The paper is devoted to studying the Dirichlet problem for the fully nonlinear parabolic equation
with a nonnegative coefficient d and exponents , .
The fully nonlinear equation (1.1), without lower-order terms, falls into the class of equations , which includes the dual porous medium equation
and the Barenblatt equation
Equation (1.3), with
was introduced in  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 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
with a given smooth function , and . It is proved that this problem has an integral solution in the sense of [22, Chapter 9]. When used for image restoration , 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
were studied in  for , , , and in  for the range of parameters , , . A local in time strong (maximal) solution of the Cauchy problem for equation (1.6) is obtained in  as the limit of a sequence of solutions of the regularized problems posed in expanding cylinders. The initial datum is assumed continuous and nonnegative if , while in the case , the condition in 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  for the case and extended in  to the case . 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 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.
A function is called strong solution of problem (1.1) if the following hold:
, , , ,
for every test-function ,
for every , as .
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 ( or ). It turns out that for a strong solution exists for any and , while in the case , the exponent has to satisfy the inequality . The existence theorem is proved under the natural assumptions and .
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 , 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 as . The energy satisfies the ordinary differential inequality, linear or nonlinear in dependence on the assumptions on the parameters of nonlinearity, and may be nonhomogeneous if . 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  and  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 , , and .
For every and problem (1.1) has at least one strong solution.
The solution is unique if either and , or and .
If and , then every strong solution:
vanishes in a finite time if ,
decreases exponentially if ,
has the power decay
with L being a constant independent of u.
If , 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,
under suitable regularity assumptions and the conditions and . 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
and the doubly nonlinear equation
It is proved in  that for and (the fast diffusion equation) every solution of the homogeneous Dirichlet problem for equation (1.8) vanishes at a finite moment . Moreover, as , the solution converges to a separable solution of the form
The case (the slow diffusion equation) and is studied in . It is shown that in this case every solution converges as 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 . It turns that the solutions of equation (1.9) converge, as , to a nonnegative separable solution of the same equation. The rate of convergence is different in the cases and . There exist positive constants C and β such that for ,
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  and . 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  for a class of anisotropic parabolic equations with variable exponents of nonlinearity which includes, as a special case, the equation
with constant exponents , , and the coefficients , . Set
vanishes in a finite time if either and , or and ,
decreases exponentially if , i.e., , ,
has the power decay if .
A revision of the proofs given in  shows that the same properties hold for the solutions of the isotropic equation
Observe that equations (1.1) and (1.11) coincide when , 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.
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 , decrease exponentially if , or decrease like if , while the solutions of (1.2) display the same behavior in the ranges of the exponent , , or .
Throughout the text, we denote and for the functions depending on . is the Sobolev space with the norm , with
The Hilbert space is the closure of the set (smooth functions with compact support in Ω) with respect to the norm of . Notice that is an equivalent norm of the space . Indeed, on the one hand we have , while on the other, is a solution of the Poisson equation with the right-hand side set to and with zero trace on . Thus, it satisfies the estimate
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 and , ,
2 Existence of strong solutions
Let . Assume that
Then, for every , , problem (1.1) has at least one strong solution which satisfies the energy equality
2.1 Galerkin’s approximations
Let be a natural number. By , we denote the system of eigenfunctions of the problem
Without loss of generality, we assume that forms an orthonormal basis of . A solution of problem (1.1) will be constructed as the limit of the sequence , where . The coefficients are defined from the system of ordinary differential equations
For every finite k the right-hand side in (2.3) is a Hölder-continuous function of , and by Peano’s theorem there exists at least one solution, , defined on an interval . To prove that each of can be continued to the maximal existence interval , it is sufficient to derive independent of k a priori estimates on .
2.2 A priori estimates
Let , and , . Then the functions satisfy the following uniform estimate:
with the constant C depending only on m and n.
It is easy to check that for every sufficiently smooth function u,
Applying Young’s inequality to the last term on the left-hand side, we find that for every and any , we have
Applying (1.13), we estimate
Choosing ϵ so small that , we obtain (2.4) for every . This inequality remains valid for every because the right-hand side is independent of k, and thus . ∎
Under the conditions of Theorem 1, there exists a constant C, independent of k, such that
Combining the embedding theorems in Sobolev spaces with (2.4), we find that
with if , if , or if . Then
Gathering these three estimate, we obtain (2.6). ∎
Let in the conditions of Lemma 2
with q defined in (2.6). Then , and there exists a constant C, independent of k, such that
Let us denote , and choose large enough so that . Given , we denote , with . By virtue of (2.3) and due to the orthogonality of the system in , we have
Using Hölder’s inequality, the embedding theorems in Sobolev spaces and the inequalities
we find that
Likewise, the term is estimated as follows:
The assumption on σ entails the inequality
The term is estimated by Hölder’s inequality as follows:
Combining these estimates with (2.4), we find that for every with ,
with the constant C depending on , , m, σ and , but being independent of k. ∎
By Lemmas 2 and 4, and are uniformly bounded. Since we have , with the embedding being compact, it follows, from the compactness result of Aubin , that is precompact in . The sequence contains a subsequence (for which we keep the same name), that possesses the following properties: there exist functions , and such that
2.4 Passing to the limit
Let us fix an arbitrary . By the method of construction, we have
for every and every . The convergence properties (2.7) allow one to pass to the limit as . Thus,
for every fixed , Since is dense in , the last equality is true in the limit as . Hence,
for every , whence . Let us introduce the function space
Let . If , then , after possible redefining on a set of zero measure in .
Let . Following the proof of [12, Section 5.4, Theorem 1], one may construct the linear extension operator so that a.e. in , for a.e. , , and
with C being a constant independent of u. Denote . The same arguments show that there exists a linear extension operator such that a.e. in , and
with being an absolute constant. The constructed extension coincides with u a.e. in . Set and choose a family of functions with compact supports in , approximating w in the norm of . Such a sequence can be obtained by means of mollification, see [12, Section 5.4, Theorems 2 and 3]. For every , we have
Integrating this inequality in τ and simplifying, we obtain
It follows that
that is, is a Cauchy sequence in and converges to a function . On the other hand,
whence for a.e. in . ∎
The function is defined by continuity and in (2.7), .
For every and every ,
Denote by the sequence of Steklov’s means of u, that is,
It is known that and as . Since
we may apply the Green formula (see, e.g., [25, pp. 69–70]). For every and every , we have
For every ,
Indeed, by Lemma 5, , which means that for every there exists such that
On the other hand,
as , by virtue of (2.10). ∎
and q from condition (2.1). Then
Let us make use of the representation
For every , the inclusion holds, whence, by virtue of (2.7),
Notice that the assumption is equivalent to . The convergence follows from the Vitali convergence theorem. Since a.e. in and uniformly in k, we have in , with . By Hölder’s inequality,
The second factor here is uniformly bounded in k due to (2.4), while the first factor tends to zero because . ∎
from (2.12), we obtain
whence, by virtue of (2.11),
Since is arbitrary, the last inequality holds in the limit as and remains true for every . Let us choose ϕ in the special way , where and is arbitrary. This choice yields the inequality
Dividing this inequality by λ and then letting , we find that for every ,
which is possible only if a.e. in .
The constructed function u takes the initial value in the sense of Definition 1 (iii), i.e., for every ,
By construction, , and for a.e. . By the Green identity (see, e.g., [25, pp. 69–70]),
and (2.2) follows.
3 Uniqueness of strong solutions
The strong solution of problem (1.1) is unique if one of the following conditions is fulfilled:
, if ,
, if .
by monotonicity. It follows that for all .
(ii) Let . If u is a solution of problem (1.1), then satisfies the equation
and the assertion follows as in the case .
(iii) Let us consider the case , and . Taking for the test-function in (1.7), we obtain
Let us make use of the following well-known inequality:
for every and . Then we have
whence, by the reverse Hölder inequality,
Since , we have
and (3.1) can be continued as follows:
Let us estimate I. Using the inequalities of Hölder and Cauchy, we have
and being an absolute constant. Plugging this estimate into (3.2), we transform it into the form
To estimate R, we will use another known inequality, that is,
with . For every , we have
By the embedding theorem, for all ,
under the notation
Let us show that . By (3.4), in the present case it is sufficient to claim if , or if . Let us notice that for , the condition is equivalent to , which means that there always exists satisfying the inequality . If , then we take and claim that
p can be arbitrary if . Then and
Simplifying (3.6), we obtain
It follows that satisfies the linear integral inequality
and by Gronwall’s inequality, . ∎
Let , . The strong solution of problem (1.1) is unique if one of the following conditions is true:
and the solution is bounded.
Using the inequality
we transform (3.1) to the form
Since , we may write
If , by the Lagrange finite-increments formula, we have
Recall that for every if and if .
Let . Then, by Hölder’s inequality,
Let . Take an arbitrary . Combining Hölder’s inequality and the embedding theorem, we obtain
Let . For , and , which leads to the estimate
Let us consider the function . Due to (3.7) and the estimates on J and A,
Since , it follows from Gronwall’s lemma that . ∎
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 . By the embedding theorem , which automatically implies the validity of all corresponding properties for the function for some .
4.1 Homogeneous equation
Let us consider first the case .
Let and . Assume that the exponent σ satisfies the condition
Then the following hold:
For , every strong solution of problem ( 1.1 ) vanishes at a finite moment, that is, there exists such that
with the positive constant C depending only on the data.
For , there exists a positive constant M , independent of u , such that
Combining identities (2.2) with , we obtain
Letting and applying the Lebesgue differentiation theorem leads to the equality
By Hölder’s inequality,
Since , it follows from Hölder’s inequality that
The embedding allows one to continue the last inequality as follows:
For every ,
It follows that
with an arbitrary if and if . Now it is straightforward to obtain the estimate
Substitution of this inequality into (4.1) leads to a nonlinear differential inequality for the nonnegative function , that is,
which can be explicitly integrated.
(a) Let . Then
Since the right-hand side of this inequality vanishes at the moment
it is necessary that for all .
(b) Let . In this case satisfies the linear differential inequality, whence
(c) Let . In this case integration of (4.4) gives
and the assertion follows. ∎
and it follows that
Let one of the following conditions be fulfilled:
, and ,
where in case (a), whereas in case (b), λ is the best constant from the inequality
Note that λ is the first eigenvalue of the Dirichlet problem for the Laplace operator in Ω.
Let , and let be a strong solution of problem (1.1).
By virtue of (4.1),
Applying the estimate
In the case , the differential inequality (4.4) is derived in a different way.
Let and . If
with C being a constant independent of u, and
We use the interpolation inequality of Gagliardo–Nirenberg: If there exists such that
then for every ,
with C being a constant independent of u. Let us check that such a number λ indeed exists. Define the function
It is obvious that under the conditions imposed on m and σ,
and since is linear, the equation has exactly one root in the interval . Set . Then
Combining this inequality with (4.11), we have
as required. ∎
Let m and σ satisfy (4.9). Assume that . If
then every strong solution of problem (1.1) satisfies the estimate
with K being a constant independent of u.
and the assertion follows. ∎
4.2 Nonhomogeneous equation
Let us consider the case . The energy equality (4.1) takes the form
Moving the last term to right-hand side and applying Young’s inequality with , we obtain
Theorem 8 (asymptotic behavior).
Let be a strong solution of problem (1.1) with . Assume that .
If , and , then
If , , then
with , and the constant C from ( 4.8 ).
Let , . There exist positive constants A and B , depending on m, n and , such that if
We begin with case (i) (a). By (4.12),
Set . By (4.13), satisfies the nonhomogeneous ordinary differential inequality
Estimate (4.14) is an immediate byproduct of the fact that is majorated by the solution of the problem
given by the explicit formula
In case (ii) (b), we drop the last term on the left-hand side of (4.12). Since , the resulting inequality can be transformed to the form
with , and the constant C from (4.8). The conclusion follows as in case (i) (a).
It is straightforward to check that the Cauchy problem
admits the solution , provided that the parameters , satisfy
is a majorant function for , that is,
4.3 Extinction at a prescribed moment
The assertion of Theorem (1) can be extended to the case when . Let us assume that
with and a given . We will assume that the parameters (the moment of vanishing of the source term f), (the intensity of the source) and the norm of the initial function are connected by the relation
The solution of the Cauchy problem
is a majorant function for the nonnegative solution of (4.18). The function
The effect of simultaneous vanishing of the source f and the solution is present if the given parameters , 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).
Condition (4.17) allows one to express the moment of vanishing through the data. Comparing it with the extinction moment for the solutions of the homogeneous equation (1.1) given by formula (4.5), we find that
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 .
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About the article
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.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0