On a Partial Boundary Value Condition of a Porous Medium Equation with Exponent Variable

(e initial-boundary value problem of a porous medium equation with a variable exponent is considered. Both the diffusion coefficient a(x, t) and the variable exponent p(x, t) depend on the time variable t, and this makes the partial boundary value condition not be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition matching up with the equation is based on a submanifold of zΩ × (0, T). By this innovation, the stability of weak solutions is proved.


Introduction
e porous medium equation with a constant exponent is widely used to model several real-life problems, and it has been extensively studied, one can refer to the survey books [1][2][3][4][5][6] and the references therein.
e dynamics system of a partially nonlocal and inhomogeneous nonlinear medium has been considered in [7][8][9]. e case where the exponent of nonlinearity is not constant was proposed by Antontsev and Shmarev in [10], where the existence, uniqueness, and some properties of the solution in a bounded fixed domain were researched. By using the Galerkin finite element method, Duque et al. [11] proved the convergence of a fully discrete solution for this problem in a fixed domain. Based on one of the properties proved in [11] that the solution is with the finite speed of propagation, Duque et al. [12] considered the free boundary problem by using the moving mesh method to the porous medium. However, the moving mesh method was first introduced by Huang and Russell in [13].
In this paper, we consider the initial-boundary value problem of a generalized porous medium equation with a variable exponent: where m(x, t) > 0 is a C 1 (Q T ) function, a(x, t) ≥ 0 is a C 1 (Q T ) function, b i (s) ∈ C 1 (R), and Ω ⊂ R N is a bounded domain with a smooth boundary zΩ.
Equation (1) is a special case of the reaction-diffusion equation: Because a(·, x, t) may degenerate on the boundary, how to impose a suitable boundary value condition to study the well posedness of weak solutions to equation (2) has attracted extensive attention and has been widely studied for a long time. In some details, though the initial value is always imposed, the Dirichlet boundary condition u(x, t) � 0, (x, t) ∈ zΩ ×(0, T), may not be imposed or be imposed in a weaker sense than the traditional trace. One can refer to the references [14][15][16][17][18][19] for the details. Naturally, besides the porous medium equation with variable exponents, the so-called electrorheological fluid equations with the form have been brought to the forefront by many more scholars. Since the beginning of this century, there are a great deal of papers devoting to the well-posedness problem, the intrinsic Harnack inequalities, the long-time behavior, and the Hölder regularity of weak solutions, one can refer to the literatures [20][21][22][23][24][25][26][27][28][29][30][31] and the references therein.
e existence and the stability of weak solutions to equation has been studied in [32]. It is found that the degeneracy of a(x) on boundary (6) may replace the usual boundary value condition (4). In other words, if a(x) satisfies (6), the stability of weak solutions may be proved without the usual boundary value condition (4). In this paper, we will use some ideas of [32] to study the well posedness of weak solutions to equation (1). Because both a(x, t) and p(x, t) are dependent on the time variable t, the problem becomes more difficult and the question of existence of such solutions is still open for equation (8), as well as for evolution p-Laplace equation with the exponent p depending on t [22]. Instead of condition (6), we only assume that a(x, t) ≥ 0. By this assumption, we find out a partial boundary value condition matching up with the equation. Moreover, because both a(x, t) and p(x, t) are dependent on the time variable t, the partial boundary value condition cannot be expressed as the usual Dirichlet boundary value condition. In other words, the partial boundary value condition is based on a submanifold of zΩ × (0, T). By this innovation, the stability of weak solutions is proved.

The Partial Boundary Value Condition and the Main Results
For any given t ∈ [0, T) and small enough λ > 0, we set e most important and essential improvement is that instead of the usual boundary value condition (4), the stability of weak solutions is proved based on a partial boundary value condition: where and for any given and if b i ′ (s) ≤ 0, us, if for every i ∈ 1, 2, . . . , N { }, either b i ′ (s) ≥ 0 or b i ′ (s) ≤ 0, then one can deduce an expression Σ 1 from the above discussion. For example, b i (s) ≥ 0 when 1 ≤ i ≤ k and b i ′ (s) ≤ 0 when k + 1 ≤ i ≤ N; then, e most characteristic out of the ordinary is that Σ 1 or Σ 2 is just a submanifold of zΩ × (0, T), and it cannot be expressed as a cylinder with the form Γ × (0, T) and Γ ⊂ zΩ.
and for ∀φ ∈ C 1 0 (Q T ), and then u(x, t) is said to be a weak solution of equation (1) with the initial value (3) in the sense 2 Discrete Dynamics in Nature and Society Moreover, if u(x, t) satisfies (4) or (3) in the sense of the trace in addition, then it is said to be a weak solution of the initial-boundary value problem of equation (1).

Theorem 3. Let u(x, t) and v(x, t) be two solutions of equation
It is supposed that, for every i ∈ 1, 2, . . . , and en, Hereinafter, the constant c(T) represents a constant which depends on T.
At the end of this section, we would like to suggest that if m(x, t) � m is a constant, then condition (22) in eorem 3 is naturally true.

The Proof of Theorem 1
First, we suppose that u 0 ∈ C ∞ 0 (Ω) and 0 ≤ u 0 ≤ M and consider the following regularized problem: According to the standard parabolic equation theory, there is a weak solution u n ∈ L ∞ (Q T ) satisfying Moreover, by comparison theorem, we clearly have which yields In what follows, we are able to prove that the limit function u is a weak solution of (1) with the initial value (3).
Multiplying both sides of the first equation in (25) ) and integrating it over Q T , we have Discrete Dynamics in Nature and Society For the left-hand side of (29), For the first term of the right-hand side of (29), because by a complicated calculation and using the Young inequality, we can deduce that For the second term of the right-hand side of (29), because 4 Discrete Dynamics in Nature and Society we can deduce that From (28) weakly in L 2 (Q T ). We now can prove as in a similar way as that in [32]. e last but not the least, by that b i ∈ C 1 (R), using (27), we have Letting n ⟶ ∞ in (30), by (37), (38), and (39), we know u(x, t) satisfies (18).
Secondly, if u 0 satisfies only (16), we should consider equation (9) with the initial value u 0ε which is the mollified function of u 0 , from the above that there is a weak solution u ε satisfying (18). Letting ε ⟶ 0, the limit function u(x, t) is a solution of (1) satisfying (17) and (18), but generally is not continuous at t � 0 as in the case u 0 ∈ C ∞ 0 (Ω). irdly, the initial value (4) can be proved in a similar way as that when m(x, t) � m − 1 is a constant, one can refer to [5] for the details. us, u is a solution of equation (1) with the initial value (4). us, eorem 1 is proved.

The Proof of Theorem 2
One can see that, in Definition 1, there is not any definition on the general derivative u t . At the beginning of this section, we first answer this question.
For any t ∈ [0, T), the Banach space V t (Ω) is defined by and V t ′ (Ω) is denoted as its dual space. e Banach space W(Q T ) is defined by and W ′ (Q T ) is denoted as its dual space. From [21], we have It is easy to prove the following lemmas, so we omit the details here. (1) with the initial value (3), then u t ∈ W ′ (Q T ).

Lemma 2. Suppose that u ∈ W(Q T ) and u t ∈ W ′ (Q T ). For any continuous function h(s), let H(s) �
s 0 h(s)ds. For a.e. t 1 , t 2 ∈ (0, T), there holds So, the weak solution of equation (1) u(x, t) can be defined as the homogeneous boundary value condition u| zΩ � 0 in the sense of the trace. (1) with the initial values u 0 (x) and v 0 (x) respectively, and with

Theorem 4. Let u(x, t) and v(x, t) be two solutions of equation
Proof. For any given positive integer n, let g n (s) � Because u(x, t) � v(x, t) � 0 on the boundary zΩ × [0, T), we choose g n (u m(x,t)+1 − v m(x,t)+1 ) as the test function and integrate over Q t � Ω × (0, t). en, Let us analyse every term in (47). In the first place, In the second place, we deal with the fourth term on the left-hand side of (47). For any given t ∈ [0, T), we set Based on these denotations, we have If D 0s � x ∈ Ω: |u m(x,s)+1 − v m(x,s)+1 | � 0 has 0 measure, then us, in both cases, we always have In the third place, for the first term on the left-hand side of (47), by Lemma 2, Let n ⟶ ∞ in (47). Formulas (48)-(57) yield Discrete Dynamics in Nature and Society □ Corollary 1. eorem 2 is true.

The Stability Based on the Partial Boundary Value Condition
Theorem 5 . Let u(x, t) and v(x, t) be two solutions of equation (1) with the initial values u 0 (x) and v 0 (x) respectively, and with a partial boundary value condition where and for any given and if b i ′ (s) ≤ 0, It is supposed that and u(x, t) and v(x, t) satisfy (64) Proof. According to the definition of weak solutions, for all For any t ∈ [0, T) and a small positive constant λ > 0, based on we define that Let χ τ,s (t) be the characteristic function of [τ, s] ⊂ (0, T). Because we can choose as the test function. en, 8 Discrete Dynamics in Nature and Society Firstly, we still have Secondly, for the third term on the left-hand side of (71) by that 1/λ( Ω\Ω λt a(x, t) as λ ⟶ 0. irdly, for the fourth term on the left-hand side of (71), we can show as λ ⟶ 0. e corresponding details are given below.

Conclusion
In this paper, we consider the initial-boundary value problem of a generalized porous medium equation with a variable exponent. Different from the previous related works, both the diffusion coefficient a(x, t) and variable exponent p(x, t) are dependent on the time variable t, we find out a partial boundary value condition matching up with the equation. e most important innovation is that the partial boundary value condition matching up with the equation is based on a submanifold of zΩ × (0, T). However, because there is an additional condition (23) imposed, eorem 3 has not answered the problem globally. In other words, how to obtain the same conclusion as that in eorem 3 without condition (23) is remained to be studied in the future.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that he has no conflicts of interest.