}\right)d\tau =\parallel \mathrm{\Delta }{u}^{\left(k\right)}\left(t\right){\parallel }_{m,{Q}_{t}}^{m}-a{\int }_{{Q}_{t}}|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}\mathrm{\Delta }{u}^{\left(k\right)}dz+{\int }_{{Q}_{t}}\mathrm{\Delta }{u}^{\left(k\right)}fdz$$=\parallel \mathrm{\Delta }{u}^{\left(k\right)}\left(t\right){\parallel }_{m,{Q}_{t}}^{m}+d\left(\sigma -1\right){\int }_{{Q}_{t}}|{u}^{\left(k\right)}{|}^{\sigma -2}|\nabla {u}^{\left(k\right)}{|}^{2}dz+{\int }_{{Q}_{t}}\mathrm{\Delta }{u}^{\left(k\right)}fdz.$

Applying Young’s inequality to the last term on the left-hand side, we find that for every $t\in \left(0,{T}_{k}\right)$ and any $ϵ>0$, we have

$\frac{1}{2}\parallel \nabla {u}^{\left(k\right)}\left(t\right){\parallel }_{2,\mathrm{\Omega }}^{2}+\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,{Q}_{t}}^{m}+d\left(\sigma -1\right){\int }_{{Q}_{t}}|{u}^{\left(k\right)}{|}^{\sigma -2}|\nabla {u}^{\left(k\right)}{|}^{2}dz+{\int }_{{Q}_{t}}\mathrm{\Delta }{u}^{\left(k\right)}fdz$$=\frac{1}{2}{\parallel \nabla {u}_{0}^{\left(k\right)}\parallel }_{2,\mathrm{\Omega }}^{2}+{\int }_{{Q}_{t}}\mathrm{\Delta }{u}^{\left(k\right)}f𝑑z$$\le \frac{1}{2}{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}+ϵ{\parallel {u}^{\left(k\right)}\parallel }_{m,{Q}_{t}}^{m}+{C}_{ϵ}{\parallel f\parallel }_{{m}^{\prime },{Q}_{t}}^{{m}^{\prime }}.$

Applying (1.13), we estimate

${\parallel {u}^{\left(k\right)}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\le {C}^{\prime }{\parallel \mathrm{\Delta }{u}^{\left(k\right)}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\mathit{ }\text{and}\mathit{ }{\parallel {u}^{\left(k\right)}\parallel }_{m,{Q}_{t}}^{m}\le {C}^{\prime }{\parallel \mathrm{\Delta }{u}^{\left(k\right)}\parallel }_{m,{Q}_{t}}^{m}.$

Choosing ϵ so small that $ϵ{C}^{\prime }\le \frac{1}{2}$, we obtain (2.4) for every $t\in \left(0,{T}_{k}\right)$. This inequality remains valid for every $t\in \left(0,T\right)$ because the right-hand side is independent of k, and thus $T={T}_{k}$. ∎

#### Lemma 3.

Under the conditions of Theorem 1, there exists a constant C, independent of k, such that

(2.6)

#### Proof.

Combining the embedding theorems in Sobolev spaces with (2.4), we find that

${\parallel {u}^{\left(k\right)}\parallel }_{r,{Q}_{T}}\le T{\parallel {u}^{\left(k\right)}\parallel }_{{L}^{\mathrm{\infty }}\left(0,T;{L}^{r}\left(\mathrm{\Omega }\right)\right)}\le C{\parallel \nabla {u}^{\left(k\right)}\parallel }_{{L}^{\mathrm{\infty }}\left(0,T;{L}^{2}\left(\mathrm{\Omega }\right)\right)}\le C{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }},$

with $1 if $n>2$, $r\in \left(1,\mathrm{\infty }\right)$ if $n=2$, or $r=\mathrm{\infty }$ if $n=1$. Then

$\parallel {u}^{\left(k\right)}{\parallel }_{m,{Q}_{T}}^{m}={\int }_{0}^{T}\parallel {u}^{\left(k\right)}\left(t\right){\parallel }_{m,\mathrm{\Omega }}^{m}dt\le C\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,{Q}_{T}}^{m}\le C\parallel \nabla {u}_{0}{\parallel }_{2,\mathrm{\Omega }}^{2},$

and, finally,

$\parallel {u}^{\left(k\right)}{\parallel }_{\sigma ,{Q}_{T}}^{\sigma }={\int }_{0}^{T}{\int }_{\mathrm{\Omega }}{\left(|{u}^{\left(k\right)}{|}^{\frac{\sigma }{2}}\right)}^{2}dz\le C{\int }_{0}^{T}{\int }_{\mathrm{\Omega }}{|\nabla \left(|{u}^{\left(k\right)}{|}^{\frac{\sigma }{2}}\right)|}^{2}dz\le C{\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{\sigma -2}|\nabla {u}^{\left(k\right)}{|}^{2}dz\le C\parallel \nabla {u}_{0}{\parallel }_{2,\mathrm{\Omega }}^{2}.$

Gathering these three estimate, we obtain (2.6). ∎

#### Lemma 4.

Let in the conditions of Lemma 2

$\sigma \le 1+\frac{q\left(m-1\right)}{m},$

with q defined in (2.6). Then ${u}_{t}^{\mathrm{\left(}k\mathrm{\right)}}\mathrm{\in }{L}^{{m}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{H}^{\mathrm{-}s}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\right)}$, and there exists a constant C, independent of k, such that

${\parallel {u}_{t}^{\left(k\right)}\parallel }_{{L}^{{m}^{\prime }}\left(0,T;{H}^{-s}\left(\mathrm{\Omega }\right)\right)}\le C.$

#### Proof.

Let us denote ${\mathcal{𝒫}}_{k}=\mathrm{span}\left\{{\psi }_{1},\mathrm{\dots },{\psi }_{k}\right\}$, and choose $s\in ℕ$ large enough so that ${H}_{0}^{s}\left(\mathrm{\Omega }\right)↪{L}^{m}\left(\mathrm{\Omega }\right)$. Given $\varphi \in {L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)$, we denote ${\varphi }^{\left(k\right)}={\sum }_{i=1}^{k}{\varphi }_{i}\left(t\right){\psi }_{i}\left(x\right)$, with ${\varphi }_{i}\left(t\right)={\left(\varphi ,{\psi }_{i}\right)}_{2,\mathrm{\Omega }}$. By virtue of (2.3) and due to the orthogonality of the system $\left\{{\psi }_{i}\right\}$ in ${L}^{2}\left(\mathrm{\Omega }\right)$, we have

${\int }_{{Q}_{T}}{u}_{t}^{\left(k\right)}\varphi dz={\int }_{{Q}_{T}}\left(\sum _{j=1}^{k}{c}_{j}^{\prime }\left(t\right){\psi }_{j}\left(x\right)\right)\left(\sum _{i=1}^{k}{\varphi }_{i}\left(t\right){\psi }_{i}\left(x\right)\right)dz$$={\int }_{0}^{T}\left({\int }_{\mathrm{\Omega }}\left(|\mathrm{\Delta }{u}^{\left(k\right)}{|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}{\varphi }^{\left(k\right)}-d|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}{\varphi }^{\left(k\right)}+f{\varphi }^{\left(k\right)}\right)dx\right)dt$$=:{I}_{1}+{I}_{2}+{I}_{3}.$

Using Hölder’s inequality, the embedding theorems in Sobolev spaces and the inequalities

${\parallel {\varphi }^{\left(k\right)}\left(t\right)\parallel }_{{H}_{0}^{s}\left(\mathrm{\Omega }\right)}^{2}=\sum _{i=1}^{k}{\lambda }_{i}{\varphi }_{i}^{2}\left(t\right)\le \sum _{i=1}^{\mathrm{\infty }}{\lambda }_{i}{\varphi }_{i}^{2}\left(t\right)={\parallel \varphi \left(t\right)\parallel }_{{H}_{0}^{s}\left(\mathrm{\Omega }\right)}^{2},$

we find that

$|{I}_{1}|\le {\int }_{0}^{T}\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,\mathrm{\Omega }}^{m-1}\parallel {\varphi }^{\left(k\right)}{\parallel }_{m,\mathrm{\Omega }}dt\le C{\int }_{0}^{T}\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,\mathrm{\Omega }}^{m-1}\parallel {\varphi }^{\left(k\right)}{\parallel }_{{H}_{0}^{s}\left(\mathrm{\Omega }\right)}dt$$\le C{\int }_{0}^{T}\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,\mathrm{\Omega }}^{m-1}\parallel \varphi {\parallel }_{{H}_{0}^{s}\left(\mathrm{\Omega }\right)}dt\le C\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,{Q}_{T}}^{m-1}\parallel \varphi {\parallel }_{{L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)}.$

Likewise, the term ${I}_{2}$ is estimated as follows:

$|{I}_{2}|\le d{\int }_{0}^{T}{\int }_{\mathrm{\Omega }}|{u}^{\left(k\right)}{|}^{\sigma -1}|{\varphi }^{\left(k\right)}|dz$$\le d{\int }_{0}^{T}\left(\int {}_{\mathrm{\Omega }}|{u}^{\left(k\right)}{|}^{{m}^{\prime }\left(\sigma -1\right)}dx\right){}^{\frac{1}{{m}^{\prime }}}\parallel \varphi {}^{\left(k\right)}\parallel {}_{m,\mathrm{\Omega }}dt$$\le d\left({\int }_{{Q}_{T}}{|{u}^{\left(k\right)}{|}^{{m}^{\prime }\left(\sigma -1\right)}dz\right)}^{\frac{1}{{m}^{\prime }}}\parallel {\varphi }^{\left(k\right)}{\parallel }_{{L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)}$$\le d\left({\int }_{{Q}_{T}}{|{u}^{\left(k\right)}{|}^{{m}^{\prime }\left(\sigma -1\right)}dz\right)}^{\frac{1}{{m}^{\prime }}}\parallel \varphi {\parallel }_{{L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)}.$

The assumption on σ entails the inequality

${\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{{m}^{\prime }\left(\sigma -1\right)}dz\le C\parallel {u}^{\left(k\right)}{\parallel }_{q,{Q}_{T}},$

which gives

$|{I}_{2}|\le C{\parallel {u}^{\left(k\right)}\parallel }_{q,{Q}_{T}}{\parallel {\varphi }^{\left(k\right)}\parallel }_{{L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)}\le C{\parallel {u}^{\left(k\right)}\parallel }_{q,{Q}_{T}}{\parallel \varphi \parallel }_{{L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)}.$

The term ${I}_{3}$ is estimated by Hölder’s inequality as follows:

$|{I}_{3}|\le {\int }_{0}^{T}\parallel f{\parallel }_{{m}^{\prime },\mathrm{\Omega }}\parallel {\varphi }^{\left(k\right)}{\parallel }_{m,\mathrm{\Omega }}dt\le C{\int }_{0}^{T}\parallel f{\parallel }_{{m}^{\prime },\mathrm{\Omega }}\parallel {\varphi }^{\left(k\right)}{\parallel }_{{H}_{0}^{s}\left(\mathrm{\Omega }\right)}dt\le C\parallel f{\parallel }_{{m}^{\prime },{Q}_{T}}\parallel \varphi {\parallel }_{{L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)}.$

Combining these estimates with (2.4), we find that for every $\varphi \in {L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)$ with $\parallel \varphi \parallel \le 1$,

$|{\int }_{{Q}_{T}}{u}_{t}^{\left(k\right)}\varphi dz|\le C,$

with the constant C depending on ${\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}$, ${\parallel f\parallel }_{{m}^{\prime },{Q}_{T}}$, m, σ and $|{Q}_{T}|$, but being independent of k. ∎

## 2.3 Convergence

By Lemmas 2 and 4, ${\parallel {u}^{\left(k\right)}\parallel }_{{L}^{\mathrm{\infty }}\left(0,T;{H}_{0}^{1}\left(\mathrm{\Omega }\right)\right)}$ and ${\parallel {u}_{t}^{\left(k\right)}\parallel }_{{L}^{{m}^{\prime }}\left(0,T;{H}^{-s}\left(\mathrm{\Omega }\right)\right)}$ are uniformly bounded. Since we have ${H}_{0}^{1}\left(\mathrm{\Omega }\right)\subset {L}^{2}\left(\mathrm{\Omega }\right)\subset {H}^{-s}\left(\mathrm{\Omega }\right)$, with the embedding ${H}_{0}^{1}\left(\mathrm{\Omega }\right)\subset {L}^{2}\left(\mathrm{\Omega }\right)$ being compact, it follows, from the compactness result of Aubin [5], that $\left\{{u}^{\left(k\right)}\right\}$ is precompact in $C\left(\left[0,T\right];{L}^{2}\left(\mathrm{\Omega }\right)\right)$. The sequence $\left\{{u}^{\left(k\right)}\right\}$ contains a subsequence (for which we keep the same name), that possesses the following properties: there exist functions $u\in C\left(\left[0,T\right];{L}^{2}\left(\mathrm{\Omega }\right)\right)$, $U\in {L}^{2}\left(\mathrm{\Omega }\right)$ and $\chi \in {L}^{{m}^{\prime }}\left({Q}_{T}\right)$ such that

(2.7)

## 2.4 Passing to the limit

Let us fix an arbitrary $r\in ℕ$. By the method of construction, we have

${\int }_{{Q}_{T}}\left({u}_{t}^{\left(k\right)}-{|\mathrm{\Delta }{u}^{\left(k\right)}|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}+d{|{u}^{\left(k\right)}|}^{\sigma -2}{u}^{\left(k\right)}-f\right){\varphi }^{\left(r\right)}𝑑z=0$(2.8)

for every $k\ge r$ and every ${\varphi }^{\left(r\right)}\in {\mathcal{𝒫}}_{r}\cap {L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)$. The convergence properties (2.7) allow one to pass to the limit as $k\to \mathrm{\infty }$. Thus,

${\int }_{{Q}_{T}}\left({u}_{t}-\chi +d{|u|}^{\sigma -2}u-f\right){\varphi }^{\left(r\right)}𝑑z=0$

for every fixed $r\in ℕ$, Since ${L}^{m}\left(0,T;{H}_{0}^{s}\left(\mathrm{\Omega }\right)\right)$ is dense in ${L}^{m}\left({Q}_{T}\right)$, the last equality is true in the limit as $r\to \mathrm{\infty }$. Hence,

${\int }_{{Q}_{T}}\left({u}_{t}-\chi +d{|u|}^{\sigma -2}u-f\right)\varphi 𝑑z=0$(2.9)

for every $\varphi \in {L}^{m}\left({Q}_{T}\right)$, whence ${u}_{t}\in {L}^{{m}^{\prime }}\left({Q}_{T}\right)$. Let us introduce the function space

$X\left(0,T;\mathrm{\Omega }\right)=\left\{u:{Q}_{T}↦ℝ\mid {u}_{t}\in {L}^{{m}^{\prime }}\left({Q}_{T}\right),u\in {L}^{2}\left(0,T;{H}_{0}^{1}\left(\mathrm{\Omega }\right)\right)\cap {L}^{m}\left(0,T;{W}^{2,m}\left(\mathrm{\Omega }\right)\right)\right\}.$

#### Lemma 5.

Let $\mathrm{\partial }\mathit{}\mathrm{\Omega }\mathrm{\in }{C}^{\mathrm{2}}$. If $u\mathrm{\in }X\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}\mathrm{\Omega }\mathrm{\right)}$, then $\mathrm{\nabla }\mathit{}u\mathrm{\in }C\mathit{}\mathrm{\left(}\mathrm{\left[}\mathrm{0}\mathrm{,}T\mathrm{\right]}\mathrm{;}{L}^{\mathrm{2}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\right)}$, after possible redefining on a set of zero measure in $\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{\right)}$.

#### Proof.

Let $u\in X\left(0,T;\mathrm{\Omega }\right)$. Following the proof of [12, Section 5.4, Theorem 1], one may construct the linear extension operator ${E}_{1}$ so that ${E}_{1}u=u$ a.e. in ${Q}_{T}$, $\mathrm{supp}{E}_{1}u\left(\cdot ,t\right)\subset {\mathrm{\Omega }}^{\prime }$ for a.e. $t\in \left(0,T\right)$, $\mathrm{\Omega }\subset \subset {\mathrm{\Omega }}^{\prime }$, and

${\parallel {E}_{1}u\parallel }_{X\left(0,T;{\mathrm{\Omega }}^{\prime }\right)}\le C{\parallel u\parallel }_{X\left(0,T;\mathrm{\Omega }\right)},$

with C being a constant independent of u. Denote $v={E}_{1}u$. The same arguments show that there exists a linear extension operator ${E}_{2}$ such that ${E}_{2}v=v$ a.e. in ${Q}_{T}^{\prime }={\mathrm{\Omega }}^{\prime }×\left(0,T\right)$, $\mathrm{supp}{E}_{2}v\subset \subset {\mathrm{\Omega }}^{\prime }×\left(-T,2T\right)$ and

${\parallel {E}_{2}\left({E}_{1}u\right)\parallel }_{X\left(-T,2T;{\mathrm{\Omega }}^{\prime }\right)}\le C{\parallel {E}_{2}v\parallel }_{X\left(0,T;{\mathrm{\Omega }}^{\prime }\right)}\le {C}^{\prime }{\parallel u\parallel }_{X\left(0,T;\mathrm{\Omega }\right)},$

with ${C}^{\prime }$ being an absolute constant. The constructed extension $w={E}_{2}v={E}_{2}\left({E}_{1}u\right)$ coincides with u a.e. in ${Q}_{T}$. Set ${Q}^{\prime }={\mathrm{\Omega }}^{\prime }×\left(-T,2T\right)$ and choose a family ${w}_{ϵ}\in {C}^{\mathrm{\infty }}\left({\overline{Q}}^{\prime }\right)$ of functions with compact supports in ${Q}^{\prime }$, approximating w in the norm of $X\left(-T,2T;{\mathrm{\Omega }}^{\prime }\right)$. Such a sequence can be obtained by means of mollification, see [12, Section 5.4, Theorems 2 and 3]. For every $\tau ,t\in \left[0,T\right]$, we have

${\parallel \nabla \left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{2,\mathrm{\Omega }}^{2}\left(t\right)\le {\parallel \nabla \left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{2,{\mathrm{\Omega }}^{\prime }}^{2}\left(t\right)$$={\parallel \nabla \left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{2,{\mathrm{\Omega }}^{\prime }}^{2}\left(\tau \right)+2{\int }_{\tau }^{t}{\int }_{{\mathrm{\Omega }}^{\prime }}\nabla \left({w}_{ϵ}-{w}_{\delta }\right)\cdot \nabla \left({\partial }_{t}{w}_{ϵ}-{\partial }_{t}{w}_{\delta }\right)dz$$={\parallel \nabla \left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{2,{\mathrm{\Omega }}^{\prime }}^{2}\left(\tau \right)-2{\int }_{\tau }^{t}{\int }_{{\mathrm{\Omega }}^{\prime }}\mathrm{\Delta }\left({w}_{ϵ}-{w}_{\delta }\right)\cdot {\partial }_{t}\left({w}_{ϵ}-{w}_{\delta }\right)dz$$\le {\parallel \nabla \left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{2,{\mathrm{\Omega }}^{\prime }}^{2}\left(\tau \right)+2{\parallel \mathrm{\Delta }\left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{m,{Q}^{\prime }}{\parallel {\partial }_{t}\left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{{m}^{\prime },{Q}^{\prime }}.$

Integrating this inequality in τ and simplifying, we obtain

${\parallel \nabla \left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{2,\mathrm{\Omega }}^{2}\left(t\right)\le \frac{1}{T}{\parallel \nabla \left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{2,{Q}^{\prime }}^{2}+2{\parallel \mathrm{\Delta }\left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{m,{Q}^{\prime }}{\parallel {\partial }_{t}\left({w}_{ϵ}-{w}_{\delta }\right)\parallel }_{{m}^{\prime },{Q}^{\prime }}$

It follows that

that is, $\left\{\nabla {w}_{ϵ}\right\}$ is a Cauchy sequence in $C\left(\left[0,T\right];{L}^{2}\left(\mathrm{\Omega }\right)\right)$ and converges to a function $V\in C\left(\left[0,T\right];{L}^{2}\left(\mathrm{\Omega }\right)\right)$. On the other hand,

whence $\nabla u=V$ for a.e. in ${Q}_{T}$. ∎

#### Corollary 6.

The function $\mathrm{\nabla }\mathit{}u\mathit{}\mathrm{\left(}T\mathrm{\right)}$ is defined by continuity and in (2.7), $U\mathrm{=}\mathrm{\nabla }\mathit{}u\mathit{}\mathrm{\left(}T\mathrm{\right)}$.

#### Lemma 7.

For every $u\mathrm{\in }X\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}\mathrm{\Omega }\mathrm{\right)}$ and every $\mathrm{0}\mathrm{\le }{t}_{\mathrm{1}}\mathrm{<}{t}_{\mathrm{2}}\mathrm{\le }T$,

${\int }_{{t}_{1}}^{{t}_{2}}{\int }_{\mathrm{\Omega }}{u}_{t}\mathrm{\Delta }udz+\frac{1}{2}{\int }_{\mathrm{\Omega }}{|\nabla u{|}^{2}dx|}_{t={t}_{1}}^{t={t}_{2}}=0.$

#### Proof.

Denote by $\left\{{u}^{h}\right\}$ the sequence of Steklov’s means of u, that is,

${u}^{h}=\frac{1}{h}{\int }_{0}^{h}u\left(x,t+\tau \right)𝑑\tau ,0

It is known that ${u}^{h}\in X\left(0,T-h;\mathrm{\Omega }\right)$ and ${\parallel {u}^{h}-u\parallel }_{X\left(0,T-h;\mathrm{\Omega }\right)}\to 0$ as $h\to 0$. Since

${\partial }_{t}{u}^{h}=\frac{1}{h}\left(u\left(x,t+h\right)-u\left(x,t\right)\right)\in {L}^{m}\left(0,T-h;{W}_{0}^{2,m}\left(\mathrm{\Omega }\right)\right)\cap {L}^{2}\left(0,T-h;{H}_{0}^{1}\left(\mathrm{\Omega }\right)\right),$

we may apply the Green formula (see, e.g., [25, pp. 69–70]). For every $0 and every $0<{t}_{1}<{t}_{2}, we have

${\int }_{{t}_{1}}^{{t}_{2}}{\int }_{\mathrm{\Omega }}{u}_{t}\mathrm{\Delta }u𝑑z={\int }_{{t}_{1}}^{{t}_{2}}{\int }_{\mathrm{\Omega }}\left({\partial }_{t}\left(u-{u}^{h}\right)\mathrm{\Delta }u+{\partial }_{t}{u}^{h}\left(\mathrm{\Delta }u-\mathrm{\Delta }{u}^{h}\right)+{\partial }_{t}{u}^{h}\mathrm{\Delta }{u}^{h}\right)𝑑z$$=-\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla {u}^{h}{|}^{2}dx{|}_{t={t}_{1}}^{t={t}_{2}}+{\int }_{{t}_{1}}^{{t}_{2}}{\int }_{\mathrm{\Omega }}\left({\partial }_{t}\left(u-{u}^{h}\right)\mathrm{\Delta }u+{\partial }_{t}{u}^{h}\left(\mathrm{\Delta }u-\mathrm{\Delta }{u}^{h}\right)\right)dz.$(2.10)

For every $t\in \left(0,T-h\right)$,

$\underset{h\to 0}{lim}\parallel \nabla {u}^{h}\left(t\right){\parallel }_{2,\mathrm{\Omega }}^{2}=\parallel \nabla u\left(t\right){\parallel }_{2,\mathrm{\Omega }}^{2}.$

Indeed, by Lemma 5, $\nabla u\left(t\right)\in C\left(\left[0,T\right];{L}^{2}\left(\mathrm{\Omega }\right)\right)$, which means that for every $t\in \left(0,T-h\right)$ there exists $\xi \in \left(0,h\right)$ such that

${|{\parallel \nabla {u}^{h}\left(t\right)\parallel }_{2,\mathrm{\Omega }}-{\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}|}^{2}\le {\parallel \nabla {u}^{h}\left(t\right)-\nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}$$=\frac{1}{{h}^{2}}{\int }_{\mathrm{\Omega }}\left({\int }_{0}^{h}\left(\nabla u\left(x,t+\tau \right)-\nabla u\left(x,t\right)\right)d\tau \right){}^{2}dx$$\le \frac{1}{{h}^{2}}{\int }_{\mathrm{\Omega }}\left(\sqrt{h}\left({\int }_{0}^{h}{|\nabla u\left(x,t+h\right)-\nabla u\left(x,t\right){|}^{2}d\tau \right)}^{\frac{1}{2}}\right){}^{2}dx$$=\frac{1}{h}{\int }_{0}^{h}\parallel \nabla u\left(\cdot ,t+\tau \right)-\nabla u\left(\cdot ,t\right){\parallel }_{2,\mathrm{\Omega }}^{2}d\tau$

On the other hand,

$|{\int }_{{t}_{1}}^{{t}_{2}}{\int }_{\mathrm{\Omega }}{u}_{t}\mathrm{\Delta }udz+\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla {u}^{h}{|}^{2}dx{|}_{t={t}_{1}}^{t={t}_{2}}|\le \parallel {\partial }_{t}\left(u-{u}^{h}\right){\parallel }_{{m}^{\prime },{Q}_{T-h}}\parallel \mathrm{\Delta }u{\parallel }_{m,{Q}_{T-h}}+\parallel {\partial }_{t}{u}^{h}{\parallel }_{{m}^{\prime },{Q}_{T}}\parallel \mathrm{\Delta }u-\mathrm{\Delta }{u}^{h}{\parallel }_{m,{Q}_{T-h}}\to 0$

as $h\to 0$, by virtue of (2.10). ∎

#### Corollary 8.

Let $u\mathrm{=}\mathrm{lim}\mathit{}{u}^{\mathrm{\left(}k\mathrm{\right)}}$. The function $\mathrm{\Delta }\mathit{}u\mathrm{\in }{L}^{m}\mathit{}\mathrm{\left(}{Q}_{T}\mathrm{\right)}$ is an admissible test-function in (2.9). Choosing $\varphi \mathrm{=}\mathrm{\Delta }\mathit{}u$ and applying Lemma 7, from (2.9), we obtain

$\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla u{|}^{2}dx{|}_{t=0}^{t=T}=-{\int }_{{Q}_{T}}\left(\chi -d|u{|}^{\sigma -2}u+f\right)\mathrm{\Delta }udz.$(2.11)

#### Lemma 9.

Let $u\mathrm{\in }X\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}\mathrm{\Omega }\mathrm{\right)}\mathrm{\cap }{L}^{q}\mathit{}\mathrm{\left(}{Q}_{T}\mathrm{\right)}$ with

$m>1,1<\sigma <1+\frac{q\left(m-1\right)}{m}$

and q from condition (2.1). Then

$\underset{k\to \mathrm{\infty }}{lim}{\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}\mathrm{\Delta }{u}^{\left(k\right)}dz={\int }_{{Q}_{T}}|u{|}^{\sigma -2}u\mathrm{\Delta }udz.$

#### Proof.

Let us make use of the representation

${\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}\mathrm{\Delta }{u}^{\left(k\right)}dz={\int }_{{Q}_{T}}\left(|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}-|u{|}^{\sigma -2}u\right)\mathrm{\Delta }{u}^{\left(k\right)}dz+{\int }_{{Q}_{T}}|u{|}^{\sigma -2}u\mathrm{\Delta }{u}^{\left(k\right)}dz$$=:{I}_{1}^{\left(k\right)}+{I}_{2}^{\left(k\right)}.$

For every $u\in {L}^{q}\left({Q}_{T}\right)$, the inclusion ${|u|}^{\sigma -2}u\in {L}^{{m}^{\prime }}\left({Q}_{T}\right)$ holds, whence, by virtue of (2.7),

Notice that the assumption $\sigma <1+\frac{q\left(m-1\right)}{m}$ is equivalent to ${m}^{\prime }\left(\sigma -1\right). The convergence ${I}_{1}^{\left(k\right)}\to 0$ follows from the Vitali convergence theorem. Since ${v}^{\left(k\right)}={|{u}^{\left(k\right)}|}^{\sigma -2}{u}^{\left(k\right)}\to v={|u|}^{\sigma -2}u$ a.e. in ${Q}_{T}$ and ${\parallel {v}^{\left(k\right)}\parallel }_{\frac{q}{\sigma -1},{Q}_{T}}\le C$ uniformly in k, we have ${v}^{\left(k\right)}\to v$ in ${L}^{r}\left({Q}_{T}\right)$, with $1\le r<\frac{q}{\sigma -1}$. By Hölder’s inequality,

$|{I}_{1}^{\left(k\right)}|\le {\int }_{{Q}_{T}}|{v}^{\left(k\right)}-v||\mathrm{\Delta }{u}^{\left(k\right)}|dz\le C\parallel {v}^{\left(k\right)}-v{\parallel }_{{m}^{\prime },{Q}_{T}}\parallel \mathrm{\Delta }{u}^{\left(k\right)}{\parallel }_{m,{Q}_{T}}.$

The second factor here is uniformly bounded in k due to (2.4), while the first factor tends to zero because ${m}^{\prime }\left(\sigma -1\right). ∎

Let us identify the function $\chi \in {L}^{{m}^{\prime }}\left({Q}_{T}\right)$. Choosing in (2.8) ${\varphi }^{\left(r\right)}=\mathrm{\Delta }{u}^{\left(k\right)}\in {L}^{m}\left({Q}_{T}\right)$ and applying Lemma 5, we find that for every $\varphi \in {\mathcal{𝒫}}_{l}$ with $l\le k$, we have

$0=\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla {u}^{\left(k\right)}{|}^{2}dx{|}_{t=0}^{t=T}+{\int }_{{Q}_{T}}|\mathrm{\Delta }{u}^{\left(k\right)}{|}^{m}dz-d{\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}\mathrm{\Delta }{u}^{\left(k\right)}dz+{\int }_{{Q}_{T}}f\mathrm{\Delta }{u}^{\left(k\right)}dz$$=\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla {u}^{\left(k\right)}{|}^{2}dx{|}_{t=0}^{t=T}+{\int }_{{Q}_{T}}|\mathrm{\Delta }{u}^{\left(k\right)}{|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}\left(\mathrm{\Delta }{u}^{\left(k\right)}-\varphi \right)dz$$+{\int }_{{Q}_{T}}|\mathrm{\Delta }{u}^{\left(k\right)}{|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}\varphi dz-d{\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}\mathrm{\Delta }{u}^{\left(k\right)}dz+{\int }_{{Q}_{T}}f\mathrm{\Delta }{u}^{\left(k\right)}dz.$(2.12)

Since

${|\mathrm{\Delta }{u}^{\left(k\right)}|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}\left(\mathrm{\Delta }{u}^{\left(k\right)}-\varphi \right)=\left({|\mathrm{\Delta }{u}^{\left(k\right)}|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}-{|\varphi |}^{m-2}\varphi \right)\left(\mathrm{\Delta }{u}^{\left(k\right)}-\varphi \right)+{|\varphi |}^{m-2}\varphi \left(\mathrm{\Delta }{u}^{\left(k\right)}-\varphi \right)$$\ge {|\varphi |}^{m-2}\varphi \left(\mathrm{\Delta }{u}^{\left(k\right)}-\varphi \right),$

from (2.12), we obtain

$0\ge \frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla {u}^{\left(k\right)}{|}^{2}dx{|}_{t=0}^{t=T}+{\int }_{{Q}_{T}}|\varphi {|}^{m-2}\varphi \left(\mathrm{\Delta }{u}^{\left(k\right)}-\varphi \right)dz$$+{\int }_{{Q}_{T}}|\mathrm{\Delta }{u}^{\left(k\right)}{|}^{m-2}\mathrm{\Delta }{u}^{\left(k\right)}\varphi dz-d{\int }_{{Q}_{T}}|{u}^{\left(k\right)}{|}^{\sigma -2}{u}^{\left(k\right)}\mathrm{\Delta }{u}^{\left(k\right)}dz+{\int }_{{Q}_{T}}f\mathrm{\Delta }{u}^{\left(k\right)}dz.$

Letting $k\to \mathrm{\infty }$ and applying (2.7), (2.11) and Lemma 9, we conclude that for every fixed $\varphi \in {\mathcal{𝒫}}_{l}$,

$0\ge \frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla u{|}^{2}dx{|}_{t=0}^{t=T}+{\int }_{{Q}_{T}}\chi \varphi dz+{\int }_{{Q}_{T}}|\varphi {|}^{m-2}\varphi \left(\mathrm{\Delta }u-\varphi \right)dz-d{\int }_{{Q}_{T}}|u{|}^{\sigma -2}u\mathrm{\Delta }udz+{\int }_{{Q}_{T}}f\mathrm{\Delta }udz,$

whence, by virtue of (2.11),

$0\ge -{\int }_{{Q}_{T}}\left(\chi -d|u{|}^{\sigma -2}u\right)\mathrm{\Delta }udz+{\int }_{{Q}_{T}}\chi \varphi dz+{\int }_{{Q}_{T}}|\varphi {|}^{m-2}\varphi \left(\mathrm{\Delta }u-\varphi \right)dz-d{\int }_{{Q}_{T}}|u{|}^{\sigma -2}u\mathrm{\Delta }udz$$={\int }_{{Q}_{T}}\left({|\varphi |}^{m-2}\varphi -\chi \right)\left(\mathrm{\Delta }u-\varphi \right)𝑑z.$

Since $l\in ℕ$ is arbitrary, the last inequality holds in the limit as $l\to \mathrm{\infty }$ and remains true for every $\varphi \in {L}^{m}\left({Q}_{T}\right)$. Let us choose ϕ in the special way $\varphi =\mathrm{\Delta }u+\lambda \psi$, where $\lambda =\text{const}>0$ and $\psi \in {L}^{m}\left({Q}_{T}\right)$ is arbitrary. This choice yields the inequality

$\lambda {\int }_{{Q}_{T}}\left({|\mathrm{\Delta }u+\lambda \psi |}^{m-2}\left(\mathrm{\Delta }u+\lambda \psi \right)-\chi \right)\psi 𝑑z\ge 0.$

Dividing this inequality by λ and then letting $\lambda \to 0$, we find that for every $\psi \in {L}^{m}\left({Q}_{T}\right)$,

${\int }_{{Q}_{T}}\left({|\mathrm{\Delta }u|}^{m-2}\mathrm{\Delta }u-\chi \right)\psi 𝑑z\ge 0,$

which is possible only if $\chi ={|\mathrm{\Delta }u|}^{m-2}\mathrm{\Delta }u$ a.e. in ${Q}_{T}$.

The constructed function u takes the initial value in the sense of Definition 1 (iii), i.e., for every $\varphi \in {L}^{m}\left(\mathrm{\Omega }\right)$,

To complete the proof of Theorem 1, we have to derive the energy equality (2.2) from (2.11), which can be written in the form

$\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla u\left(\tau \right){|}^{2}dx{|}_{\tau =0}^{\tau =t}=-{\int }_{{Q}_{t}}\left(|\mathrm{\Delta }u{|}^{m}-d|u{|}^{\sigma -2}u\mathrm{\Delta }u+f\mathrm{\Delta }u\right)dz.$

By construction, ${|u|}^{\sigma -2}{|\nabla u|}^{2}$, ${|u|}^{\sigma -1}|\mathrm{\Delta }u|\in {L}^{1}\left(\mathrm{\Omega }\right)$ and ${|u|}^{\sigma -2}u{D}_{{x}_{i}}u\in {W}_{0}^{1,1}\left(\mathrm{\Omega }\right)$ for a.e. $t\in \left(0,T\right)$. By the Green identity (see, e.g., [25, pp. 69–70]),

$0={\int }_{{Q}_{t}}\mathrm{div}\left(|u{|}^{\sigma -2}u\nabla u\right)dz=\left(\sigma -1\right){\int }_{{Q}_{T}}|u{|}^{\sigma -2}|\nabla u{|}^{2}dz+{\int }_{{Q}_{t}}|u{|}^{\sigma -2}u\mathrm{\Delta }udz,$

and (2.2) follows.

## 3 Uniqueness of strong solutions

#### Theorem 1.

The strong solution of problem (1.1) is unique if one of the following conditions is fulfilled:

• (i)

$d=0$,

• (ii)

$d\in ℝ$ and $\sigma =2$,

• (iii)

$d>0$ and

• $1, $2\le \sigma <\mathrm{\infty }$ if $n\le 2$,

• $\frac{2n}{n+2}, $2\le \sigma \le 2+\frac{m\left(n+2\right)-2n}{m\left(n-2\right)}$ if $n\ge 3$.

#### Proof.

(i) Let $d=0$. Assume that problem (1.1) admits two different solutions ${u}_{1}$ and ${u}_{2}$. Let us subtract equalities (2.2) for ${u}_{i}$ with the test-function $\mathrm{\Delta }\left({u}_{1}-{u}_{2}\right)$. By Lemma 7, for every $t\in \left(0,T\right]$,

${\int }_{0}^{t}{\int }_{\mathrm{\Omega }}{\partial }_{t}\left({u}_{1}-{u}_{2}\right)\mathrm{\Delta }\left({u}_{1}-{u}_{2}\right)dz=-\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla \left({u}_{1}-{u}_{2}\right){|}^{2}\left(\tau \right)dz{|}_{\tau =0}^{\tau =t},$

while

${\int }_{{Q}_{t}}\left({|\mathrm{\Delta }{u}_{1}|}^{m-2}\mathrm{\Delta }{u}_{1}-{|\mathrm{\Delta }{u}_{2}|}^{m-2}\mathrm{\Delta }{u}_{2}\right)\mathrm{\Delta }\left({u}_{1}-{u}_{2}\right)𝑑z\ge 0$

by monotonicity. It follows that ${\parallel {u}_{1}-{u}_{2}\parallel }_{2,\mathrm{\Omega }}\left(t\right)=0$ for all $t\in \left[0,T\right]$.

(ii) Let $\sigma =2$. If u is a solution of problem (1.1), then $v=u{\mathrm{e}}^{dt}$ satisfies the equation

${v}_{t}={\mathrm{e}}^{-d\left(m-1\right)t}{|\mathrm{\Delta }v|}^{m-1}\mathrm{\Delta }v,$

and the assertion follows as in the case $d=0$.

(iii) Let us consider the case $d>0$, $m\in \left(1,2\right]$ and $\sigma \ge 2$. Taking $\mathrm{\Delta }w=\mathrm{\Delta }\left({u}_{1}-{u}_{2}\right)$ for the test-function in (1.7), we obtain

$\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla w{|}^{2}\left(t\right)dx+{\int }_{{Q}_{t}}\left(|\mathrm{\Delta }{u}_{1}{|}^{m-2}\mathrm{\Delta }{u}_{1}-|\mathrm{\Delta }{u}_{2}{|}^{m-2}\mathrm{\Delta }{u}_{2}\right)\mathrm{\Delta }wdz=d{\int }_{{Q}_{t}}\left(|{u}_{1}{|}^{\sigma -2}{u}_{1}-|{u}_{2}{|}^{\sigma -2}{u}_{2}\right)\mathrm{\Delta }wdz=:I.$(3.1)

Let us make use of the following well-known inequality:

$\left({|b|}^{m-2}b-{|a|}^{m-2}a\right)\left(b-a\right)\ge \left(m-1\right){\left(1+{a}^{2}+{b}^{2}\right)}^{\frac{m-2}{2}}{\left(b-a\right)}^{2}$

for every $a,b\in ℝ$ and $m\in \left(1,2\right]$. Then we have

$\left(m-1\right){\int }_{{Q}_{t}}|\mathrm{\Delta }w{|}^{2}{\left(1+|\mathrm{\Delta }{u}_{1}{|}^{m}+|\mathrm{\Delta }{u}_{2}{|}^{m}\right)}^{\frac{m-2}{m}}dz\le {\int }_{{Q}_{t}}\left(|\mathrm{\Delta }{u}_{1}{|}^{m-2}\mathrm{\Delta }{u}_{1}-|\mathrm{\Delta }{u}_{2}{|}^{m-2}\mathrm{\Delta }{u}_{2}\right)\mathrm{\Delta }wdz,$

whence, by the reverse Hölder inequality,

$\left(m-1\right){\parallel \mathrm{\Delta }w\parallel }_{m,{Q}_{t}}^{2}{\left({\int }_{{Q}_{t}}\left(1+{|\mathrm{\Delta }{u}_{1}|}^{m}+{|\mathrm{\Delta }{u}_{2}|}^{m}\right)𝑑z\right)}^{\frac{m-2}{m}}\le {\int }_{{Q}_{t}}\left({|\mathrm{\Delta }{u}_{1}|}^{m-2}\mathrm{\Delta }{u}_{1}-{|\mathrm{\Delta }{u}_{2}|}^{m-2}\mathrm{\Delta }{u}_{2}\right)\mathrm{\Delta }w𝑑z.$

Since ${\parallel \mathrm{\Delta }{u}_{i}\parallel }_{m,{Q}_{T}}^{m}\le C$, we have

${\left(1+\parallel \mathrm{\Delta }{u}_{1}{\parallel }_{m,{Q}_{t}}^{m}+\parallel \mathrm{\Delta }{u}_{2}{\parallel }_{m,{Q}_{t}}^{m}\right)}^{\frac{m-2}{m}}\ge \frac{1}{{\left(1+2C\right)}^{\frac{2-m}{m}}}=:{C}^{\prime },$

and (3.1) can be continued as follows:

$\frac{1}{2}{\parallel \nabla w\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\left(t\right)+{C}^{\prime }\left(m-1\right){\parallel \mathrm{\Delta }w\parallel }_{m,{Q}_{t}}^{2}\le I.$(3.2)

Let us estimate I. Using the inequalities of Hölder and Cauchy, we have

$|I|\le d{\parallel \mathrm{\Delta }w\parallel }_{m,{Q}_{t}}{\parallel {|{u}_{1}|}^{\sigma -2}{u}_{1}-{|{u}_{2}|}^{\sigma -2}{u}_{2}\parallel }_{{m}^{\prime },{Q}_{t}}\le \frac{1}{2}{C}^{\prime }\left(m-1\right){\parallel \mathrm{\Delta }w\parallel }_{m,{Q}_{t}}^{2}+{C}^{\prime \prime }R,$

with

$R={\parallel {|{u}_{1}|}^{\sigma -2}{u}_{1}-{|{u}_{2}|}^{\sigma -2}{u}_{2}\parallel }_{{m}^{\prime },{Q}_{t}}^{2}$

and ${C}^{\prime \prime }$ being an absolute constant. Plugging this estimate into (3.2), we transform it into the form

${\int }_{\mathrm{\Omega }}|\nabla w{|}^{2}\left(t\right)dx+C\left(m-1\right){\left({\int }_{{Q}_{t}}{|\mathrm{\Delta }w|}^{m}dz\right)}^{\frac{2}{m}}\le 2{C}^{\prime \prime }R.$(3.3)

To estimate R, we will use another known inequality, that is,

with $C=C\left(\sigma \right)$. For every $p>1$, we have

$R\le C\left({\int }_{0}^{t}{\int }_{\mathrm{\Omega }}{|w{|}^{{m}^{\prime }}{\left(|{u}_{1}{|}^{\sigma -2}+|{u}_{2}{|}^{\sigma -2}\right)}^{{m}^{\prime }}dxdt\right)}^{\frac{2}{{m}^{\prime }}}$$\le C\left({\int }_{0}^{t}{\parallel |w{|}^{{m}^{\prime }}{\parallel }_{p,\mathrm{\Omega }}\parallel {\left(|{u}_{1}{|}^{\sigma -2}+|{u}_{2}{|}^{\sigma -2}\right)}^{{m}^{\prime }}{\parallel }_{{p}^{\prime },\mathrm{\Omega }}d\tau \right)}^{\frac{2}{{m}^{\prime }}}$$=C\left({\int }_{0}^{t}{\parallel w{\parallel }_{{m}^{\prime }p,\mathrm{\Omega }}^{{m}^{\prime }}\parallel |{u}_{1}{|}^{\sigma -2}+|{u}_{2}{|}^{\sigma -2}{\parallel }_{{m}^{\prime }{p}^{\prime },\mathrm{\Omega }}^{\frac{{m}^{\prime }}{{p}^{\prime }}}d\tau \right)}^{\frac{2}{{m}^{\prime }}}.$

By the embedding theorem, for all $w\in {H}_{0}^{1}\left(\mathrm{\Omega }\right)$,

(3.4)

whence

$R\le {\left({\int }_{0}^{t}{\left({\parallel \nabla w\parallel }_{2,\mathrm{\Omega }}^{2}\right)}^{\frac{{m}^{\prime }}{2}}{\parallel {\left({|{u}_{1}|}^{\sigma -2}+{|{u}_{2}|}^{\sigma -2}\right)}^{{m}^{\prime }}\parallel }_{{p}^{\prime },\mathrm{\Omega }}𝑑\tau \right)}^{\frac{2}{{m}^{\prime }}}={\left({\int }_{0}^{t}A\left(\tau \right){Y}^{\mu }\left(\tau \right)𝑑\tau \right)}^{\frac{1}{\mu }},$(3.5)

under the notation

$Y\left(t\right)={\parallel \nabla w\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2},\mu =\frac{{m}^{\prime }}{2}>1,A\left(t\right)={\parallel {\left({|{u}_{1}|}^{\sigma -2}+{|{u}_{2}|}^{\sigma -2}\right)}^{{m}^{\prime }}\parallel }_{{p}^{\prime },\mathrm{\Omega }}.$

Let us show that $A\left(t\right)\in {L}^{\mathrm{\infty }}\left(0,T\right)$. By (3.4), in the present case it is sufficient to claim $\left(\sigma -2\right){m}^{\prime }{p}^{\prime }<\mathrm{\infty }$ if $n=1,2$, or $\left(\sigma -2\right){m}^{\prime }{p}^{\prime }<\frac{2n}{n-2}$ if $n\ge 3$. Let us notice that for $n\ge 3$, the condition $\frac{2n}{n+2} is equivalent to ${m}^{\prime }<\frac{2n}{n-2}$, which means that there always exists $p>1$ satisfying the inequality ${m}^{\prime }p\le \frac{2n}{n-2}$. If $n\ge 3$, then we take $p=\frac{2n}{n-2}\frac{1}{{m}^{\prime }}$ and claim that

${m}^{\prime }{p}^{\prime }\left(\sigma -2\right)=\frac{2nm}{mn-2n+2m}\left(\sigma -2\right)\le \frac{2n}{n-2},$(3.6)

p can be arbitrary if $n=1,2$. Then ${\parallel {|{u}_{i}|}^{\left(\sigma -2\right){m}^{\prime }}\parallel }_{{p}^{\prime },\mathrm{\Omega }}\le C{\parallel \nabla {u}_{i}\parallel }_{2,\mathrm{\Omega }}$ and

$\parallel A\left(t\right){\parallel }_{\mathrm{\infty },\left(0,T\right)}=\underset{\left(0,T\right)}{\mathrm{ess}\mathrm{sup}}{\parallel {\left(|{u}_{1}\left(t\right){|}^{\sigma -2}+|{u}_{2}\left(t\right){|}^{\sigma -2}\right)}^{{m}^{\prime }}\parallel }_{{p}^{\prime },\mathrm{\Omega }}$$\le C\left(\underset{\left(0,T\right)}{\mathrm{ess}\mathrm{sup}}{\parallel \nabla {u}_{1}\left(t\right){\parallel }_{2,\mathrm{\Omega }}+\underset{\left(0,T\right)}{\mathrm{ess}\mathrm{sup}}\parallel \nabla {u}_{2}\left(t\right){\parallel }_{2,\mathrm{\Omega }}\right)}^{{m}^{\prime }\left(\sigma -2\right)}$$\le \stackrel{~}{C}.$

Simplifying (3.6), we obtain

$\sigma \le 2+\frac{m\left(n+2\right)-2n}{m\left(n-2\right)}.$

Plugging (3.5) into (3.3), we arrive at

$Y\left(t\right)\le 2{C}^{\prime \prime }{\left({\int }_{0}^{t}A\left(\tau \right){Y}^{\mu }\left(\tau \right)𝑑\tau \right)}^{\frac{1}{\mu }}.$

It follows that $U\left(t\right)={Y}^{\mu }\left(t\right)$ satisfies the linear integral inequality

$U\left(t\right)\le {\left(2{C}^{\prime \prime }\right)}^{\mu }{\int }_{0}^{t}A\left(\tau \right)U\left(\tau \right)𝑑\tau ,U\left(0\right)=0,$

and by Gronwall’s inequality, $U\left(t\right)=0$. ∎

#### Theorem 2.

Let $d\mathrm{>}\mathrm{0}$, $\mathrm{2}\mathrm{<}m\mathrm{<}\mathrm{\infty }$. The strong solution of problem (1.1) is unique if one of the following conditions is true:

• (i)

$2\le n<2m$ and $3\le \sigma \le m+2$,

• (ii)

$3\le \sigma <\mathrm{\infty }$ and $n=1$,

• (iii)

$3\le \sigma <\mathrm{\infty }$ and the solution is bounded.

#### Proof.

Using the inequality

we transform (3.1) to the form

$\frac{1}{2}{\int }_{\mathrm{\Omega }}|\nabla w{|}^{2}\left(t\right)dx+{2}^{2-m}{\int }_{{Q}_{t}}|\mathrm{\Delta }w{|}^{m}dz\le d{\int }_{{Q}_{t}}\mathrm{\Delta }w\left(|{u}_{1}{|}^{\sigma -2}{u}_{1}-|{u}_{2}{|}^{\sigma -2}{u}_{2}\right)dz$$=-d{\int }_{{Q}_{t}}\nabla w\nabla \left(|{u}_{1}{|}^{\sigma -2}{u}_{1}-|{u}_{2}{|}^{\sigma -2}{u}_{2}\right)dz=:I.$(3.7)

Since $\nabla \left({|v|}^{\sigma -2}v\right)=\left(\sigma -1\right){|v|}^{\sigma -2}\nabla v$, we may write

$I=-d\left(\sigma -1\right){\int }_{{Q}_{t}}\nabla w\left({|{u}_{1}|}^{\sigma -2}\nabla {u}_{1}-{|{u}_{2}|}^{\sigma -2}\nabla {u}_{2}\right)𝑑z$$=-d\left(\sigma -1\right){\int }_{{Q}_{T}}|\nabla w{|}^{2}|{u}_{1}{|}^{\sigma -2}dz+d\left(\sigma -1\right){\int }_{{Q}_{t}}\nabla w\nabla {u}_{2}\left(|{u}_{1}{|}^{\sigma -2}-|{u}_{2}{|}^{\sigma -2}\right)dz$$\le d\left(\sigma -1\right){\int }_{{Q}_{t}}\nabla w\nabla {u}_{2}\left(|{u}_{1}{|}^{\sigma -2}-|{u}_{2}{|}^{\sigma -2}\right)dz=:J.$

If $\sigma \ge 3$, by the Lagrange finite-increments formula, we have

$|{|{u}_{1}|}^{\sigma -2}-{|{u}_{2}|}^{\sigma -2}|=\left(\sigma -2\right){|\theta {u}_{1}+\left(1-\theta \right){u}_{2}|}^{\sigma -3}$$\le \left(\sigma -2\right)|{u}_{1}-{u}_{2}|{\left(|{u}_{1}|+|{u}_{2}|\right)}^{\sigma -3}$$\le C\left(\sigma \right)|w|\left({|{u}_{1}|}^{\sigma -3}+{|{u}_{2}|}^{\sigma -3}\right),$

whence

$|J|\le C{\int }_{0}^{t}\parallel \nabla w\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}\left({\int }_{\mathrm{\Omega }}{|w{|}^{2}|\nabla {u}_{2}{|}^{2}\left(|{u}_{1}{|}^{2\left(\sigma -3\right)}+|{u}_{2}{|}^{2\left(\sigma -3\right)}\right)dx\right)}^{\frac{1}{2}}d\tau .$

Recall that ${\parallel w\parallel }_{2p,\mathrm{\Omega }}\le C{\parallel \nabla w\parallel }_{2,\mathrm{\Omega }}$ for every $p>1$ if $n=1,2$ and $p=\frac{n}{n-2}$ if $n\ge 3$.

Let $n\ge 3$. Then, by Hölder’s inequality,

$|J|\le C{\int }_{0}^{t}\parallel \nabla w\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}\parallel w\left(\tau \right){\parallel }_{\frac{2n}{n-2},\mathrm{\Omega }}\left({\int }_{\mathrm{\Omega }}{|\nabla {u}_{2}{|}^{n}\left(|{u}_{1}{|}^{n\left(\sigma -3\right)}+|{u}_{2}{|}^{n\left(\sigma -3\right)}\right)dx\right)}^{\frac{1}{n}}d\tau =C{\int }_{0}^{t}A\left(\tau \right)\parallel \nabla w\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}^{2}d\tau ,$

with

$A\left(t\right)={\left({\int }_{\mathrm{\Omega }}{|\nabla {u}_{2}\left(t\right)|}^{n}\left({|{u}_{1}\left(t\right)|}^{n\left(\sigma -3\right)}+{|{u}_{2}\left(t\right)|}^{n\left(\sigma -3\right)}\right)𝑑x\right)}^{\frac{1}{n}}.$

#### Proposition 3.

If ${u}_{\mathrm{1}}\mathrm{,}{u}_{\mathrm{2}}\mathrm{\in }{L}^{m}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{W}_{\mathrm{0}}^{\mathrm{2}\mathrm{,}m}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\right)}$ with $n\mathrm{\ge }\mathrm{3}$, $\mathrm{3}\mathrm{\le }\sigma \mathrm{\le }m\mathrm{+}\mathrm{2}$, $\frac{n}{\mathrm{2}}\mathrm{<}m$, then $A\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{\in }{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{\right)}$.

#### Proof.

By the embedding theorem, for $2m>n$ and $v\in {W}^{2,m}\left(\mathrm{\Omega }\right)\cap {H}_{0}^{1}\left(\mathrm{\Omega }\right)$,

$\underset{\mathrm{\Omega }}{sup}|v|+\parallel \nabla v{\parallel }_{n,\mathrm{\Omega }}\le {C}^{\prime }\parallel v{\parallel }_{{W}^{2,m}\left(\mathrm{\Omega }\right)}\le C\parallel \mathrm{\Delta }v{\parallel }_{m,\mathrm{\Omega }},$

with C being a constant independent of v. Applying these inequalities, for $\sigma \ge 3$, we estimate

$A\left(t\right)\le C\left({\int }_{\mathrm{\Omega }}{|\nabla {u}_{2}\left(t\right){|}^{n}\left(\parallel \mathrm{\Delta }{u}_{1}\left(t\right){\parallel }_{m,\mathrm{\Omega }}^{n\left(\sigma -3\right)}+\parallel \mathrm{\Delta }{u}_{2}\left(t\right){\parallel }_{m,\mathrm{\Omega }}^{n\left(\sigma -3\right)}\right)dx\right)}^{\frac{1}{n}}$$\le C{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}{\left({\parallel \mathrm{\Delta }{u}_{1}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}+{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\right)}^{\frac{\left(\sigma -3\right)}{m}}$$\le C{\left({\parallel \mathrm{\Delta }{u}_{1}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}+{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\right)}^{\frac{1}{m}+\frac{\left(\sigma -3\right)}{m}}.$

It follows that $A\left(t\right)\in {L}^{1}\left(0,T\right)$ if $\sigma \ge 3$ and $\frac{1}{m}+\frac{\left(\sigma -3\right)}{m}\le 1$, that is, if $3\le \sigma \le m+2$. ∎

Let $n=2$. Take an arbitrary $p\in \left(1,\mathrm{\infty }\right)$. Combining Hölder’s inequality and the embedding theorem, we obtain

$|J|\le C{\int }_{0}^{t}\parallel \nabla w\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}\parallel w\left(\tau \right){\parallel }_{2p,\mathrm{\Omega }}\left({\int }_{\mathrm{\Omega }}{|\nabla {u}_{2}{|}^{2{p}^{\prime }}{\left(|{u}_{1}{|}^{2\left(\sigma -3\right)}+|{u}_{2}{|}^{2\left(\sigma -3\right)}\right)}^{{p}^{\prime }}dx\right)}^{\frac{1}{2{p}^{\prime }}}d\tau =C{\int }_{0}^{t}A\left(\tau \right)\parallel \nabla w\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}^{2}d\tau ,$

with

$A\left(t\right)=\left({\int }_{\mathrm{\Omega }}{|\nabla {u}_{2}\left(t\right){|}^{2{p}^{\prime }}\left(|{u}_{1}\left(t\right){|}^{{p}^{\prime }\left(\sigma -3\right)}+|{u}_{2}\left(t\right){|}^{{p}^{\prime }\left(\sigma -3\right)}\right)dx\right)}^{\frac{1}{2{p}^{\prime }}}.$

#### Proposition 4.

If ${u}_{\mathrm{1}}\mathrm{,}{u}_{\mathrm{2}}\mathrm{\in }{L}^{m}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{W}_{\mathrm{0}}^{\mathrm{2}\mathrm{,}m}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\right)}$, with $n\mathrm{=}\mathrm{2}$, $m\mathrm{>}\mathrm{2}$, $\mathrm{3}\mathrm{\le }\sigma \mathrm{\le }m\mathrm{+}\mathrm{2}$, then $A\mathit{}\mathrm{\left(}t\mathrm{\right)}\mathrm{\in }{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{\right)}$.

#### Proof.

For $n=2$ and $m>2$, we have ${sup}_{\mathrm{\Omega }}|{u}_{i}\left(t\right)|\le C\parallel \mathrm{\Delta }{u}_{i}\left(t\right){\parallel }_{m,\mathrm{\Omega }}$ (see (1.13)), whence, for every $p>1$,

$A\left(t\right)\le C{\left({\parallel \mathrm{\Delta }{u}_{1}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}+{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\right)}^{\frac{\sigma -3}{m}}{\parallel \nabla {u}_{2}\left(t\right)\parallel }_{2{p}^{\prime },\mathrm{\Omega }}.$

Since ${\parallel \nabla {u}_{2}\left(t\right)\parallel }_{2{p}^{\prime },\mathrm{\Omega }}\le C{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}$ for $n=2$, $m>2$ and any finite $p>1$, we have

$A\left(t\right)\le C{\left({\parallel \mathrm{\Delta }{u}_{1}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}+{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\right)}^{\frac{\sigma -3}{m}}{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}\le C{\left({\parallel \mathrm{\Delta }{u}_{1}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}+{\parallel \mathrm{\Delta }{u}_{2}\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\right)}^{\frac{1}{m}+\frac{\sigma -3}{m}},$

and the conclusion follows. ∎

Let $n=1$. For $m>2$, ${\parallel v\parallel }_{{C}^{1+\alpha }\left(\mathrm{\Omega }\right)}\le C{\parallel {v}_{xx}\parallel }_{m,\mathrm{\Omega }}$ and ${\parallel v\parallel }_{\mathrm{\infty },\mathrm{\Omega }}\le C{\parallel {v}_{x}\parallel }_{2,\mathrm{\Omega }}$, which leads to the estimate

$|J|\le C{\int }_{0}^{t}\parallel {w}_{x}\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}\parallel w\left(\tau \right){\parallel }_{\mathrm{\infty },\mathrm{\Omega }}\left({\int }_{\mathrm{\Omega }}{|{u}_{2x}{|}^{2}\left(|{u}_{1}{|}^{2\left(\sigma -3\right)}+|{u}_{2}{|}^{2\left(\sigma -3\right)}\right)dx\right)}^{\frac{1}{2}}d\tau =C{\int }_{0}^{t}A\left(\tau \right)\parallel {w}_{x}\left(\tau \right){\parallel }_{2,\mathrm{\Omega }}^{2}d\tau ,$

with

$A\left(t\right)=\left({\int }_{\mathrm{\Omega }}{|{u}_{2x}\left(t\right){|}^{2}\left(|{u}_{1}\left(t\right){|}^{2\left(\sigma -3\right)}+|{u}_{2}\left(t\right){|}^{2\left(\sigma -3\right)}\right)dx\right)}^{\frac{1}{2}}\le C{\left(\parallel {u}_{1xx}{\parallel }_{m,\mathrm{\Omega }}+\parallel {u}_{2xx}{\parallel }_{m,\mathrm{\Omega }}\right)}^{\sigma -2}\le C.$

If the solutions are bounded, a revision of the estimates on $|J|$ and $A\left(t\right)$ shows that the conclusions of Propositions 3 and 4 remain valid for $\frac{1}{m}\le 1$, which is true by assumption.

Let us consider the function $Y\left(t\right)={\parallel \nabla w\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}$. Due to (3.7) and the estimates on J and A,

$Y\left(t\right)\le C{\int }_{0}^{t}A\left(\tau \right)Y\left(\tau \right)𝑑\tau ,Y\left(0\right)=0.$

Since $A\left(t\right)\in {L}^{1}\left(0,T\right)$, it follows from Gronwall’s lemma that $Y\left(t\right)=0$. ∎

## 4 Extinction in a finite time and asymptotic behavior

Prior to formulating the results, let us notice that the conclusions concerning extinction and the decay rates of strong solutions are derived for ${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}$. By the embedding theorem ${H}_{0}^{1}\left(\mathrm{\Omega }\right)\subset {L}^{r}\left(\mathrm{\Omega }\right)$, which automatically implies the validity of all corresponding properties for the function ${\parallel u\left(t\right)\parallel }_{r,\mathrm{\Omega }}$ for some $r\ge 2$.

## 4.1 Homogeneous equation

Let us consider first the case $f\left(x,t\right)\equiv 0$.

#### Theorem 1.

Let $d\mathrm{>}\mathrm{0}$ and $f\mathrm{\equiv }\mathrm{0}$. Assume that the exponent σ satisfies the condition

Then the following hold:

• (a)

For $m+\sigma <4$ , every strong solution of problem ( 1.1 ) vanishes at a finite moment, that is, there exists ${t}^{\ast }\equiv {t}^{\ast }\left(m,\sigma ,{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}\right)$ such that

• (b)

For $m+\sigma =4$,

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}\le {\mathrm{e}}^{-Ct}{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2},$

with the positive constant C depending only on the data.

• (c)

For $m+\sigma >4$ , there exists a positive constant M , independent of u , such that

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le \frac{{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}}{{\left(1+Mt{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{\frac{2}{\alpha }}\right)}^{\alpha }}\le \frac{1}{{\left(Mt\right)}^{\alpha }},\alpha =\frac{m-\sigma +4}{m+\sigma -4}.$

#### Proof.

Combining identities (2.2) with $t,t+h\in \left(0,T\right)$, we obtain

$\frac{1}{2h}{\int }_{\mathrm{\Omega }}|\nabla u\left(\tau \right){|}^{2}dx{|}_{\tau =t}^{\tau =t+h}+\frac{1}{h}{\int }_{t}^{t+h}{\int }_{\mathrm{\Omega }}\left(|\mathrm{\Delta }u{|}^{m}+d\left(\sigma -1\right)|u{|}^{\sigma -2}|\nabla u{|}^{2}\right)dz=0.$

Letting $h\to 0$ and applying the Lebesgue differentiation theorem leads to the equality

(4.1)

By Hölder’s inequality,

${\int }_{\mathrm{\Omega }}|\nabla u{|}^{2}dx={\int }_{\mathrm{\Omega }}\left(|u{|}^{\frac{\sigma -2}{2}}|\nabla |\right)\left(|{u}^{\left(k\right)}{|}^{\frac{2-\sigma }{2}}|\nabla u|\right)dx\le \left({\int }_{\mathrm{\Omega }}{|u{|}^{\sigma -2}|\nabla u{|}^{2}dx\right)}^{\frac{1}{2}}\left({\int }_{\mathrm{\Omega }}{|u{|}^{2-\sigma }|\nabla u{|}^{2}dx\right)}^{\frac{1}{2}}.$

Since $0<2-\sigma <1$, it follows from Hölder’s inequality that

$\left({\int }_{\mathrm{\Omega }}{|u{|}^{2-\sigma }|\nabla u{|}^{2}dx\right)}^{\frac{1}{2}}\le \parallel u{\parallel }_{1,\mathrm{\Omega }}^{\frac{2-\sigma }{2}}\left({\int }_{\mathrm{\Omega }}{|\nabla u{|}^{\frac{2}{\sigma -1}}dx\right)}^{\frac{\sigma -1}{2}}=\parallel u{\parallel }_{1,\mathrm{\Omega }}^{\frac{2-\sigma }{2}}\parallel \nabla u{\parallel }_{\frac{2}{\sigma -1},\mathrm{\Omega }}.$

The embedding ${L}^{1}\left(\mathrm{\Omega }\right)\subset {W}_{0}^{1,\frac{2}{\sigma -1}}\left(\mathrm{\Omega }\right)$ allows one to continue the last inequality as follows:

$\left({\int }_{\mathrm{\Omega }}{|u{|}^{2-\sigma }|\nabla u{|}^{2}dx\right)}^{\frac{1}{2}}\le C\parallel \nabla u{\parallel }_{\frac{2}{\sigma -1},\mathrm{\Omega }}^{1+\frac{2-\sigma }{2}}.$(4.2)

For every $u\in {W}^{2,m}\left(\mathrm{\Omega }\right)\cap {H}_{0}^{1}\left(\mathrm{\Omega }\right)$,

(4.3)

with C being a constant independent of u. Combining (4.3) with (4.2), we have

$\left({\int }_{\mathrm{\Omega }}{|u{|}^{2-\sigma }|\nabla u{|}^{2}dx\right)}^{\frac{1}{2}}\le C\parallel \mathrm{\Delta }u{\parallel }_{m,\mathrm{\Omega }}^{1+\frac{2-\sigma }{2}}.$

It follows that

${\parallel \nabla u\parallel }_{2,\mathrm{\Omega }}^{2}={\int }_{\mathrm{\Omega }}\left({|u|}^{\frac{\sigma -2}{2}}|\nabla u|\right)\left({|u|}^{\frac{2-\sigma }{2}}|\nabla u|\right)𝑑x$$\le C\left({\int }_{\mathrm{\Omega }}{|u{|}^{\sigma -2}|\nabla u{|}^{2}dx\right)}^{\frac{1}{2}}\parallel \mathrm{\Delta }u{\parallel }_{m,\mathrm{\Omega }}^{1+\frac{2-\sigma }{2}}$$\le C{\left({\int }_{\mathrm{\Omega }}\left({|\mathrm{\Delta }u|}^{m}+{|u|}^{\sigma -2}{|\nabla u|}^{2}\right)𝑑x\right)}^{\frac{1}{2}+\frac{4-\sigma }{2m}},$

with an arbitrary $\sigma \in \left(1,2\right)$ if $m\ge n$ and $\sigma >1+2\left(\frac{1}{m}-\frac{1}{n}\right)$ if $m. Now it is straightforward to obtain the estimate

${\parallel \nabla u\parallel }_{2,\mathrm{\Omega }}^{\frac{4m}{m+4-\sigma }}\le C{\int }_{\mathrm{\Omega }}\left({|\mathrm{\Delta }u|}^{m}+{|u|}^{\sigma -2}{|\nabla u|}^{2}\right)𝑑x.$

Substitution of this inequality into (4.1) leads to a nonlinear differential inequality for the nonnegative function $y\left(t\right)={\parallel \nabla u\parallel }_{2,\mathrm{\Omega }}^{2}$, that is,

${y}^{\prime }+C{y}^{\beta }\left(t\right)\le 0,y\left(0\right)={\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2},\beta =\frac{2m}{m-\sigma +4},$(4.4)

which can be explicitly integrated.

(a) Let $\beta <1$. Then

$0\le {y}^{1-\beta }\left(t\right)\le {y}^{1-\beta }\left(0\right)-C\left(1-\beta \right)t={\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2\left(1-\beta \right)}-C\left(1-\beta \right)t.$

Since the right-hand side of this inequality vanishes at the moment

${t}^{\ast }=\frac{1}{C\left(1-\beta \right)}{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2\left(1-\beta \right)},$(4.5)

it is necessary that ${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\equiv 0$ for all $t\ge {t}^{\ast }$.

(b) Let $\beta =1$. In this case $y\left(t\right)$ satisfies the linear differential inequality, whence

$y\left(t\right)\le {\parallel {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}{\mathrm{e}}^{-Ct}.$

(c) Let $\beta >1$. In this case integration of (4.4) gives

$\frac{1}{1-\beta }\left({y}^{1-\beta }-{y}_{0}^{1-\beta }\right)\le -Ct,$

and the assertion follows. ∎

#### Remark 2.

Let us assume that under the conditions of Theorem 1, the solution is bounded, i.e., $|u|\le M$ a.e in ${Q}_{T}$. Then (4.1) leads to

$\frac{d}{dt}\left(\parallel \nabla u\left(t\right){\parallel }_{2,\mathrm{\Omega }}^{2}\right)+2\parallel \mathrm{\Delta }u\left(t\right){\parallel }_{m,\mathrm{\Omega }}^{m}+2d\left(\sigma -1\right){M}^{\sigma -2}{\int }_{\mathrm{\Omega }}|\nabla u\left(t\right){|}^{2}dx\le 0,$

and it follows that

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le {\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}{\mathrm{e}}^{-Ct},C=2d\left(\sigma -1\right){M}^{\sigma -2}.$

#### Remark 3.

Let one of the following conditions be fulfilled:

• (a)

$d>0$, $\sigma =2$ and $m\in \left(1,\mathrm{\infty }\right)$,

• (b)

$d\ge 0$, $m=2$.

Then

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le {\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}{\mathrm{e}}^{-\lambda t},$(4.6)

where $\lambda =2d$ in case (a), whereas in case (b), λ is the best constant from the inequality

$\lambda {\parallel \nabla u\parallel }_{2,\mathrm{\Omega }}^{2}\le {\parallel \mathrm{\Delta }u\parallel }_{2,\mathrm{\Omega }}^{2},u\in {W}^{2,2}\left(\mathrm{\Omega }\right)\cap {H}_{0}^{1}\left(\mathrm{\Omega }\right).$(4.7)

Note that λ is the first eigenvalue of the Dirichlet problem for the Laplace operator in Ω.

#### Theorem 4.

Let $d\mathrm{\ge }\mathrm{0}$, $\sigma \mathrm{\in }\mathrm{\left(}\mathrm{1}\mathrm{,}\mathrm{\infty }\mathrm{\right)}$ and let $u\mathit{}\mathrm{\left(}x\mathrm{,}t\mathrm{\right)}$ be a strong solution of problem (1.1).

• (a)

If $\mathrm{max}\left\{1,\frac{2n}{n+2}\right\} , then $u\left(x,t\right)$ extincts in a finite time. The moment of extinction depends on m, n and ${\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}$.

• (b)

If $m=2$ , then $u\left(x,t\right)$ satisfies inequality ( 4.6 ).

• (c)

If $m>2$ , then $u\left(x,t\right)$ satisfies the estimate

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le {\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}{\left(1+\frac{\left(m-2\right)}{2}Ct{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{m-2}\right)}^{-\frac{2}{m-2}}\le \frac{1}{{\left(Mt\right)}^{\frac{2}{m-2}}},$

where $M=\frac{1}{2}{\lambda }^{\frac{2}{m}}\left(m-2\right)$ and λ is the constant from ( 4.7 ).

#### Proof.

By virtue of (4.1),

$\frac{1}{2}\frac{d}{dt}\left({\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\right)+{\parallel \mathrm{\Delta }u\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{m}\le 0.$

Applying the estimate

${\parallel \nabla u\parallel }_{2,\mathrm{\Omega }}^{2}\le C{\parallel \mathrm{\Delta }u\parallel }_{m,\mathrm{\Omega }},u\in {W}^{2,m}\left(\mathrm{\Omega }\right)\cap {H}_{0}^{1}\left(\mathrm{\Omega }\right),$(4.8)

we obtain inequality (4.4) for $Y\left(t\right)={\left({\parallel \nabla u\parallel }_{2,\mathrm{\Omega }}^{2}\right)}^{\frac{m}{2}}$. The conclusion follows as in the proof of Theorem 1. ∎

In the case $\sigma >2$, the differential inequality (4.4) is derived in a different way.

#### Lemma 5.

Let $u\mathrm{\in }{W}^{\mathrm{2}\mathrm{,}m}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\cap }{H}_{\mathrm{0}}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$ and ${\mathrm{|}u\mathrm{|}}^{\sigma \mathrm{-}\mathrm{2}}\mathit{}{\mathrm{|}\mathrm{\nabla }\mathit{}u\mathrm{|}}^{\mathrm{2}}\mathrm{\in }{L}^{\mathrm{1}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}$. If

$\sigma >2,\mathrm{max}\left\{1,\frac{2n}{n+2}\right\}(4.9)

then

$\parallel \nabla u{\parallel }_{2,\mathrm{\Omega }}^{2}\le C\left({\int }_{\mathrm{\Omega }}|\mathrm{\Delta }u{|}^{m}dx+{\int }_{\mathrm{\Omega }}{|u{|}^{\sigma -2}|\nabla u{|}^{2}dx\right)}^{2\left(\frac{\lambda }{m}+\frac{1-\lambda }{\sigma }\right)},$(4.10)

with C being a constant independent of u, and

$\lambda =\frac{\frac{1}{2}-\frac{1}{n}-\frac{2}{\sigma }}{\frac{1}{m}-\frac{2}{n}-\frac{2}{\sigma }}\in \left(\frac{1}{2},1\right).$

#### Proof.

We use the interpolation inequality of Gagliardo–Nirenberg: If there exists $\lambda \in \left(\frac{1}{2},1\right)$ such that

$\frac{1}{2}=\frac{1}{n}+\lambda \left(\frac{1}{m}-\frac{2}{n}\right)+\left(1-\lambda \right)\frac{2}{\sigma },$

then for every $u\in {W}^{2,m}\left(\mathrm{\Omega }\right)\cap {H}_{0}^{1}\left(\mathrm{\Omega }\right)$,

${\parallel \nabla u\parallel }_{2,\mathrm{\Omega }}\le {C}^{\prime }{\parallel u\parallel }_{{W}^{2,m}\left(\mathrm{\Omega }\right)}^{\lambda }{\parallel u\parallel }_{\frac{\sigma }{2},\mathrm{\Omega }}^{1-\lambda }\le C{\parallel \mathrm{\Delta }u\parallel }_{m,\mathrm{\Omega }}^{\lambda }{\parallel u\parallel }_{\frac{\sigma }{2},\mathrm{\Omega }}^{1-\lambda },$(4.11)

with C being a constant independent of u. Let us check that such a number λ indeed exists. Define the function

$\mathcal{𝒫}\left(\lambda \right)=\frac{1}{2}-\frac{1}{n}-\lambda \left(\frac{1}{m}-\frac{2}{n}\right)-\left(1-\lambda \right)\frac{2}{\sigma }.$

It is obvious that under the conditions imposed on m and σ,

$\mathcal{𝒫}\left(1\right)=-\frac{1}{m}+\frac{n+2}{2n}>0,\mathcal{𝒫}\left(\frac{1}{2}\right)=\frac{1}{2}\left(\frac{\sigma -2}{\sigma }-\frac{1}{m}\right)<0,$

and since $\mathcal{𝒫}\left(\lambda \right)$ is linear, the equation $\mathcal{𝒫}\left(\lambda \right)=0$ has exactly one root in the interval $\left(\frac{1}{2},1\right)$. Set $v={|u|}^{\frac{\sigma }{2}}$. Then

$\parallel u{\parallel }_{\frac{\sigma }{2},\mathrm{\Omega }}=\parallel v{\parallel }_{1,\mathrm{\Omega }}^{\frac{2}{\sigma }}\le C\parallel \nabla v{\parallel }_{2,\mathrm{\Omega }}^{\frac{2}{\sigma }}=C\left({\int }_{\mathrm{\Omega }}{|u{|}^{\sigma -2}|\nabla u{|}^{2}dx\right)}^{\frac{1}{\sigma }}.$

Combining this inequality with (4.11), we have

$\parallel \nabla u{\parallel }_{2,\mathrm{\Omega }}\le C\parallel \mathrm{\Delta }u{\parallel }_{m,\mathrm{\Omega }}^{\lambda }\left({\int }_{\mathrm{\Omega }}{|u{|}^{\sigma -2}|\nabla u{|}^{2}dx\right)}^{\frac{1-\lambda }{\sigma }}\le C\left({\int }_{\mathrm{\Omega }}|\mathrm{\Delta }u{|}^{m}dx+{\int }_{\mathrm{\Omega }}{|u{|}^{\sigma -2}|\nabla u{|}^{2}dx\right)}^{\frac{\lambda }{m}+\frac{1-\lambda }{\sigma }},$

as required. ∎

#### Theorem 6.

Let m and σ satisfy (4.9). Assume that $d\mathrm{>}\mathrm{0}$. If

$\frac{1}{\beta }=2\left(\frac{\lambda }{m}+\frac{1-\lambda }{\sigma }\right)<1,$

then every strong solution of problem (1.1) satisfies the estimate

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le \frac{K}{{t}^{\frac{1}{\beta -1}}}$

with K being a constant independent of u.

#### Proof.

The proof imitates the proof of Theorem 1. Combining estimate (4.10) with the energy equality (4.1), we arrive at the inequality

${y}^{\prime }\left(t\right)+C{y}^{\beta }\left(t\right)\le 0,y\left(t\right)={\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2},$

and the assertion follows. ∎

#### Remark 7.

It is shown in Theorem 4 that for $m>2$ and $d\ge 0$, $y\left(t\right)={\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}$ decreases like ${t}^{-\alpha }$, with $\alpha =\frac{2}{m-2}$. By Theorem 6, in the case $d>0$, $y\left(t\right)$ decreases like ${t}^{-\frac{1}{\beta -1}}$. If $m>\sigma$, in the latter case the rate of decay is higher.

## 4.2 Nonhomogeneous equation

Let us consider the case $f\left(x,t\right)\ne 0$. The energy equality (4.1) takes the form

$\frac{1}{2}\frac{d}{dt}\parallel \nabla u\left(t\right){\parallel }_{2,\mathrm{\Omega }}^{2}+\parallel \mathrm{\Delta }u\left(t\right){\parallel }_{m,\mathrm{\Omega }}^{m}+d\left(\sigma -1\right){\int }_{\mathrm{\Omega }}|u\left(t\right){|}^{\sigma -2}|\nabla u\left(t\right){|}^{2}dx+{\int }_{\mathrm{\Omega }}f\mathrm{\Delta }u\left(t\right)dx=0.$

Moving the last term to right-hand side and applying Young’s inequality with $ϵ=\frac{1}{2}$, we obtain

$\frac{d}{dt}\left(\parallel \nabla u\left(t\right){\parallel }_{2,\mathrm{\Omega }}^{2}\right)+\parallel \mathrm{\Delta }u\left(t\right){\parallel }_{m,\mathrm{\Omega }}^{m}+2d\left(\sigma -1\right){\int }_{\mathrm{\Omega }}|u\left(t\right){|}^{\sigma -2}|\nabla u\left(t\right){|}^{2}dx\le {2}^{{m}^{\prime }}\frac{m-1}{{m}^{{m}^{\prime }}}\parallel f\left(t\right){\parallel }_{m,\mathrm{\Omega }}^{{m}^{\prime }}.$(4.12)

#### Theorem 8 (asymptotic behavior).

Let $u\mathit{}\mathrm{\left(}x\mathrm{,}t\mathrm{\right)}$ be a strong solution of problem (1.1) with $f\mathrm{\not\equiv }\mathrm{0}$. Assume that $f\mathrm{\in }{L}^{{m}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}{Q}_{T}\mathrm{\right)}\mathrm{\cap }{L}^{\mathrm{\infty }}\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{,}T\mathrm{;}{L}^{{m}^{\mathrm{\prime }}}\mathit{}\mathrm{\left(}\mathrm{\Omega }\mathrm{\right)}\mathrm{\right)}$.

• (i)

Let

${\parallel f\left(t\right)\parallel }_{{m}^{\prime },\mathrm{\Omega }}^{{m}^{\prime }}\le C{\mathrm{e}}^{-\alpha t},\alpha >0.$(4.13)

• (a)(a)

If $d>0$, $\sigma =2$ and $m\in \left(1,\mathrm{\infty }\right)$ , then

(4.14)

• (a)(b)

If $d\ge 0$, $m=2$ , then

with $\mu =\frac{1}{C}$ , and the constant C from ( 4.8 ).

• (ii)

Let $d\ge 0$, $m>2$ . There exist positive constants A and B , depending on m, n and $|\mathrm{\Omega }|$ , such that if

${\parallel f\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{{m}^{\prime }}\le \frac{A}{{\left(1+Bt\right)}^{\frac{m}{m-2}}},$

then

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le \frac{{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}}{{\left(1+Bt\right)}^{\frac{2}{m-2}}}.$

#### Proof.

We begin with case (i) (a). By (4.12),

$\frac{1}{2}\frac{d}{dt}\left({\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\right)+d{\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le {2}^{\frac{1}{m-1}}\left(m-1\right){m}^{-{m}^{\prime }}{\parallel f\left(t\right)\parallel }_{m,\mathrm{\Omega }}^{{m}^{\prime }}.$

Set $y\left(t\right)={\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}$. By (4.13), $y\left(t\right)$ satisfies the nonhomogeneous ordinary differential inequality

${y}^{\prime }+2dy\le {C}^{\prime }{\mathrm{e}}^{-\alpha t},y\left(0\right)={\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2},{C}^{\prime }=C{2}^{\frac{1}{m-1}}\left(m-1\right){m}^{-{m}^{\prime }}.$(4.15)

Estimate (4.14) is an immediate byproduct of the fact that $y\left(t\right)$ is majorated by the solution of the problem

${z}^{\prime }+2dz={C}^{\prime }{\mathrm{e}}^{-\alpha t},z\left(0\right)={\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2},$

given by the explicit formula

In case (ii) (b), we drop the last term on the left-hand side of (4.12). Since $m=2$, the resulting inequality can be transformed to the form

${y}^{\prime }+\mu y\le {C}^{\prime }{\mathrm{e}}^{-\alpha t},y\left(t\right)={\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2},$

with $\mu =\frac{1}{C}$, and the constant C from (4.8). The conclusion follows as in case (i) (a).

Let us consider case (ii). Using the assumption on f and (4.8), from (4.12), we derive the following differential inequality for the function $y\left(t\right)$:

${y}^{\prime }+C{y}^{\frac{m}{2}}\le {C}^{\prime }\frac{A}{{\left(1+Bt\right)}^{\frac{m}{m-2}}},y\left(0\right)={\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}.$

It is straightforward to check that the Cauchy problem

${v}^{\prime }+C{v}^{\frac{m}{2}}=\frac{{C}^{\prime }A}{{\left(1+Bt\right)}^{\frac{m}{m-2}}},v\left(0\right)=y\left(0\right),$

admits the solution $v=v\left(0\right){\left(1+Bt\right)}^{\frac{-2}{m-2}}$, provided that the parameters $A>0$, $B>0$ satisfy

$-\frac{2AB}{m-2}+C{A}^{\frac{m}{2}}={C}^{\prime }.$

$v\left(t\right)$ is a majorant function for $y\left(t\right)$, that is,

${\parallel \nabla u\left(t\right)\parallel }_{2,\mathrm{\Omega }}^{2}\le v\left(t\right)=\frac{{\parallel \nabla {u}_{0}\parallel }_{2,\mathrm{\Omega }}^{2}}{{\left(1+Bt\right)}^{\frac{2}{m-2}}}.\mathit{∎}$

## 4.3 Extinction at a prescribed moment

The assertion of Theorem (1) can be extended to the case when $f\not\equiv 0$. Let us assume that

$f\in {L}^{{m}^{\prime }}\left({Q}_{T}\right)\cap {L}^{\mathrm{\infty }}\left(0,T;{L}^{{m}^{\prime }}\left(\mathrm{\Omega }\right)\right),{\parallel f\left(t\right)\parallel }_{{m}^{\prime },\mathrm{\Omega }}^{{m}^{\prime }}\le ϵ{\left[1-\frac{t}{{t}_{f}}\right]}_{+}^{\frac{\beta }{1-\beta }},ϵ>0,{\left[v\right]}_{+}=\mathrm{max}\left\{0,v\right\},$(4.16)

with $\beta \in \left(0,1\right)$ and a given ${t}_{f}<\mathrm{\infty }$. We will assume that the parameters ${t}_{f}$ (the moment of vanishing of the source term f), $ϵ>0$ (the intensity of the source) and the norm of the initial function $y\left(0\right)={\parallel \nabla u\left(0\right)\parallel }_{2,\mathrm{\Omega }}^{2}$ are connected by the relation

$-\frac{1}{1-\beta }y\left(0\right)\frac{1}{{t}_{f}}+C{y}^{\beta }\left(0\right)=ϵ{C}^{\prime },$(4.17)

where $\beta =\frac{2m}{m-\sigma +4}$, C and ${C}^{\prime }$ are the constants in (4.4) and (4.15).

#### Theorem 9.

Assume that $d\mathrm{>}\mathrm{0}$, the exponents m, σ satisfy the conditions of Theorem 1 with $m\mathrm{+}\sigma \mathrm{<}\mathrm{4}$, and f satisfies (4.16), with $\beta \mathrm{=}\frac{\mathrm{2}\mathit{}m}{m\mathrm{-}\sigma \mathrm{+}\mathrm{4}}\mathrm{\in }\mathrm{\left(}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$. If the data ${t}_{f}$, ϵ and ${\mathrm{\parallel }\mathrm{\nabla }\mathit{}u\mathit{}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{\parallel }}_{\mathrm{2}\mathrm{,}\mathrm{\Omega }}^{\mathrm{2}}$ satisfy (4.17), then

#### Proof.

Starting from the energy inequality (4.12) and arguing as in the derivation of (4.4) in the proof of Theorem 1, for $y\left(t\right)$, we obtain

${y}^{\prime }\left(t\right)+C{y}^{\beta }\left(t\right)\le ϵ{C}^{\prime }{\left[1-\frac{t}{{t}_{f}}\right]}_{+}^{\frac{\beta }{1-\beta }},y\left(0\right)={\parallel \nabla u\left(0\right)\parallel }_{2,\mathrm{\Omega }}^{2}.$(4.18)

The solution of the Cauchy problem

${v}^{\prime }\left(t\right)+C{v}^{\beta }\left(t\right)=ϵ{C}^{\prime }{\left[1-\frac{t}{{t}_{f}}\right]}_{+}^{\frac{\beta }{1-\beta }},v\left(0\right)=y\left(0\right),$(4.19)

is a majorant function for the nonnegative solution of (4.18). The function

$v\left(t\right)=v\left(0\right){\left[1-\frac{t}{{t}_{f}}\right]}_{+}^{\frac{\beta }{1-\beta }}$

solves (4.19), provided that $y\left(0\right)$, ϵ and ${t}_{f}$ satisfy (4.17). ∎

#### Remark 10.

The effect of simultaneous vanishing of the source f and the solution is present if the given parameters $y\left(0\right)$, ${t}_{f}$ and ϵ satisfy (4.17). The assertion of Theorem 9 remains true if any of the three parameters is fixed while the other may vary in order to fulfill (4.17).

#### Remark 11.

Condition (4.17) allows one to express the moment of vanishing ${t}_{f}$ through the data. Comparing it with the extinction moment ${t}^{\ast }$ for the solutions of the homogeneous equation (1.1) given by formula (4.5), we find that

${t}_{f}=\frac{y\left(0\right)}{\left(1-\beta \right)\left(C{y}^{\beta }\left(0\right)-ϵ{C}^{\prime }\right)}>\frac{{y}^{1-\beta }\left(0\right)}{C\left(1-\beta \right)}={t}^{\ast },y\left(0\right)={\parallel \nabla u\left(0\right)\parallel }_{2,\mathrm{\Omega }}^{2}.$

According to this inequality, it is necessary that the solutions of the nonhomogeneous equation vanish after the solution of the homogeneous equation with the same initial data. In other words, it is necessary that in condition (4.16), we have ${t}_{f}>{t}^{\ast }$.

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Revised: 2016-08-04

Accepted: 2016-09-27

Published Online: 2016-11-23

Funding Source: Russian Science Foundation

Award identifier / Grant number: 15-11-20019

Funding Source: Ministerio de Ciencia e Innovación

Award identifier / Grant number: MTM2013-43671

Funding Source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

Award identifier / Grant number: 88887.059583/2014-00

The first author was partially supported by the Research Project 15-11-20019 of the Russian Science Foundation (50% of all results of this paper). The second author acknowledges the support of the Research Grant MTM2013-43671-P, MICINN, Spain, and the program “Science Without Borders”, CSF-CAPES-PVE-Process 88887.059583/2014-00, Brasil.

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 79–100, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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