The Partial Second Boundary Value Problem of an Anisotropic Parabolic Equation

Abstract

Consider an anisotropic parabolic equation with the variable exponents , where , , , , . If is degenerate on , then the second boundary value condition is imposed on the remaining part . The uniqueness of weak solution can be proved without the boundary value condition on .

1. Introduction

The evolutionary Laplacian equation,possesses some interesting mechanical properties in the presence of an electromagnetic field [1, 2], the well-posedness of weak solutions to the first initial-boundary value problem of (1) had been researched in [3–7], etc. Here, is a bounded domain with the smooth boundary , and is a function. Sobolev spaces play an important role in the theory of evolutionary Laplacian equation. In recent years, the generalized Orlicz-Lebesgue spaces and the corresponding generalized Orlicz-Sobolev spaces have attracted more and more attention. The spaces are special cases of the generalized Orlicz-Spaces originated by Nakano and developed by Musielak and Orlicz (see [8–10]). We refer to [11–13] for properties of the spaces and such as reflexivity, denseness of smooth functions, and Sobolev type embeddings. The study of these spaces has been stimulated by problems of elasticity, fluid dynamics, calculus of variations, and differential equations with -growth conditions. Roughly speaking, the interest in variable exponent spaces comes not only from their mathematical curiosity but also from their relevance in many applications such as fluid dynamics, elasticity theory, differential equations with nonstandard growth conditions, and image restoration. In addition, the study of the weak solution in other spaces such as Orlicz-Morrey space and space is a research problem (see [14–18]).

The so-called anisotropic evolutionary Laplacian equation,comes closer to the truth than equation (1) [19–21], where . Recently, Zhan et al. [22–24] considered the first initial-boundary value problem to the equation:where satisfiesWe have shown that this condition may act as the role of the Dirichlet boundary conditionto assure the stability of weak solutions to (3).

In this paper, we will consider the anisotropic parabolic equationwith the initial valueand with a partial second boundary value conditionwhere and at least there is a point such that , is a bounded domain with a smooth boundary . We first assume thatwhere and are the interior of and , which are relatively open subset of . We mainly assume thatdenote that , for any , and let for any . As for the anisotropic function spaces and their applications to anisotropic equations, one can refer to [25, 26] and the references therein.

Definition 1. If a function satisfies thatfor any function , , ,and, for any ,then we say is a weak solution of equation (6) with the initial value (7) and with the partial second value condition (8).

The main results are the following theorems.

Theorem 2. Suppose that , satisfies (10) and are two functions satisfyingLet Then we have a positive constant such that there exists a weak solution of (6) with the initial condition (7) and with the partial second boundary value condition (8).

We would like to suggest that, since and at least there is a point such that , then the weak solution generally blows up in a finite time [27]. However, the uniqueness of weak solution still may be true.

Theorem 3. If satisfies (10) and is a function, and when is near to , and are two solutions of (6) with the same partial second boundary value conditionand with the same initial valuesthen

Here is the blow-up time of the weak solutions.

At the end of the introduction, we would like to suggest that it is an interesting research problem to study the anisotropic parabolic equation (6) for either -Laplacian or -biharmonic (see [28–30]).

2. The Existence of Solutions

Consider the following asymptotic problem:with the initial valueand a partial second boundary value conditionwhere , such thatand in . This is possible; only we assume that and every has the logarithmic Hölder continuity [11–13]. Then similar as the usual Laplacian equation, one can show that problem (21)-(24) has a classical solution by the classical theory for parabolic equations, and provided that and at least there is a point such that ([31], Theorem 4.1). We first give a lemma in a similar way as Lemma 2.1 in [32].

Lemma 4. If and , then there exists a such thatwhere represents dependent on .

Proof. Let be the solution of the ordinary differential equationIt is well known that there is a local solution , where ([33], Chapter 5). Let . One hasSince , one has From (29),with the mixed boundary conditionandBy the Hopf maximum principle, one has Hence, for any given , one has

By multiplying (21) with and integrating over , one has

Lemma 5. If and , then there exists a such that

Proof. By multiplying (21) with and integrating over , one hasSinceaccordinglyThus by (37), using Young’s inequality, one has

Proof of Theorem 2. By multiplying (21) with and integrating over , one hasBy (37), (39), (43), and (44), there is a function that satisfiesIn addition, if one notices that by (43) and (44) and by that , one hasThen, there exists , such thatUsing the similar method as that of the evolutionary Laplacian equation [34], we can deduce thatfor any .
At last, the initial value in the sense of (14) can be found in [3]. Consequently, is the solution of (6).

3. The Stability

One can refer to [11–13] for the definitions of the exponent variable spaces, , , and . Also, one can find other details and recent applications to partial differential equations in [35–39].

Lemma 6 (see [11–13]). If and are real functions with and , then, for any and , we haveMoreover,

We let be an odd function, andThen,

Let be a function satisfyingandDefineThen and

Theorem 7. Let and be two solutions of (6) with the same partial second boundary value condition (8) and with the initial values and . If satisfies (9), satisfies (10),and, for large enough ,then

Proof. Let be the characteristic function of . By a process of limit, we can choose the test function as . ThenFirst of all,Secondly, since , by the Lebesgue dominated convergence theorem, we haveBy (59), we have where , .
Then we haveIn addition, by (58)At last, let in (61). ThenBy the arbitrary of , we have the conclusion.

Proof of Theorem 3. We only need to choosein Theorem 7, the conclusion is clear.

4. Conclusion

Comparing with the isotropic case, the anisotropic parabolic equations are closer to the truth about the applications. They reflect in the mathematical modeling of physical and mechanical processes in anisotropic continuous medium. Different from the elliptic anisotropic equations with the variable exponent that has attracted much attentions recently, more or less beyond my imagination, only a few references related to the anisotropic parabolic equations can be found. Even there are not any papers to discuss the fundamental solution of the anisotropic equation. This paper has taken this one step further. We think the contributions are mainly in two aspects. The first one lies in the fact that if the diffusion coefficient is degenerate on a part of the boundary , then, only under the partial second boundary value condition, one can study the well-posedness of solutions to the anisotropic parabolic equations. The second one lies in the fact that even if the weak solutions may be blow-up in finite time provided that the source , the uniqueness of weak solution still may be true.

Data Availability

There is not any data in the paper.

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The paper is supported by NSF of Fujian Province and SF of Xiamen University of Technology, China.