Research Article | Open Access
The Partial Second Boundary Value Problem of an Anisotropic Parabolic Equation
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 .
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 . However, the uniqueness of weak solution still may be true.
Here is the blow-up time of the weak solutions.
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 (, Theorem 4.1). We first give a lemma in a similar way as Lemma 2.1 in .
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 (, 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 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 , we can deduce thatfor any .
At last, the initial value in the sense of (14) can be found in . Consequently, is the solution of (6).
3. The Stability
We let be an odd function, andThen,
Let be a function satisfyingandDefineThen and
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