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 bðx, tÞ on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω × ð0, TÞ. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.


Introduction
In this paper, we are concerned with the initial-boundary value problem of a parabolic equation with nonstandard growth condition where bðx, tÞ ≥ 0 is a CðQTÞ function, rðxÞ > 0 is a C 1 ð ΩÞ function, a ! ðs, x, tÞ = fa i ðs, x, tÞg, a i ðs, x, tÞ ∈ Cðℝ × Q T Þ, and Ω ⊂ ℝ N is a bounded domain with a smooth boundary ∂Ω. 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 b ðx, tÞ ≡ 1, a i = 0, rðxÞ = r is a constant; equation (1) is with the type which includes the so-called porous medium equation (PME), the fast diffusion equation (FDE) when r > 1, and the slow diffusion equation (SDE) when r < 1: Correspondingly, if r − = min x∈ Ω rðxÞ > 1, then equation (1) is a degenerate parabolic equation and has a slow diffusion characteristic. If r + = max x∈ Ω rðxÞ < 1, 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][2][3][4][5][6][7][8][9][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 bðx, tÞ ≥ 0, equation (1) is a special case of the usual reaction-diffusion equation If bðu, x, tÞ is degenerate in the interior of Ω, 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][12][13][14][15][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 [17]. The existence, uniqueness, and localization properties of solutions to equation had been studied. Another property proved in [17] is the finite speed of propagation, which permits one to consider the free boundary problem. The work [18] 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 [19] and the moving mesh for partial differential equations by Huang and Russell [20] were used there.
Recently, when bðx, tÞ = bðxÞ with bðxÞ > 0, x ∈ Ω, and the well-posedness of weak solutions to equation (1) has been studied by the author [21]. Roughly speaking, if bðxÞ satisfies (6) and the uniqueness of weak solution to equation (1) with the initial value has been proved in [21]. This result can be comprehended as that, the degeneracy of diffusion coefficient bðxÞ on the boundary (6) acts as a role as the usual boundary value condition In this paper, different from [21], the diffusion coefficient bðx, tÞ is dependent on the time variable t; moreover, we merely assume that bðx, tÞ ≥ 0 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 ∂Ω × ð0, TÞ, is a cylinder, Σ ⊂ ∂Ω × ð0, TÞ appearing in (10) is just a submanifold and generally can not be expressed as a cylinder type as Γ × ð0, TÞ with Γ ⊆ ∂Ω. 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 u is with the Hölder continuity [5]. However, for a porous medium equation with nonstandard growth conditions, weak solutions in [17,18,[21][22][23][24][25][26][27][28][29][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).
If for any function φ ∈ C 1 0 ðQ T Þ, there is then uðx, tÞ is said to be a weak solution of the initialboundary 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 uðx, tÞ be a nonnegative function satisfying (14) and Journal of Function Spaces then uðx, tÞ is said to be a strong solution of the initialboundary 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 bðx, tÞ satisfies (4), (16) means that u rðxÞ t and ∇u rðxÞ exist almost everywhere in QT. This is the reason that we call uðx, tÞ is a strong solution of equation (1).
From Definition 2, for all φðx, tÞ ∈ L 2 ð0, T ; W 1,2 0 ðΩÞ, we have which implies that Thus, if uðx, tÞ is a strong solution of equation (1), then it is a weak solution.
0 ≤ bðx, tÞ ∈ CðQ T Þ satisfies (4), f ðs, x, tÞ is a continuous function on ℝ × Q T and when s < 0 then the initial boundary value problem of equation (1) has a nonnegative weak solution.
for any jsj ≤ M + 1, M = ku 0 ðxÞk L ∞ ðΩÞ , f ðs, x, tÞ 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 c may be different from one to another, cðMÞ represents the constant c depends on M, and cðTÞ represents the constant c depends on T.
Theorem 5. Let uðx, tÞ be a weak solution of equation (1) with the initial value (8) and with the partial homogenous value condition (10), If 0 <r − ≤ rðxÞ ∈ C 1 ðΩÞ, f ðs, x, tÞ is a Lipschitz function on ℝ × Q T , for every given t ∈ ½0, TÞ, bðx, tÞ is a differential function on the spatial variables then the solution uðx, tÞ is unique.
One can see that there are functions satisfying condition (24). For example, if Γ ⊂ ∂Ω is a relatively open set of ∂Ω, and let d Γ ðxÞ = distðx, ΓÞ. Then, for any f ðtÞ ∈ C 1 ½0, T, is a function satisfying (24).
We note that, if rðxÞ ≡ r is a constant, then condition (22) is naturally true. By the way, just as one reviewer has suggested, one can study Theorem 3 and Theorem 4 when rðxÞ ≥ r − > 0 and f ðs, x, tÞ is just a Carathodory function.

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 [5], since f ðs, x, tÞ > 0 when s < 0, there is a nonnegative weak solution u n ðx, tÞ of the following initial-boundary value problem 3 Journal of Function Spaces which satisfies Multiplying (26) by ϕ = u rðxÞ n , then we obtain In the first place, by a similar calculation as that in [21], we can extrapolate that weakly in L 2 ðQ T Þ.
The last but not the least, by a process of limit, in a similar way as that in [17], we can choose the test function φðx, tÞ = χ ½t 1 ,t 2 ϕðxÞ in which ϕðxÞ ∈ C ∞ 0 ðΩÞ and χ ½t 1 ,t 2 is the characteristic function of ½t 1 , t 2 ⊂ ð0, TÞ. Then Let t = t 2 and t 1 ⟶ 0. Then we have (15). Finally, since bðx, tÞ > 0 when x ∈ Σ in (10), by (32), we know that the trace of uðx, tÞ on Σ is well defined, the details are given in Section 4 as follows. Till now, we have shown that u is a weak solution of equation (1) with the initial value (6) and the partial boundary value condition (10).

Journal of Function Spaces
Proof of Theorem 4. We consider the regularized problem (26) and obtain (27)- (32) as in the proof Theorem 3. Let us multiply u rðxÞ nt on both sides of equation (26), integrate it over Q t = Ω × ð0, tÞ. Then Since u n ≥ 0, we have Since b t ðx, tÞ ≤ 0, by the assumption rðxÞ ≥ r − > 1, we have By the assumption jð∂/∂u n Þa i ðu rðxÞ n , x, tÞj 2 bðx, tÞ −1 ≤ c and when ju n j ≤ M + 1, we have Now, combining with (44)-(51), we have Then Since bðx, tÞ > 0 when x ∈ Ω, by Sobolev embedding theorem, (32) and (53) implies that u rðxÞ n ⟶ u 1 a.e. in Q T . Since u n ⇀ u weakly star in L ∞ ðQ T Þ, by the uniqueness property of the weak convergence, we know that u 1 = u rðxÞ .

The Stability Based on the Partial Boundary Value Condition
In this section, we will prove Theorem 5.

Conclusion
Since the beginning of this century, the evolutionary pðxÞ -Laplacian equations u t = div ðbðu, x, tÞj∇uj pðxÞ−2 ∇uÞ + gð∇ u, u, x, tÞ have been studied by many mathematicians in [32][33][34][35][36][37]. So far, however, there has been little discussion about the porous medium equation with nonstandard growth conditions. Therefore, this study makes a major contribution to research in the related fields by imposing a reasonable partial boundary value condition matching up with the equation. The unusual is that this partial boundary value condition is based on a submanifold Σ ⊂ ∂Ω × ð0, TÞ. In addition, since the equation is with nonstandard growth conditions, how to deal with the nonlinearity becomes difficult, especially when we try to prove that ffiffiffiffiffiffiffiffiffiffiffiffi u rðxÞ−1 p u t ∈ L 2 ðQ T Þ.

Data Availability
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Conflicts of Interest
The author declares that he has no competing interests.