On the Boundary Value Condition of an Isotropic Parabolic Equation

Ou, Qitong; Zhan, Huashui

doi:https://doi.org/10.1155/2020/2180830

Journal of Function Spaces

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

Research Article | Open Access

Volume 2020 | Article ID 2180830 | https://doi.org/10.1155/2020/2180830

On the Boundary Value Condition of an Isotropic Parabolic Equation

Qitong Ou¹and Huashui Zhan¹

Academic Editor: Wilfredo Urbina

Received12 Jun 2020

Revised19 Sept 2020

Accepted04 Oct 2020

Published27 Nov 2020

Abstract

The well-posedness problem of anisotropic parabolic equation with variable exponents is studied in this paper. The weak solutions and the strong solutions are introduced, respectively. By a generalized Gronwall inequality, the stability of strong solutions to this equation is established, and the uniqueness of weak solutions is proved. Compared with the related works, a new boundary value condition, , is introduced the first time and has been proved that it can take place of the Dirichlet boundary value condition in some way.

1. Introduction

In this paper, we mainly pay attention on the stability of solutions to the following anisotropic parabolic equation with variable exponents: with the initial value and the boundary value condition

Here, , , , and is a bounded domain with the smooth boundary , .

When is a constant, , equation (1) arises in the mathematical modelling of various physical processes such as flows of incompressible turbulent fluids, gases in pipes, and processes of filtration in glaciology. The equation in this case has been studied widely [1–5]. When is a measurable function, , equation (1) is similar with the equation with the type which arises in the phenomena of electrorheological fluids [6, 7]. The existence of solutions of the initial-boundary value problem to this equation can be found in [8–11]. Also, one can refer to [12–18] for some other related works.

If and satisfies the well-posedness problem of the following equations has been studied by the second author in recent years [19–21]. Instead of boundary value condition (3), only a partial boundary value condition is imposed, where is a relatively open subset which has different expression according to different kinds of and sometimes is just an empty set [19–21].

Compared with [19, 20] and [21], since the diffusion coefficient and the variable exponent both depend on the time variable , equation (1) has a wider applications than equation (6), and in mathematical theory, there are some essential difficulties to be overcome. More than that, instead of (5), we only assume that and do not require that which is similar as (5) in [19–21].

To see the essential difference between (9) and (10), let us give a special case of equation when , , , and and are relatively open subset of , . Consider the equation where then (9) is true, i.e.,

More precisely, for example,

However, in (12) and (13),

This fact makes us feel that only under the boundary value condition (3), the uniqueness (or the stability) of weak solutions to equation (11) can be true. The following works seem to supply more evidences. One is [22] in which the equation is studied. The others are the equations arising from the double phase obstacle problems where , which have gained a wide attention in recent years, one can refer to [23, 24] and the references therein. In these papers, the boundary value condition (3) is imposed without exception.

The main dedication of this paper is that the stability of weak solutions to equation (11)(in general, (1)) can be established independent of boundary value condition (3). Such a conclusion totally overthrows our imagination. In theory, condition (9) can take place of boundary value condition (3) is found the first time. In applications, condition (9) reflects a synthesized effect of an anisotropic diffusion process.

This paper is arranged as follows. In Section 1, we have given a simple introduction. In Section 2, we will introduce the definitions of weak solution and strong weak solution, respectively, quote some basic lemmas, and give the main results. In Section 3, we will study the stability of weak solutions to equation (1) with the new boundary value condition (9). In Section 4, we will study the uniqueness of weak solution to equation (1) independent of the boundary value condition (7). In Section 5, we will give the outline of the proof on the existence of strong solutions.

2. Definitions and Main Results

We denote

First of all, let us introduce the definition of solutions.

Definition 1. A function is said to be a weak solution of equation (1), if and for any function , This definition of weak solution is similar as that defined in [20], where is a constant. Also, we can prove the existence of weak solutions similar as that defined in [20], so we do not repeat the details in this paper. As an improvement from the existing result in [20], we introduce the following definition.

Definition 2. A function is said to be a strong solution of equation (1) with the initial value (2), if and for any function , satisfies (21) and the initial value is satisfied in the sense The proof of the existence of strong solution will be given at Section 5 of this paper. Since is positive when , (22) means that and exist almost everywhere in . This is the reason that we call as a strong solution of equation (1). Moreover, from Definition 2, for all , we still have the integral equality (21), which implies that Thus, if is a strong solution of equation (1), then it is a weak solution.
Secondly, in order to make the paper sufficiently self-contained and present our discussions in a straightforward manner, let us briefly recall some preliminary results on properties of variable exponent Lebesgue spaces and variable exponent Sobolev spaces [24–26].
Set For any , we define For any , let consist of all measurable real-valued functions satisfying endowed with the Luxemburg norm Define endowed with the norm Let be the closure space of in .

From [25–27], we have

Lemma 3. The following three statements are true. (i)The space , , and are reflexive Banach spaces(ii)(-Hölder’s inequality), let and be real functions satisfying with . Then, the conjugate space of is . For any and , we have (iii)There are the following properties: (1)if , then (2)if , then (3)if , then Basing on Lemma 3, by generalizing the Gronwall inequality, we will prove the following stability theorems, in which the initial values satisfy

Theorem 4. Let and be two strong solutions of (1) with the initial values and , respectively; and satisfy (8) and (9) and satisfies If for small enough, then where for any , .
In this paper, the constant represents that depends on . If we only want to prove the uniqueness of weak solutions, condition (34) is not necessary; we have the following result.

Theorem 5. Let satisfy (8) and (9) and be a Lipschitz function on . If and are two strong solutions of equation (1) with the initial values and , respectively, then for any , which implies that the uniqueness of weak solution is true.
One can see that both Theorem 4 and Theorem 5 imply the uniqueness of solution is true. However, in Theorem 5, the convection function is independent of the diffusion coefficient , so as a uniqueness theorem, it is a better than Theorem 4.

3. The Stability of Strong Solutions Independent of the Boundary Value Condition

For small , let

Obviously, , and

In addition, if we denote , then

At first, we give a generalization of the Gronwall inequality.

Lemma 6. If and where , then,

Proof. Define that . Without loss of the generality, we may assume that there exists , . Then, for any , . Denoting then, , and Since , there exists a constant such that Combing (41) with (45), we obtain Using the Gronwall inequality, we have If , this lemma has been proved in [19] recently.
Secondly, we give the proof of Theorem 4.

Proof of Theorem 4. Let and be two strong solutions of equation (1) with the initial values and , respectively.
For any , set , and By a process of limit, we can choose as the test function. Here, is the characteristic function of . Then, In the first place, using the dominated convergence theorem, we have and for any , In the second place, we notice when ; in the other places, it is identical to zero. Since we assume that by Lemma 3, we have where and .
(53) implies In the third place, since satisfies condition (34), we can deduce that In details, If the set has a positive measure, by condition (34), we have where or according to or from Lemma 3.
If the set only has zero measure, since , Finally, by condition (34), we have since .
Now, let in (42). We easily obtain that where .
Using Lemma 6, we have Then by the arbitrary of ,

4. The Uniqueness of Weak Solution

Lemma 7. Let . For any continuous function , , , it holds This lemma can be generalized from of Lemma 2.2 in [28] simply; we do not give the details here.

Theorem 8. Let satisfy (8)(9) and , be a Lipschitz function on . If and are two weak solutions of equation (1) with the initial values and , respectively, then, there exists a positive constant such that

Proof. Let and be two weak solutions of equation (1) with the initial values and , respectively. By a process of limitation, we may choose as a test function. Denoting that , then, At first, we have Here, we have used the fact that and or according to or has a similar meaning. Now, if we denote that and , by that , we have and by the Hölder inequality, where .
Combing (67)-(71), we obtain where .
Secondly, For the first term on the right hand side of (73), by that , , using the Hölder inequality, we have For the second term on the right hand side of (73), since , denoting as usual, we have By this inequality, we deduce Defining as above, if , then . By the Hölder inequality, where .
If , then , From (76)-(78), we obtain where .
Once more, by Lemma 7, we have According to (65), (66), (72), (74), (79), and (80), since and , we have where . By (81), using the generalized Gronwall inequality, we deduce Thus, by the arbitrary of , we have By (83), we have the conclusion. The proof is complete.

By this theorem, Theorem 5 is true clearly.

5. The Strong Solutions Dependent on the Initial Value

For the completeness of the paper, we will give a basic theorem about the existence of strong solutions.

Theorem 9. If , satisfies (8) and (9), is a function on , then equation (1) with initial value (2) has a weak solution.
Here, . Before we give the proof of Theorem 9, we would like to point out that condition (86) is just a sufficient condition; we also can use other conditions to replace them. For example, when , by the conditions Theorem 9 had been obtained in [19].

Proof of Theorem 9. Consider the following regularized problem Here, , , and converges to in . It is well-known that the above problem has a unique weak solution and [8], and where .
Multiplying (88) by and integrating it over , then If , by that , then Accordingly, by (92), we have Multiplying (88) by , integrating it over , then it yields Note that , and by the Young inequality, We have If , Combining (95), (98), and (99), we have By the above inequality, we have Thus, by (91), (94), and (101), there exist a function and a dimensional vector function satisfying that a.e. in , and Similar as with Theorem 2.1 of Chapter 2 in [1] (also, one can refer to [19] in which is just a constant), we are able to show that for any . At last, by a process of limit, we can choose the test function in which and is the characteristic function of . Then, Let and . Then, we have (23) and is a strong solution of equation (1) with the initial value (2) in the sense of Definition 2.
At last, we give a simple comment. The condition can not assure the boundary value condition is imposed in the sense of the trace. In fact, we have the following proposition.

Proposition 10. If satisfies (85), i.e., Then, Thus, if satisfies (85), the partial boundary value condition (105) can be imposed in the sense of trace. However, in this paper, we pay our attention on the studying the stability (or the uniqueness) of solutions independent of the boundary value condition (105), so Proposition 10 is not important.

6. Conclusion

It is clear of that Lemma 6 has a wider applications than the classical Gronwall inequality. Moreover, compared with reference [18], there is at least the essential difference in two aspects. The first one is that condition (9), i.e., is much weaker than condition (5) appearing in [19], i.e.

Such a degeneracy is the special nature of the anisotropic equation. The second one is that we have not used boundary value condition (3) throughout this paper; in other words, condition (9) may replace boundary value condition (3) in some way. Moreover, using some techniques developed by the second author in his work [10], in which the well-posedness of weak solutions to equation, has been discussed; the method used in this paper can be applied to study a more general equation in the future.

Data Availability

There is not any data in the paper.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors read and approve the final manuscript.

Acknowledgments

The paper is supported by the Natural Science Foundation of Fujian Province of China (no. 2019J01858). The authors would like to thank everyone for their help.

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Copyright © 2020 Qitong Ou and 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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