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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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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
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  • 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

ut=|Δu|m-2Δu-d|u|σ-2u+fin QT=Ω×(0,T),

with the parameters m>1, σ>1 and d0. At the points where Δu=0, the equation degenerates if m>2, or becomes singular if m(1,2). We derive conditions of existence and uniqueness of strong solutions, and study the asymptotic behavior of strong solutions as t. Sufficient conditions for exponential or power decay of u(t)2,Ω are derived. It is proved that for certain ranges of the exponents m and σ, every strong solution vanishes in a finite time.

Keywords: Fully nonlinear parabolic equation; strong solution; asymptotic behavior; extinction 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

{ut=|Δu|m-2Δu-d|u|σ-2u+fin QT=Ω×(0,T),u=0on Ω×(0,T),u(x,0)=u0(x)in Ω,(1.1)

with a nonnegative coefficient d and exponents m>1, σ>1.

The fully nonlinear equation (1.1), without lower-order terms, falls into the class of equations ut=β(Δu), which includes the dual porous medium equation

ut=|Δu|m-2Δu,m>1,(1.2)

and the Barenblatt equation

F(ut)=Δu.(1.3)

Equation (1.3), with

F(s)={(1+γ)sif s0,(1-γ)sif s<0,  γ(-1,1),

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(tu)+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

ut=Φ(u)|Δu|m-1Δu+Ψ(u)(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(x,0)C2+α(n), u(x,0)0, and the following structural assumptions are fulfilled:

m1,Φ(s)0,Φ(s)C0[0,),Ψ(s)C1[0,),Ψ(0)=0.

In [26], local in time existence of a classical solution to the Dirichlet problem for (1.4) is proved under the assumptions that Φ(s)>0, Φ and Ψ are Lipschitz-continuous and bounded, and u0C2+α(Ω¯).

At the points where Δu=0 equation (1.4) degenerates if m>2, or becomes singular if m(1,2). Li et al. [19] consider the Dirichlet problem for the singular equation

ut=g(x)|Δu|m-2Δu+k(u0-u)(1.5)

with a given smooth function g(x)>0, k=const>0 and m(1,2). It is proved that this problem has an integral solution uC(0,T;Cα(Ω¯)) 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

(uq)t=|Δu|m-1Δu+up(1.6)

were studied in [13] for q=1, m>1, p>1, and in [18] for the range of parameters m1, q1, 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 u0 is assumed continuous and nonnegative if pq1, while in the case 0<p<q, the condition 0<ϵu0 in 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 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(x,t) is called strong solution of problem (1.1) if the following hold:

  • (i)

    uC([0,T];H01(Ω))Lσ(QT)Lm(0,T;W2,m(Ω)), utLm(QT), |u|σ-1Lm(QT), m=mm-1,

  • (ii)

    for every test-function ϕLm(QT),

    QT(ut-|Δu|m-2Δu+d|u|σ-2u-f)ϕ𝑑z=0,(1.7)

  • (iii)

    for every ϕ(x)Lm(Ω), (u(t)-u0,ϕ)2,Ω0 as t0+.

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(1,2) or m>2). It turns out that for n=1,2 a strong solution exists for any m>1 and σ(1,), while in the case n3, the exponent σ>1 has to satisfy the inequality σ<max{m,2nn-2}. The existence theorem is proved under the natural assumptions u0H01(Ω) and fLm(QT).

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, 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(t)=u(t)2,Ω2 as t. The energy y(t) satisfies the ordinary differential inequality, linear or nonlinear in dependence on the assumptions on the parameters of nonlinearity, and may be nonhomogeneous if f0. 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, f0 and u0H01(Ω).

  • (a)

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

  • (b)

    The solution is unique if either m(1,2] and σ2, or m>2 and 3σm+2.

  • (c)

    If m2 and σ(1,2), then every strong solution:

    • vanishes in a finite time if m+σ<4,

    • decreases exponentially if m+σ=4,

    • has the power decay

      u(t)2,Ω2Lt-α,α=m-σ+4m+σ-4if m+σ>4,

      with L being a constant independent of u.

  • (d)

    If m(1,2), 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,

ut=a(x,t)|Δu|m-2Δu-d(x,t)|u|σ-2u+f,

under suitable regularity assumptions and the conditions 0<a0a< and 0d<. 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

ut=div(|u|m-1u)+μu,m>0,μ0,(1.8)

and the doubly nonlinear equation

ut=div(|u|m-1|u|p-2u),p>1,m+p>3.(1.9)

It is proved in [10] that for μ=0 and m(0,1) (the fast diffusion equation) every solution of the homogeneous Dirichlet problem for equation (1.8) vanishes at a finite moment T. Moreover, as tT, the solution converges to a separable solution of the form

v(x,t)=K(T-t)m1-mw(x),K=const>0.

The case m>1 (the slow diffusion equation) and μ0 is studied in [4]. It is shown that in this case every solution converges as t 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 a nonnegative separable solution v=θ(t)w(x) 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(x,t)|Ct13-m-pw(x)if m+p>3,|u(x,t)|Ce-βtw(x)if m+p=3.

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

ut=i=1nDi(a(x,t,u)|Diu|p-2Diu)+c(x,t)|u|σ-2u,(1.10)

with constant exponents p>1, σ>1, and the coefficients 0<a0a(x,t,u)a1, 0c0c(x,t)c1. Set

1ν={1σ+1pif c0>0,2pif c0=0.

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 c0>0 and ν<1, or c0=0 and 1<1ν1+2n,

  • decreases exponentially if ν=1, i.e., u(t)2,Ω2Le-Ct, L,C=const>0,

  • has the power decay u(t)2,Ω2C(1+t)-νν-1 if ν>1.

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

ut=div(a(x,t,u)|u|p-2u)+c(x,t)|u|σ-2u.(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

ut=div(|u|m-1u),(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(0,1), decrease exponentially if m=1, or decrease like t11-m if m>1, while the solutions of (1.2) display the same behavior in the ranges of the exponent m(1,2), m=2, or m>2.

Throughout the text, we denote up,Ω=uLp(Ω) and u(t)p,Ω=u(,t)p,Ω for the functions depending on (x,t)Ω×(0,T). Wk,p(Ω) is the Sobolev space with the norm uWk,p(Ω), with

uWk,p(Ω)p=0|α|kDαup,Ωp.

The Hilbert space H0k(Ω)=W0k,2(Ω) is the closure of the set Cc(Ω) (smooth functions with compact support in Ω) with respect to the norm of Wk,2(Ω). Notice that Δum,Ω is an equivalent norm of the space W2,m(Ω)H01(Ω). Indeed, on the one hand we have Δum,Ωmi=1nDi2um,ΩmnuW2,m(Ω)m, while on the other, uW2,m(Ω)H01(Ω) is a solution of the Poisson equation with the right-hand side set to Δu and with zero trace on Ω. Thus, it satisfies the estimate

uW2,m(Ω)CΔum,Ω,(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,b0 and ϵ>0, p>1,

abϵap+Cϵbq,Cϵ=(p-1)ϵ-1p-1pq,q=pp-1.

2 Existence of strong solutions

Theorem 1.

Let ΩC2. Assume that

m>1,1<σ<1+q(m-1)m,{q=max{m,2nn-2}if n3,q(1,)if n=2,q=if n=1.(2.1)

Then, for every u0H01(Ω), fLm(QT), problem (1.1) has at least one strong solution which satisfies the energy equality

12u(t)2,Ω2+Qt|Δu|mdz+d(σ-1)Qt|u|σ-2|u|2dz=12u02,Ω2+QtfΔudzfor a.e. t(0,T).(2.2)

2.1 Galerkin’s approximations

Let s1 be a natural number. By {ψi}, we denote the system of eigenfunctions of the problem

(ψ,ϕ)H0s(Ω)=λ(ψ,ϕ)2,Ωfor all ϕH0s(Ω),s1.

Without loss of generality, we assume that {ψi} forms an orthonormal basis of L2(Ω). A solution of problem (1.1) will be constructed as the limit of the sequence {u(k)}, where u(k)=i=1kci(t)ψi(x). The coefficients ci(t) are defined from the system of ordinary differential equations

{ci(t)=Ωψi(x)|Δu(k)|m-2Δu(k)dx-dΩ|u(k)|σ-2u(k)ψi(x)dx+Ωψi(x)fdx,ci(0)=(u0,ψi)2,Ω,i=1,2,,k.(2.3)

For every finite k the right-hand side in (2.3) is a Hölder-continuous function of c1,c2,,ck, and by Peano’s theorem there exists at least one solution, 𝐜(t)=(c1,,ck), defined on an interval [0,Tk]. To prove that each of 𝐜(t) can be continued to the maximal existence interval [0,T], it is sufficient to derive independent of k a priori estimates on u(k).

2.2 A priori estimates

Lemma 2.

Let m>1, σ>1 and u0H01(Ω), fLm(QT). Then the functions u(k) satisfy the following uniform estimate:

12esssup(0,T)u(k)(t)2,Ω2+12Δu(k)m,QTm+d(σ-1)QT|u(k)|σ-2|u(k)|2dzu0(k)2,Ω2+Cfm,QTm,(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,

-Ω|u|σ-2uΔudx=-Ω(u2)σ-22uΔudx=σ-22Ω(u2)σ-422u2|u|2dx+Ω|u|σ-2|u|2dx=(σ-1)Ω|u|σ-2|u|2dx.(2.5)

Multiplying each one of equations (2.3) by λici(t), summing up from 1 to k, integrating the result over an interval (0,t)(0,Tk), and then applying (2.5), we obtain

-120tddt(u(k)(τ)2,Ω2)dτ=Δu(k)(t)m,Qtm-aQt|u(k)|σ-2u(k)Δu(k)dz+QtΔu(k)fdz=Δu(k)(t)m,Qtm+d(σ-1)Qt|u(k)|σ-2|u(k)|2dz+QtΔu(k)fdz.

Applying Young’s inequality to the last term on the left-hand side, we find that for every t(0,Tk) and any ϵ>0, we have

12u(k)(t)2,Ω2+Δu(k)m,Qtm+d(σ-1)Qt|u(k)|σ-2|u(k)|2dz+QtΔu(k)fdz=12u0(k)2,Ω2+QtΔu(k)f𝑑z12u02,Ω2+ϵu(k)m,Qtm+Cϵfm,Qtm.

Applying (1.13), we estimate

u(k)(t)m,ΩmCΔu(k)(t)m,Ωmandu(k)m,QtmCΔu(k)m,Qtm.

Choosing ϵ so small that ϵC12, we obtain (2.4) for every t(0,Tk). This inequality remains valid for every t(0,T) because the right-hand side is independent of k, and thus T=Tk. ∎

Lemma 3.

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

u(k)σ,QT+u(k)q,QTC,q=max{m,r},{r=2nn-2if n3,r(1,)if n=2,r=if n=1.(2.6)

Proof.

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

u(k)r,QTTu(k)L(0,T;Lr(Ω))Cu(k)L(0,T;L2(Ω))Cu02,Ω,

with 1<r2nn-2 if n>2, r(1,) if n=2, or r= if n=1. Then

u(k)m,QTm=0Tu(k)(t)m,ΩmdtCΔu(k)m,QTmCu02,Ω2,

and, finally,

u(k)σ,QTσ=0TΩ(|u(k)|σ2)2dzC0TΩ|(|u(k)|σ2)|2dzCQT|u(k)|σ-2|u(k)|2dzCu02,Ω2.

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

Lemma 4.

Let in the conditions of Lemma 2

σ1+q(m-1)m,

with q defined in (2.6). Then ut(k)Lm(0,T;H-s(Ω)), and there exists a constant C, independent of k, such that

ut(k)Lm(0,T;H-s(Ω))C.

Proof.

Let us denote 𝒫k=span{ψ1,,ψk}, and choose s large enough so that H0s(Ω)Lm(Ω). Given ϕLm(0,T;H0s(Ω)), we denote ϕ(k)=i=1kϕi(t)ψi(x), with ϕi(t)=(ϕ,ψi)2,Ω. By virtue of (2.3) and due to the orthogonality of the system {ψi} in L2(Ω), we have

QTut(k)ϕdz=QT(j=1kcj(t)ψj(x))(i=1kϕi(t)ψi(x))dz=0T(Ω(|Δu(k)|m-2Δu(k)ϕ(k)-d|u(k)|σ-2u(k)ϕ(k)+fϕ(k))dx)dt=:I1+I2+I3.

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

ϕ(k)(t)H0s(Ω)2=i=1kλiϕi2(t)i=1λiϕi2(t)=ϕ(t)H0s(Ω)2,

we find that

|I1|0TΔu(k)m,Ωm-1ϕ(k)m,ΩdtC0TΔu(k)m,Ωm-1ϕ(k)H0s(Ω)dtC0TΔu(k)m,Ωm-1ϕH0s(Ω)dtCΔu(k)m,QTm-1ϕLm(0,T;H0s(Ω)).

Likewise, the term I2 is estimated as follows:

|I2|d0TΩ|u(k)|σ-1|ϕ(k)|dzd0T(Ω|u(k)|m(σ-1)dx)1mϕ(k)m,Ωdtd(QT|u(k)|m(σ-1)dz)1mϕ(k)Lm(0,T;H0s(Ω))d(QT|u(k)|m(σ-1)dz)1mϕLm(0,T;H0s(Ω)).

The assumption on σ entails the inequality

QT|u(k)|m(σ-1)dzCu(k)q,QT,

which gives

|I2|Cu(k)q,QTϕ(k)Lm(0,T;H0s(Ω))Cu(k)q,QTϕLm(0,T;H0s(Ω)).

The term I3 is estimated by Hölder’s inequality as follows:

|I3|0Tfm,Ωϕ(k)m,ΩdtC0Tfm,Ωϕ(k)H0s(Ω)dtCfm,QTϕLm(0,T;H0s(Ω)).

Combining these estimates with (2.4), we find that for every ϕLm(0,T;H0s(Ω)) with ϕ1,

|QTut(k)ϕdz|C,

with the constant C depending on u02,Ω, fm,QT, m, σ and |QT|, but being independent of k. ∎

2.3 Convergence

By Lemmas 2 and 4, u(k)L(0,T;H01(Ω)) and ut(k)Lm(0,T;H-s(Ω)) are uniformly bounded. Since we have H01(Ω)L2(Ω)H-s(Ω), with the embedding H01(Ω)L2(Ω) being compact, it follows, from the compactness result of Aubin [5], that {u(k)} is precompact in C([0,T];L2(Ω)). The sequence {u(k)} contains a subsequence (for which we keep the same name), that possesses the following properties: there exist functions uC([0,T];L2(Ω)), UL2(Ω) and χLm(QT) such that

{u(k)ua.e. in QT and in C([0,T];L2(Ω)),u(k)uin Lq(QT), with q defined in (2.6),u(k)(T)Uin L2(Ω),u(k)u-weakly in L(0,T;L2(Ω)),Δu(k)Δuin Lm(QT),|Δu(k)|m-2Δu(k)χin Lm(QT),ut(k)utin Lm(0,T;H-s(Ω)).(2.7)

2.4 Passing to the limit

Let us fix an arbitrary r. By the method of construction, we have

QT(ut(k)-|Δu(k)|m-2Δu(k)+d|u(k)|σ-2u(k)-f)ϕ(r)𝑑z=0(2.8)

for every kr and every ϕ(r)𝒫rLm(0,T;H0s(Ω)). The convergence properties (2.7) allow one to pass to the limit as k. Thus,

QT(ut-χ+d|u|σ-2u-f)ϕ(r)𝑑z=0

for every fixed r, Since Lm(0,T;H0s(Ω)) is dense in Lm(QT), the last equality is true in the limit as r. Hence,

QT(ut-χ+d|u|σ-2u-f)ϕ𝑑z=0(2.9)

for every ϕLm(QT), whence utLm(QT). Let us introduce the function space

X(0,T;Ω)={u:QTutLm(QT),uL2(0,T;H01(Ω))Lm(0,T;W2,m(Ω))}.

Lemma 5.

Let ΩC2. If uX(0,T;Ω), then uC([0,T];L2(Ω)), after possible redefining on a set of zero measure in (0,T).

Proof.

Let uX(0,T;Ω). Following the proof of [12, Section 5.4, Theorem 1], one may construct the linear extension operator E1 so that E1u=u a.e. in QT, suppE1u(,t)Ω for a.e. t(0,T), ΩΩ, and

E1uX(0,T;Ω)CuX(0,T;Ω),

with C being a constant independent of u. Denote v=E1u. The same arguments show that there exists a linear extension operator E2 such that E2v=v a.e. in QT=Ω×(0,T), suppE2vΩ×(-T,2T) and

E2(E1u)X(-T,2T;Ω)CE2vX(0,T;Ω)CuX(0,T;Ω),

with C being an absolute constant. The constructed extension w=E2v=E2(E1u) coincides with u a.e. in QT. Set Q=Ω×(-T,2T) and choose a family wϵC(Q¯) of functions with compact supports in Q, approximating w in the norm of X(-T,2T;Ω). Such a sequence can be obtained by means of mollification, see [12, Section 5.4, Theorems 2 and 3]. For every τ,t[0,T], we have

(wϵ-wδ)2,Ω2(t)(wϵ-wδ)2,Ω2(t)=(wϵ-wδ)2,Ω2(τ)+2τtΩ(wϵ-wδ)(twϵ-twδ)dz=(wϵ-wδ)2,Ω2(τ)-2τtΩΔ(wϵ-wδ)t(wϵ-wδ)dz(wϵ-wδ)2,Ω2(τ)+2Δ(wϵ-wδ)m,Qt(wϵ-wδ)m,Q.

Integrating this inequality in τ and simplifying, we obtain

(wϵ-wδ)2,Ω2(t)1T(wϵ-wδ)2,Q2+2Δ(wϵ-wδ)m,Qt(wϵ-wδ)m,QCwϵ-wδX(-T,2T;Ω)20as ϵ,δ0.

It follows that

sup(0,T)(wϵ-wδ)2,Ω2(t)0as ϵ,δ0,

that is, {wϵ} is a Cauchy sequence in C([0,T];L2(Ω)) and converges to a function VC([0,T];L2(Ω)). On the other hand,

wϵuin L2(QT),

whence u=V for a.e. in QT. ∎

Corollary 6.

The function u(T) is defined by continuity and in (2.7), U=u(T).

Lemma 7.

For every uX(0,T;Ω) and every 0t1<t2T,

t1t2ΩutΔudz+12Ω|u|2dx|t=t1t=t2=0.

Proof.

Denote by {uh} the sequence of Steklov’s means of u, that is,

uh=1h0hu(x,t+τ)𝑑τ,0<h<T.

It is known that uhX(0,T-h;Ω) and uh-uX(0,T-h;Ω)0 as h0. Since

tuh=1h(u(x,t+h)-u(x,t))Lm(0,T-h;W02,m(Ω))L2(0,T-h;H01(Ω)),

we may apply the Green formula (see, e.g., [25, pp. 69–70]). For every 0<h<T and every 0<t1<t2<T-h, we have

t1t2ΩutΔu𝑑z=t1t2Ω(t(u-uh)Δu+tuh(Δu-Δuh)+tuhΔuh)𝑑z=-12Ω|uh|2dx|t=t1t=t2+t1t2Ω(t(u-uh)Δu+tuh(Δu-Δuh))dz.(2.10)

For every t(0,T-h),

limh0uh(t)2,Ω2=u(t)2,Ω2.

Indeed, by Lemma 5, u(t)C([0,T];L2(Ω)), which means that for every t(0,T-h) there exists ξ(0,h) such that

|uh(t)2,Ω-u(t)2,Ω|2uh(t)-u(t)2,Ω2=1h2Ω(0h(u(x,t+τ)-u(x,t))dτ)2dx1h2Ω(h(0h|u(x,t+h)-u(x,t)|2dτ)12)2dx=1h0hu(,t+τ)-u(,t)2,Ω2dτu(,t+ξ)-u(,t)2,Ω20as h0.

On the other hand,

|t1t2ΩutΔudz+12Ω|uh|2dx|t=t1t=t2|t(u-uh)m,QT-hΔum,QT-h+tuhm,QTΔu-Δuhm,QT-h0

as h0, by virtue of (2.10). ∎

Corollary 8.

Let u=limu(k). The function ΔuLm(QT) is an admissible test-function in (2.9). Choosing ϕ=Δu and applying Lemma 7, from (2.9), we obtain

12Ω|u|2dx|t=0t=T=-QT(χ-d|u|σ-2u+f)Δudz.(2.11)

Lemma 9.

Let uX(0,T;Ω)Lq(QT) with

m>1,1<σ<1+q(m-1)m

and q from condition (2.1). Then

limkQT|u(k)|σ-2u(k)Δu(k)dz=QT|u|σ-2uΔudz.

Proof.

Let us make use of the representation

QT|u(k)|σ-2u(k)Δu(k)dz=QT(|u(k)|σ-2u(k)-|u|σ-2u)Δu(k)dz+QT|u|σ-2uΔu(k)dz=:I1(k)+I2(k).

For every uLq(QT), the inclusion |u|σ-2uLm(QT) holds, whence, by virtue of (2.7),

I2(k)QT|u|σ-2uΔudzas k.

Notice that the assumption σ<1+q(m-1)m is equivalent to m(σ-1)<q. The convergence I1(k)0 follows from the Vitali convergence theorem. Since v(k)=|u(k)|σ-2u(k)v=|u|σ-2u a.e. in QT and v(k)qσ-1,QTC uniformly in k, we have v(k)v in Lr(QT), with 1r<qσ-1. By Hölder’s inequality,

|I1(k)|QT|v(k)-v||Δu(k)|dzCv(k)-vm,QTΔu(k)m,QT.

The second factor here is uniformly bounded in k due to (2.4), while the first factor tends to zero because m(σ-1)<q. ∎

Let us identify the function χLm(QT). Choosing in (2.8) ϕ(r)=Δu(k)Lm(QT) and applying Lemma 5, we find that for every ϕ𝒫l with lk, we have

0=12Ω|u(k)|2dx|t=0t=T+QT|Δu(k)|mdz-dQT|u(k)|σ-2u(k)Δu(k)dz+QTfΔu(k)dz=12Ω|u(k)|2dx|t=0t=T+QT|Δu(k)|m-2Δu(k)(Δu(k)-ϕ)dz+QT|Δu(k)|m-2Δu(k)ϕdz-dQT|u(k)|σ-2u(k)Δu(k)dz+QTfΔu(k)dz.(2.12)

Since

|Δu(k)|m-2Δu(k)(Δu(k)-ϕ)=(|Δu(k)|m-2Δu(k)-|ϕ|m-2ϕ)(Δu(k)-ϕ)+|ϕ|m-2ϕ(Δu(k)-ϕ)|ϕ|m-2ϕ(Δu(k)-ϕ),

from (2.12), we obtain

012Ω|u(k)|2dx|t=0t=T+QT|ϕ|m-2ϕ(Δu(k)-ϕ)dz+QT|Δu(k)|m-2Δu(k)ϕdz-dQT|u(k)|σ-2u(k)Δu(k)dz+QTfΔu(k)dz.

Letting k and applying (2.7), (2.11) and Lemma 9, we conclude that for every fixed ϕ𝒫l,

012Ω|u|2dx|t=0t=T+QTχϕdz+QT|ϕ|m-2ϕ(Δu-ϕ)dz-dQT|u|σ-2uΔudz+QTfΔudz,

whence, by virtue of (2.11),

0-QT(χ-d|u|σ-2u)Δudz+QTχϕdz+QT|ϕ|m-2ϕ(Δu-ϕ)dz-dQT|u|σ-2uΔudz=QT(|ϕ|m-2ϕ-χ)(Δu-ϕ)𝑑z.

Since l is arbitrary, the last inequality holds in the limit as l and remains true for every ϕLm(QT). Let us choose ϕ in the special way ϕ=Δu+λψ, where λ=const>0 and ψLm(QT) is arbitrary. This choice yields the inequality

λQT(|Δu+λψ|m-2(Δu+λψ)-χ)ψ𝑑z0.

Dividing this inequality by λ and then letting λ0, we find that for every ψLm(QT),

QT(|Δu|m-2Δu-χ)ψ𝑑z0,

which is possible only if χ=|Δu|m-2Δu a.e. in QT.

The constructed function u takes the initial value in the sense of Definition 1 (iii), i.e., for every ϕLm(Ω),

|(u(t)-u0,ϕ)2,Ω|Ω(0t|ut(x,τ)|dτ)|ϕ|dxutm,QTϕm,Ωt1m0as t0+.

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

12Ω|u(τ)|2dx|τ=0τ=t=-Qt(|Δu|m-d|u|σ-2uΔu+fΔu)dz.

By construction, |u|σ-2|u|2, |u|σ-1|Δu|L1(Ω) and |u|σ-2uDxiuW01,1(Ω) for a.e. t(0,T). By the Green identity (see, e.g., [25, pp. 69–70]),

0=Qtdiv(|u|σ-2uu)dz=(σ-1)QT|u|σ-2|u|2dz+Qt|u|σ-2uΔudz,

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 and σ=2,

  • (iii)

    d>0 and

    • 1<m2, 2σ< if n2,

    • 2nn+2<m2, 2σ2+m(n+2)-2nm(n-2) if n3.

Proof.

(i) Let d=0. Assume that problem (1.1) admits two different solutions u1 and u2. Let us subtract equalities (2.2) for ui with the test-function Δ(u1-u2). By Lemma 7, for every t(0,T],

0tΩt(u1-u2)Δ(u1-u2)dz=-12Ω|(u1-u2)|2(τ)dz|τ=0τ=t,

while

Qt(|Δu1|m-2Δu1-|Δu2|m-2Δu2)Δ(u1-u2)𝑑z0

by monotonicity. It follows that u1-u22,Ω(t)=0 for all t[0,T].

(ii) Let σ=2. If u is a solution of problem (1.1), then v=uedt satisfies the equation

vt=e-d(m-1)t|Δv|m-1Δv,

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

(iii) Let us consider the case d>0, m(1,2] and σ2. Taking Δw=Δ(u1-u2) for the test-function in (1.7), we obtain

12Ω|w|2(t)dx+Qt(|Δu1|m-2Δu1-|Δu2|m-2Δu2)Δwdz=dQt(|u1|σ-2u1-|u2|σ-2u2)Δwdz=:I.(3.1)

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

(|b|m-2b-|a|m-2a)(b-a)(m-1)(1+a2+b2)m-22(b-a)2

for every a,b and m(1,2]. Then we have

(m-1)Qt|Δw|2(1+|Δu1|m+|Δu2|m)m-2mdzQt(|Δu1|m-2Δu1-|Δu2|m-2Δu2)Δwdz,

whence, by the reverse Hölder inequality,

(m-1)Δwm,Qt2(Qt(1+|Δu1|m+|Δu2|m)𝑑z)m-2mQt(|Δu1|m-2Δu1-|Δu2|m-2Δu2)Δw𝑑z.

Since Δuim,QTmC, we have

(1+Δu1m,Qtm+Δu2m,Qtm)m-2m1(1+2C)2-mm=:C,

and (3.1) can be continued as follows:

12w(t)2,Ω2(t)+C(m-1)Δwm,Qt2I.(3.2)

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

|I|dΔwm,Qt|u1|σ-2u1-|u2|σ-2u2m,Qt12C(m-1)Δwm,Qt2+C′′R,

with

R=|u1|σ-2u1-|u2|σ-2u2m,Qt2

and C′′ being an absolute constant. Plugging this estimate into (3.2), we transform it into the form

Ω|w|2(t)dx+C(m-1)(Qt|Δw|mdz)2m2C′′R.(3.3)

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

||b|σ-2b-|a|σ-2a|C(|b|σ-2+|a|σ-2)|b-a|for all a,b,σ2,

with C=C(σ). For every p>1, we have

RC(0tΩ|w|m(|u1|σ-2+|u2|σ-2)mdxdt)2mC(0t|w|mp,Ω(|u1|σ-2+|u2|σ-2)mp,Ωdτ)2m=C(0twmp,Ωm|u1|σ-2+|u2|σ-2mp,Ωmpdτ)2m.

By the embedding theorem, for all wH01(Ω),

wmp,ΩCw2,Ω,mp={2nn-2if n>2,any number from (1,)if n=2,if n=1,(3.4)

whence

R(0t(w2,Ω2)m2(|u1|σ-2+|u2|σ-2)mp,Ω𝑑τ)2m=(0tA(τ)Yμ(τ)𝑑τ)1μ,(3.5)

under the notation

Y(t)=w(t)2,Ω2,μ=m2>1,A(t)=(|u1|σ-2+|u2|σ-2)mp,Ω.

Let us show that A(t)L(0,T). By (3.4), in the present case it is sufficient to claim (σ-2)mp< if n=1,2, or (σ-2)mp<2nn-2 if n3. Let us notice that for n3, the condition 2nn+2<m is equivalent to m<2nn-2, which means that there always exists p>1 satisfying the inequality mp2nn-2. If n3, then we take p=2nn-21m and claim that

mp(σ-2)=2nmmn-2n+2m(σ-2)2nn-2,(3.6)

p can be arbitrary if n=1,2. Then |ui|(σ-2)mp,ΩCui2,Ω and

A(t),(0,T)=esssup(0,T)(|u1(t)|σ-2+|u2(t)|σ-2)mp,ΩC(esssup(0,T)u1(t)2,Ω+esssup(0,T)u2(t)2,Ω)m(σ-2)C~.

Simplifying (3.6), we obtain

σ2+m(n+2)-2nm(n-2).

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

Y(t)2C′′(0tA(τ)Yμ(τ)𝑑τ)1μ.

It follows that U(t)=Yμ(t) satisfies the linear integral inequality

U(t)(2C′′)μ0tA(τ)U(τ)𝑑τ,U(0)=0,

and by Gronwall’s inequality, U(t)=0. ∎

Theorem 2.

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

  • (i)

    2n<2m and 3σm+2,

  • (ii)

    3σ< and n=1,

  • (iii)

    3σ< and the solution is bounded.

Proof.

Using the inequality

(|b|m-2b-|a|m-2a)(b-a)22-m|b-a|mfor all a,b,m>2,

we transform (3.1) to the form

12Ω|w|2(t)dx+22-mQt|Δw|mdzdQtΔw(|u1|σ-2u1-|u2|σ-2u2)dz=-dQtw(|u1|σ-2u1-|u2|σ-2u2)dz=:I.(3.7)

Since (|v|σ-2v)=(σ-1)|v|σ-2v, we may write

I=-d(σ-1)Qtw(|u1|σ-2u1-|u2|σ-2u2)𝑑z=-d(σ-1)QT|w|2|u1|σ-2dz+d(σ-1)Qtwu2(|u1|σ-2-|u2|σ-2)dzd(σ-1)Qtwu2(|u1|σ-2-|u2|σ-2)dz=:J.

If σ3, by the Lagrange finite-increments formula, we have

||u1|σ-2-|u2|σ-2|=(σ-2)|θu1+(1-θ)u2|σ-3(σ-2)|u1-u2|(|u1|+|u2|)σ-3C(σ)|w|(|u1|σ-3+|u2|σ-3),

whence

|J|C0tw(τ)2,Ω(Ω|w|2|u2|2(|u1|2(σ-3)+|u2|2(σ-3))dx)12dτ.

Recall that w2p,ΩCw2,Ω for every p>1 if n=1,2 and p=nn-2 if n3.

Let n3. Then, by Hölder’s inequality,

|J|C0tw(τ)2,Ωw(τ)2nn-2,Ω(Ω|u2|n(|u1|n(σ-3)+|u2|n(σ-3))dx)1ndτ=C0tA(τ)w(τ)2,Ω2dτ,

with

A(t)=(Ω|u2(t)|n(|u1(t)|n(σ-3)+|u2(t)|n(σ-3))𝑑x)1n.

Let n=2. Take an arbitrary p(1,). Combining Hölder’s inequality and the embedding theorem, we obtain

|J|C0tw(τ)2,Ωw(τ)2p,Ω(Ω|u2|2p(|u1|2(σ-3)+|u2|2(σ-3))pdx)12pdτ=C0tA(τ)w(τ)2,Ω2dτ,

with

A(t)=(Ω|u2(t)|2p(|u1(t)|p(σ-3)+|u2(t)|p(σ-3))dx)12p.

Let n=1. For m>2, vC1+α(Ω)Cvxxm,Ω and v,ΩCvx2,Ω, which leads to the estimate

|J|C0twx(τ)2,Ωw(τ),Ω(Ω|u2x|2(|u1|2(σ-3)+|u2|2(σ-3))dx)12dτ=C0tA(τ)wx(τ)2,Ω2dτ,

with

A(t)=(Ω|u2x(t)|2(|u1(t)|2(σ-3)+|u2(t)|2(σ-3))dx)12C(u1xxm,Ω+u2xxm,Ω)σ-2C.

If the solutions are bounded, a revision of the estimates on |J| and A(t) shows that the conclusions of Propositions 3 and 4 remain valid for 1m1, which is true by assumption.

Let us consider the function Y(t)=w(t)2,Ω2. Due to (3.7) and the estimates on J and A,

Y(t)C0tA(τ)Y(τ)𝑑τ,Y(0)=0.

Since A(t)L1(0,T), it follows from Gronwall’s lemma that Y(t)=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 u(t)2,Ω. By the embedding theorem H01(Ω)Lr(Ω), which automatically implies the validity of all corresponding properties for the function u(t)r,Ω for some r2.

4.1 Homogeneous equation

Let us consider first the case f(x,t)0.

Theorem 1.

Let d>0 and f0. Assume that the exponent σ satisfies the condition

2>σ>{1if mn,1+2(1m-1n)if m<n.

Then the following hold:

  • (a)

    For m+σ<4 , every strong solution of problem ( 1.1 ) vanishes at a finite moment, that is, there exists tt(m,σ,u02,Ω) such that

    u(t)2,Ω2=0for all tt,

  • (b)

    For m+σ=4,

    u(t)2,Ωe-Ctu02,Ω2,

    with the positive constant C depending only on the data.

  • (c)

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

    u(t)2,Ω2u02,Ω2(1+Mtu02,Ω2α)α1(Mt)α,α=m-σ+4m+σ-4.

Proof.

Combining identities (2.2) with t,t+h(0,T), we obtain

12hΩ|u(τ)|2dx|τ=tτ=t+h+1htt+hΩ(|Δu|m+d(σ-1)|u|σ-2|u|2)dz=0.

Letting h0 and applying the Lebesgue differentiation theorem leads to the equality

12ddtu(t)2,Ω2+Δu(t)m,Ωm+d(σ-1)Ω|u(t)|σ-2|u(t)|2dx=0for a.e. t(0,T).(4.1)

By Hölder’s inequality,

Ω|u|2dx=Ω(|u|σ-22||)(|u(k)|2-σ2|u|)dx(Ω|u|σ-2|u|2dx)12(Ω|u|2-σ|u|2dx)12.

Since 0<2-σ<1, it follows from Hölder’s inequality that

(Ω|u|2-σ|u|2dx)12u1,Ω2-σ2(Ω|u|2σ-1dx)σ-12=u1,Ω2-σ2u2σ-1,Ω.

The embedding L1(Ω)W01,2σ-1(Ω) allows one to continue the last inequality as follows:

(Ω|u|2-σ|u|2dx)12Cu2σ-1,Ω1+2-σ2.(4.2)

For every uW2,m(Ω)H01(Ω),

u2σ-1,ΩCΔum,Ω,2σ-1<{nmn-mif n>m,if mn,(4.3)

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

(Ω|u|2-σ|u|2dx)12CΔum,Ω1+2-σ2.

It follows that

u2,Ω2=Ω(|u|σ-22|u|)(|u|2-σ2|u|)𝑑xC(Ω|u|σ-2|u|2dx)12Δum,Ω1+2-σ2C(Ω(|Δu|m+|u|σ-2|u|2)𝑑x)12+4-σ2m,

with an arbitrary σ(1,2) if mn and σ>1+2(1m-1n) if m<n. Now it is straightforward to obtain the estimate

u2,Ω4mm+4-σCΩ(|Δu|m+|u|σ-2|u|2)𝑑x.

Substitution of this inequality into (4.1) leads to a nonlinear differential inequality for the nonnegative function y(t)=u2,Ω2, that is,

y+Cyβ(t)0,y(0)=u02,Ω2,β=2mm-σ+4,(4.4)

which can be explicitly integrated.

(a) Let β<1. Then

0y1-β(t)y1-β(0)-C(1-β)t=u02,Ω2(1-β)-C(1-β)t.

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

t=1C(1-β)u02,Ω2(1-β),(4.5)

it is necessary that u(t)2,Ω20 for all tt.

(b) Let β=1. In this case y(t) satisfies the linear differential inequality, whence

y(t)u02,Ω2e-Ct.

(c) Let β>1. In this case integration of (4.4) gives

11-β(y1-β-y01-β)-Ct,

and the assertion follows. ∎

Remark 2.

Let us assume that under the conditions of Theorem 1, the solution is bounded, i.e., |u|M a.e in QT. Then (4.1) leads to

ddt(u(t)2,Ω2)+2Δu(t)m,Ωm+2d(σ-1)Mσ-2Ω|u(t)|2dx0,

and it follows that

u(t)2,Ω2u02,Ω2e-Ct,C=2d(σ-1)Mσ-2.

Remark 3.

Let one of the following conditions be fulfilled:

  • (a)

    d>0, σ=2 and m(1,),

  • (b)

    d0, m=2.

Then

u(t)2,Ω2u02,Ω2e-λt,(4.6)

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

λu2,Ω2Δu2,Ω2,uW2,2(Ω)H01(Ω).(4.7)

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

Theorem 4.

Let d0, σ(1,) and let u(x,t) be a strong solution of problem (1.1).

  • (a)

    If max{1,2nn+2}<m<2 , then u(x,t) extincts in a finite time. The moment of extinction depends on m, n and u02,Ω.

  • (b)

    If m=2 , then u(x,t) satisfies inequality ( 4.6 ).

  • (c)

    If m>2 , then u(x,t) satisfies the estimate

    u(t)2,Ω2u02,Ω2(1+(m-2)2Ctu02,Ωm-2)-2m-21(Mt)2m-2,

    where M=12λ2m(m-2) and λ is the constant from ( 4.7 ).

Proof.

By virtue of (4.1),

12ddt(u(t)2,Ω2)+Δu(t)m,Ωm0.

Applying the estimate

u2,Ω2CΔum,Ω,uW2,m(Ω)H01(Ω),(4.8)

we obtain inequality (4.4) for Y(t)=(u2,Ω2)m2. The conclusion follows as in the proof of Theorem 1. ∎

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

Lemma 5.

Let uW2,m(Ω)H01(Ω) and |u|σ-2|u|2L1(Ω). If

σ>2,max{1,2nn+2}<m<σσ-2,(4.9)

then

u2,Ω2C(Ω|Δu|mdx+Ω|u|σ-2|u|2dx)2(λm+1-λσ),(4.10)

with C being a constant independent of u, and

λ=12-1n-2σ1m-2n-2σ(12,1).

Proof.

We use the interpolation inequality of Gagliardo–Nirenberg: If there exists λ(12,1) such that

12=1n+λ(1m-2n)+(1-λ)2σ,

then for every uW2,m(Ω)H01(Ω),

u2,ΩCuW2,m(Ω)λuσ2,Ω1-λCΔum,Ωλuσ2,Ω1-λ,(4.11)

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

𝒫(λ)=12-1n-λ(1m-2n)-(1-λ)2σ.

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

𝒫(1)=-1m+n+22n>0,𝒫(12)=12(σ-2σ-1m)<0,

and since 𝒫(λ) is linear, the equation 𝒫(λ)=0 has exactly one root in the interval (12,1). Set v=|u|σ2. Then

uσ2,Ω=v1,Ω2σCv2,Ω2σ=C(Ω|u|σ-2|u|2dx)1σ.

Combining this inequality with (4.11), we have

u2,ΩCΔum,Ωλ(Ω|u|σ-2|u|2dx)1-λσC(Ω|Δu|mdx+Ω|u|σ-2|u|2dx)λm+1-λσ,

as required. ∎

Theorem 6.

Let m and σ satisfy (4.9). Assume that d>0. If

1β=2(λm+1-λσ)<1,

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

u(t)2,Ω2Kt1β-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(t)+Cyβ(t)0,y(t)=u(t)2,Ω2,

and the assertion follows. ∎

Remark 7.

It is shown in Theorem 4 that for m>2 and d0, y(t)=u(t)2,Ω2 decreases like t-α, with α=2m-2. By Theorem 6, in the case d>0, y(t) decreases like t-1β-1. If m>σ, in the latter case the rate of decay is higher.

4.2 Nonhomogeneous equation

Let us consider the case f(x,t)0. The energy equality (4.1) takes the form

12ddtu(t)2,Ω2+Δu(t)m,Ωm+d(σ-1)Ω|u(t)|σ-2|u(t)|2dx+ΩfΔu(t)dx=0.

Moving the last term to right-hand side and applying Young’s inequality with ϵ=12, we obtain

ddt(u(t)2,Ω2)+Δu(t)m,Ωm+2d(σ-1)Ω|u(t)|σ-2|u(t)|2dx2mm-1mmf(t)m,Ωm.(4.12)

Theorem 8 (asymptotic behavior).

Let u(x,t) be a strong solution of problem (1.1) with f0. Assume that fLm(QT)L(0,T;Lm(Ω)).

  • (i)

    Let

    f(t)m,ΩmCe-αt,α>0.(4.13)

    • (a)(a)

      If d>0, σ=2 and m(1,) , then

      u(t)2,Ω2u02,Ω2e-2dt+{C2d-αe-αtif 2dα,Cte-αtif 2d=α.(4.14)

    • (a)(b)

      If d0, m=2 , then

      u(t)2,Ω2u02,Ω2e-μt+{Cμ-αe-αtif μα,Cte-αtif μ=α

      with μ=1C , and the constant C from ( 4.8 ).

  • (ii)

    Let d0, m>2 . There exist positive constants A and B , depending on m, n and |Ω| , such that if

    f(t)m,ΩmA(1+Bt)mm-2,

    then

    u(t)2,Ω2u02,Ω2(1+Bt)2m-2.

Proof.

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

12ddt(u(t)2,Ω2)+du(t)2,Ω221m-1(m-1)m-mf(t)m,Ωm.

Set y(t)=u(t)2,Ω2. By (4.13), y(t) satisfies the nonhomogeneous ordinary differential inequality

y+2dyCe-αt,y(0)=u(t)2,Ω2,C=C21m-1(m-1)m-m.(4.15)

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

z+2dz=Ce-αt,z(0)=u(t)2,Ω2,

given by the explicit formula

z(t)=z(0)e-2dt+{C2d-αe-αtif 2dα,Cte-αtif 2d=α.

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+μyCe-αt,y(t)=u(t)2,Ω2,

with μ=1C, 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(t):

y+Cym2CA(1+Bt)mm-2,y(0)=u02,Ω2.

It is straightforward to check that the Cauchy problem

v+Cvm2=CA(1+Bt)mm-2,v(0)=y(0),

admits the solution v=v(0)(1+Bt)-2m-2, provided that the parameters A>0, B>0 satisfy

-2ABm-2+CAm2=C.

v(t) is a majorant function for y(t), that is,

u(t)2,Ω2v(t)=u02,Ω2(1+Bt)2m-2.

4.3 Extinction at a prescribed moment

The assertion of Theorem (1) can be extended to the case when f0. Let us assume that

fLm(QT)L(0,T;Lm(Ω)),f(t)m,Ωmϵ[1-ttf]+β1-β,ϵ>0,[v]+=max{0,v},(4.16)

with β(0,1) and a given tf<. We will assume that the parameters tf (the moment of vanishing of the source term f), ϵ>0 (the intensity of the source) and the norm of the initial function y(0)=u(0)2,Ω2 are connected by the relation

-11-βy(0)1tf+Cyβ(0)=ϵC,(4.17)

where β=2mm-σ+4, C and C are the constants in (4.4) and (4.15).

Theorem 9.

Assume that d>0, the exponents m, σ satisfy the conditions of Theorem 1 with m+σ<4, and f satisfies (4.16), with β=2mm-σ+4(0,1). If the data tf, ϵ and u(0)2,Ω2 satisfy (4.17), then

y(t)=u(t)2,Ω2=0for all t>tf.

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(t), we obtain

y(t)+Cyβ(t)ϵC[1-ttf]+β1-β,y(0)=u(0)2,Ω2.(4.18)

The solution of the Cauchy problem

v(t)+Cvβ(t)=ϵC[1-ttf]+β1-β,v(0)=y(0),(4.19)

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

v(t)=v(0)[1-ttf]+β1-β

solves (4.19), provided that y(0), ϵ and tf satisfy (4.17). ∎

Remark 10.

The effect of simultaneous vanishing of the source f and the solution is present if the given parameters y(0), tf 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 tf through the data. Comparing it with the extinction moment t for the solutions of the homogeneous equation (1.1) given by formula (4.5), we find that

tf=y(0)(1-β)(Cyβ(0)-ϵC)>y1-β(0)C(1-β)=t,y(0)=u(0)2,Ω2.

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 tf>t.

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About the article

Received: 2016-03-10

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, DOI: https://doi.org/10.1515/anona-2016-0055.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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