Research Article | Open Access
Huashui Zhan, "On Solutions of a Parabolic Equation with Nonstandard Growth Condition", Journal of Function Spaces, vol. 2020, Article ID 9397620, 10 pages, 2020. https://doi.org/10.1155/2020/9397620
On Solutions of a Parabolic Equation with Nonstandard Growth Condition
A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient on the time variable , and the partial boundary value condition based on a submanifold of . How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.
In this paper, we are concerned with the initial-boundary value problem of a parabolic equation with nonstandard growth condition where is a function, is a function, , , and is a bounded domain with a smooth boundary . These kinds of equations can be regarded as nonlinear variation of the classical heat equation, the well-known paradigm to explain the diffusion process. In particular, when , is a constant; equation (1) is with the type which includes the so-called porous medium equation (PME), the fast diffusion equation (FDE) when , and the slow diffusion equation (SDE) when Correspondingly, if , then equation (1) is a degenerate parabolic equation and has a slow diffusion characteristic. If , then equation (1) is a singular parabolic equation and has a fast diffusion characteristic.
In fact, equation (2) itself has a wide number of applications, ranging from plasma physics to filtration in porous media, thin films, Riemannian geometry, relativistic physics, and many others. It has at the same time tested as the testing ground from the development of new methods of analytical investigation, since it offers a variety of surprising phenomena that strongly deviate from the heat equation standard, for example, free boundary, limited regularity, mass loss, and extinction or quenching. There exists an abundant literature on these topics, one can refer to [1–10] and the references therein. In this literature, the nonlinearities are with power types that lead to degenerate or singular parabolicity, and we can gather these collectively under the name “the porous medium equation with standard growth conditions.” Since, in equation (1), the nonlinearity is with variable exponent, so it is called as “with nonstandard growth condition.”
From another perspective, since , equation (1) is a special case of the usual reaction-diffusion equation
If is degenerate in the interior of , this equation is a hyperbolic-parabolic mixed type equation theoretically. To ensure the uniqueness of a weak solution to this equation, the entropy condition and the corresponding entropy solution are required, one can refer to references [11–16] for the details. In this paper, we assume that
As a consequence, equation (1) is a pure nonlinear parabolic equation and has not the hyperbolic characteristic.
The porous medium equation with nonstandard growth conditions was first proposed by Antontsev and Shmarev in . The existence, uniqueness, and localization properties of solutions to equation had been studied. Another property proved in  is the finite speed of propagation, which permits one to consider the free boundary problem. The work  by Duque etc. implemented a finite element method with adaptive mesh to approximate the solution of the porous medium equation with nonstandard growth conditions in 2D domains with free boundary. The equidistribution principle deduced by de Boor  and the moving mesh for partial differential equations by Huang and Russell  were used there. Recently, when with , and the well-posedness of weak solutions to equation (1) has been studied by the author . Roughly speaking, if satisfies (6) and the uniqueness of weak solution to equation (1) with the initial value has been proved in . This result can be comprehended as that, the degeneracy of diffusion coefficient on the boundary (6) acts as a role as the usual boundary value condition
In this paper, different from , the diffusion coefficient is dependent on the time variable ; moreover, we merely assume that satisfies (4) and do not restrict the similar condition as (7). As a compensation, a partial boundary value condition is imposed as follows. One can see that, unlike the usual Dirichlet boundary value condition (9), in which , is a cylinder, appearing in (10) is just a submanifold and generally can not be expressed as a cylinder type as with . The first aim of this paper is to study the stability of weak solutions based on this partial boundary value condition. Another aim of this paper is to study the existence of a strong solution to equation (1). The details are provided below. It is well-known that, for the usual porous medium equation there holds and is with the Hölder continuity . However, for a porous medium equation with nonstandard growth conditions, weak solutions in [17, 18, 21–30] do not satisfy the regularity as (12). Thus, in this paper, we will try to make up for some gaps in such a field, considering that the weak solutions of equation (1) have the property (12).
Definition 1. Let be a nonnegative function satisfying If for any function , there is then is said to be a weak solution of the initial-boundary value problem of equation (1), provided that initial value (6) is true in the sense and the partial boundary value condition (10) is true in the sense of the trace.
Definition 2. Let be a nonnegative function satisfying (14) and
then is said to be a strong solution of the initial-boundary value problem of equation (1), provided that initial value (6) is true in the sense of (15), and the partial boundary value condition (10) is true in the sense of the trace.
Since satisfies (4), (16) means that and exist almost everywhere in . This is the reason that we call is a strong solution of equation (1).
From Definition 2, for all , we have which implies that
Thus, if is a strong solution of equation (1), then it is a weak solution.
Theorem 4. If is a function, , , , satisfies (19), satisfies (4) and
for any , , is a continuous function satisfies (20), then the initial boundary value problem of equation (1) has a nonnegative strong solution.
Throughout this paper, the constant may be different from one to another, represents the constant depends on , and represents the constant depends on .
Theorem 5. Let be a weak solution of equation (1) with the initial value (8) and with the partial homogenous value condition (10),
If 0 , is a Lipschitz function on , for every given , is a differential function on the spatial variables , , and satisfies
then the solution is unique.
One can see that there are functions satisfying condition (24). For example, if is a relatively open set of , and let . Then, for any , is a function satisfying (24).
2. The Proof of Theorems 3-4
In this section, we give the proof of the existence theorems.
Proof of Theorem 3. By the monotone convergent method , since when , there is a nonnegative weak solution of the following initial-boundary value problem
Multiplying (26) by , then we obtain
In the first place, by a similar calculation as that in , we can extrapolate that
and show that
weakly in .
In the second place, for all , we have from this equality, by (27) and (31), it is easy to get Moreover, from (27), there is a nonnegative function, weakly star in . For any , by (34), we can show that Since when , then . Accordingly, In addition, we have where . Thus, Since , Aubin’s compactness theorem in  yields that there is a function such that strongly in . Then, almost everywhere in , by that weakly star in , we know that . The arbitraries of yields almost everywhere in .
Thus, since and are continuous functions on ), we have Let in (29). By (32), (39), and (40), we know satisfies (13) and (14).
The last but not the least, by a process of limit, in a similar way as that in , we can choose the test function in which and is the characteristic function of . Then Let and . Then we have (15).
Finally, since when in (10), by (32), we know that the trace of on is well defined, the details are given in Section 4 as follows. Till now, we have shown that is a weak solution of equation (1) with the initial value (6) and the partial boundary value condition (10).
Proof of Theorem 4. We consider the regularized problem (26) and obtain (27)–(32) as in the proof Theorem 3.
Let us multiply on both sides of equation (26), integrate it over . Then Since , we have Since , by the assumption , we have By the assumption and when , we have Moreover, we have Now, combining with (44)–(51), we have Then Since when , by Sobolev embedding theorem, (32) and (53) implies that a.e. in . Since weakly star in , by the uniqueness property of the weak convergence, we know that .
Accordingly, , we have as well as by that is a contunuous function.
Moreover, for any , Thus, Let in (29). By (32), (35), (39), (56), and (57), we know satisfies (14) and (16). The initial value (6) and the partial boundary value condition have the same meaning as those in Theorem 3.
3. The Stability Based on the Partial Boundary Value Condition
In this section, we will prove Theorem 5.
Lemma 6. Let for any , . For any continuous function , , ,
This Lemma can be seen as a generalization of Lemma 2.2 in . We postpone the reader to it for details.
Theorem 7. Let , be a Lipschitz function on , with that if and for every given , be a differential function on the spatial variables , and satisfy If are two nonnegative weak solutions of equation (1) with the initial value , respectively, with the same homogeneous partial boundary value condition and satisfying then
Proof. Let , . Here and the after, is a positive integer. Supposed that is the characteristic function of , since and are with the same homogeneous partial boundary value condition (61), we then choose as the test function. Thus In the first place