On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

Zhan, Huashui

doi:https://doi.org/10.1155/2020/2929045

Discrete Dynamics in Nature and Society

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 2929045 | https://doi.org/10.1155/2020/2929045

On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

Huashui Zhan^1,2

Academic Editor: Genni Fragnelli

Received09 Aug 2019

Accepted11 Feb 2020

Published14 Mar 2020

Abstract

The initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient and the variable exponent depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition matching up with the equation is based on a submanifold of . By this innovation, the stability of weak solutions is proved.

1. Introduction

The porous medium equation with a constant exponent is widely used to model several real-life problems, and it has been extensively studied, one can refer to the survey books [1–6] and the references therein. The dynamics system of a partially nonlocal and inhomogeneous nonlinear medium has been considered in [7–9]. The case where the exponent of nonlinearity is not constant was proposed by Antontsev and Shmarev in [10], where the existence, uniqueness, and some properties of the solution in a bounded fixed domain were researched. By using the Galerkin finite element method, Duque et al. [11] proved the convergence of a fully discrete solution for this problem in a fixed domain. Based on one of the properties proved in [11] that the solution is with the finite speed of propagation, Duque et al. [12] considered the free boundary problem by using the moving mesh method to the porous medium. However, the moving mesh method was first introduced by Huang and Russell in [13].

In this paper, we consider the initial-boundary value problem of a generalized porous medium equation with a variable exponent:where is a function, is a function, , and is a bounded domain with a smooth boundary .

Equation (1) is a special case of the reaction-diffusion equation:

Because may degenerate on the boundary, how to impose a suitable boundary value condition to study the well posedness of weak solutions to equation (2) has attracted extensive attention and has been widely studied for a long time. In some details, though the initial valueis always imposed, the Dirichlet boundary conditionmay not be imposed or be imposed in a weaker sense than the traditional trace. One can refer to the references [14–19] for the details.

Naturally, besides the porous medium equation with variable exponents, the so-called electrorheological fluid equations with the formhave been brought to the forefront by many more scholars. Since the beginning of this century, there are a great deal of papers devoting to the well-posedness problem, the intrinsic Harnack inequalities, the long-time behavior, and the Hölder regularity of weak solutions, one can refer to the literatures [20–31] and the references therein.

If , then satisfies

The existence and the stability of weak solutions to equationhas been studied in [32]. It is found that the degeneracy of on boundary (6) may replace the usual boundary value condition (4). In other words, if satisfies (6), the stability of weak solutions may be proved without the usual boundary value condition (4).

In this paper, we will use some ideas of [32] to study the well posedness of weak solutions to equation (1). Because both and are dependent on the time variable t, the problem becomes more difficult and the question of existence of such solutions is still open for equation (8), as well as for evolution p-Laplace equation with the exponent p depending on t [22]. Instead of condition (6), we only assume that . By this assumption, we find out a partial boundary value condition matching up with the equation. Moreover, because both and are dependent on the time variable t, the partial boundary value condition cannot be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition is based on a submanifold of . By this innovation, the stability of weak solutions is proved.

2. The Partial Boundary Value Condition and the Main Results

For any given and small enough , we set

The most important and essential improvement is that instead of the usual boundary value condition (4), the stability of weak solutions is proved based on a partial boundary value condition:whereand for any given , if ,and if ,

Thus, if for every , either or , then one can deduce an expression from the above discussion. For example, when and when ; then,

The most characteristic out of the ordinary is that or is just a submanifold of , and it cannot be expressed as a cylinder with the form and .

Definition 1. If and satisfiesand for , and then is said to be a weak solution of equation (1) with the initial value (3) in the senseMoreover, if satisfies (4) or (3) in the sense of the trace in addition, then it is said to be a weak solution of the initial-boundary value problem of equation (1).

Theorem 1. If is a function, satisfies

If satisfies (16), then equation (1) with initial value (3) has a nonnegative solution.

Theorem 2. Let be a function, then satisfy (19):

Then, the initial-boundary value problem (1), (3), and (4) has a uniqueness solution.

Theorem 3. Let be two solutions of equation (1) with the initial values , respectively, and with a partial boundary value condition

It is supposed that, for every , either or , satisfiesand and satisfy

Then,

Hereinafter, the constant represents a constant which depends on T.

At the end of this section, we would like to suggest that if is a constant, then condition (22) in Theorem 3 is naturally true.

3. The Proof of Theorem 1

First, we suppose that and and consider the following regularized problem:

According to the standard parabolic equation theory, there is a weak solution satisfying

Moreover, by comparison theorem, we clearly havewhich yields

In what follows, we are able to prove that the limit function u is a weak solution of (1) with the initial value (3).

Multiplying both sides of the first equation in (25) by and integrating it over , we have

For the left-hand side of (29),

For the first term of the right-hand side of (29), becauseby a complicated calculation and using the Young inequality, we can deduce that

For the second term of the right-hand side of (29), becausewe can deduce that

From (28)–(34), we extrapolate

Accordingly, there is such thatweakly in . We now can proveas in a similar way as that in [32].

The last but not the least, by that , using (27), we have

Letting in (30), by (37), (38), and (39), we know satisfies (18).

Secondly, if satisfies only (16), we should consider equation (9) with the initial value which is the mollified function of , from the above that there is a weak solution satisfying (18). Letting , the limit function is a solution of (1) satisfying (17) and (18), but generally is not continuous at as in the case .

Thirdly, the initial value (4) can be proved in a similar way as that when is a constant, one can refer to [5] for the details. Thus, u is a solution of equation (1) with the initial value (4). Thus, Theorem 1 is proved.

4. The Proof of Theorem 2

One can see that, in Definition 1, there is not any definition on the general derivative . At the beginning of this section, we first answer this question.

For any , the Banach space is defined byand is denoted as its dual space. The Banach space is defined byand is denoted as its dual space. From [21], we have

It is easy to prove the following lemmas, so we omit the details here.

Lemma 1. If is a weak solution of equation (1) with the initial value (3), then .

Lemma 2. Suppose that and . For any continuous function , let . For , there holds

Lemma 3. If , then . So, the weak solution of equation (1) can be defined as the homogeneous boundary value condition in the sense of the trace.

Theorem 4. Let and be two solutions of equation (1) with the initial values respectively, and with

If , then

Proof. For any given positive integer n, let and . Then, , and we haveBecause on the boundary , we choose as the test function and integrate over . Then,Let us analyse every term in (47). In the first place,In the second place, we deal with the fourth term on the left-hand side of (47). For any given , we setClearly,Based on these denotations, we haveBecause ,If has 0 measure, thenIf has a positive measure, thenThus, in both cases, we always haveIn the third place, for the first term on the left-hand side of (47), by Lemma 2,Let in (47). Formulas (48)–(57) yield

Corollary 1. Theorem 2 is true.

5. The Stability Based on the Partial Boundary Value Condition

Theorem 5. Let and be two solutions of equation (1) with the initial values and respectively, and with a partial boundary value conditionwhereand for any given , if ,and if ,

It is supposed thatand and satisfy

Then,

Proof. According to the definition of weak solutions, for all , we haveFor any and a small positive constant , based onwe define thatLet be the characteristic function of . Becausewe can chooseas the test function. Then,Firstly, we still haveSecondly, for the third term on the left-hand side of (71) by that , we haveas . Thirdly, for the fourth term on the left-hand side of (71), we can showas . The corresponding details are given below.
If is with 0 measure, thenand by (64),we obtainWhen has a positive measure,Also by (64),Fourthly, when , using the partial boundary value condition (59),When , using the partial boundary value condition (59), we haveThe last but not the least, as in the proof of Theorem 2, we can show thatThen,Letting and in (73), by (74) and (80)–(85), we obtainBy the arbitraries of τ, we have

Proof of Theorem 2. Because for every , either or , by checking the process of the proof of (80) or (81), we easily obtain Theorem 3.

6. Conclusion

In this paper, we consider the initial-boundary value problem of a generalized porous medium equation with a variable exponent. Different from the previous related works, both the diffusion coefficient and variable exponent are dependent on the time variable t, we find out a partial boundary value condition matching up with the equation. The most important innovation is that the partial boundary value condition matching up with the equation is based on a submanifold of . However, because there is an additional condition (23) imposed, Theorem 3 has not answered the problem globally. In other words, how to obtain the same conclusion as that in Theorem 3 without condition (23) is remained to be studied in the future.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

This study was supported by the Natural Science Foundation of Fujian Province (2019J01858), China.

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Copyright

Copyright © 2020 Huashui Zhan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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