On the Solutions of a Porous Medium Equation with Exponent Variable

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.


Introduction
Let be the density, let be the velocity, and let be the pressure of the ideal barotropic gas through a porous medium. The motion is governed by the mass conservation law the Darcy law and the equation of stage where ( ) is a given matrix. One of the most common cases is ( ) = with , = const. Then we obtain a semilinear parabolic equation on the density If we additionally assume that may explicitly depend on and has the form = ( ) , then equation for becomes and can be written as = div ( ( ) ( ) ( ) ∇ ) + log ( ) ∇ ) . (6) If ( ) = ( ) , where ( ) is a function and is the unit matrix, then (4) becomes = 1 + div ( ( ) ∇ 1+ ) = div ( ( ) ∇ ) , (7) and (6) has the form = div ( ( ) ( ) ( ) ∇ + log ( ) ⋅ ∇ ) . (8) In this paper, we generalized (8) to the following type: ( , ) ∈ = Ω × (0, ) , and consider the initial-boundary value problem, where ( ) > 0 is a 1 (Ω) function, ( ) ∈ 1 (R), Ω ⊂ R is a bounded domain with a smooth boundary Ω.
If ( ) ≡ 1, = 0, ( ) = − 1 is a constant, (9) is equivalent to the so-called porous medium equation 2 Discrete Dynamics in Nature and Society In this case, there exists an abundant literature; one can refer to the survey books [1][2][3][4][5][6] and the references therein. If ( ) ≥ 0, in one way, (9) can be regarded as a special case of reaction-diffusion equation = div ( ( , , ) ∇ ) + div ( → ( )) , (11) there are also many papers devoted to its well-posedness problem. The most striking part of this equation is that if there is an interior point of the set { ∈ Ω : (⋅, , ) = 0} , then the uniqueness of weak solution can be proved only under the entropy condition; one can refer to [7][8][9][10][11][12][13][14][15]. Moreover, if (⋅, , ) is degenerate on the boundary, how to impose a suitable boundary value condition to study the wellposedness of weak solutions to (11) has attracted extensive attentions and has been widely studied for a long time. In the other word, though the initial value is always imposed, the Dirchilet boundary condition ( , ) = 0, ( , ) ∈ = Ω × (0, ) , may not be imposed or be imposed in a weaker sense than the traditional trace. One can refer to [7][8][9][10][11][12] for the details. In another way, the evolutionary equations with variable exponents, especially the so-called electrorheological fluids equations with the form have been brought to the forefront by many scholars since the beginning of this century; one can refer to [16][17][18][19][20][21][22][23] and the references therein. But we noticed that, compared with (15), the papers devoted to the equations with the type seem much fewer. The existence, uniqueness, and localization properties of solutions to (16) have been studied by Antontsev-Ahmarev in [24]. The free boundary problem and the numerical study were researched in [25] by Duque et al. Different from these papers [16-20, 24, 25], we enable the diffusion ( ) in (9) to be degenerate on the boundary. In detail, we suppose that ( ) is a 1 (R function, and Definition 1. If a nonnegative function ( , ) satisfies and for any function ∈ 1 ( ), | = = 0, | Ω = 0, there holds then we say ( , ) is weak solution of (9) with the initial value (13) in the sense If ( , ) satisfies (14) in the sense of the trace in addition, then we say it is a weak solution of the initial-boundary value problem of (9).
Based on the usual Dirichlet boundary value condition, we have the following.
In some cases, we can establish the stability of the weak solutions without any boundary value condition. (17) and (22) ( , ) is a solution of (9) with the initial value (13) but without the boundary value condition, ( , ) satisfies then ( , ) is the unique solution.
At last, we assume that and probe the stability of weak solutions based on a partial boundary value condition.
Discrete Dynamics in Nature and Society 3 Theorem 5. Let , V be two solutions of (9) with the initial values 0 ( ), V 0 ( ), respectively, and with a partial boundary value condition It is supposed that, for every ∈ {1, 2, . . . , }, either ( ) ≥ 0 or ( ) ≤ 0, ( ) satisfies (17) and and V satisfy Then Here, if ( ) ≥ 0, 1 ≤ ≤ , then However, if ( ) ≥ 0, 1 ≥ ≤ , then To show that the partial boundary value condition (26) with the expression (30) or (31) is reasonable, let us review the equation According to Fichera-Oleinik theory [26][27][28][29], the boundary value condition matching up with (32) is with that where → = { } is the inner normal vector of Ω. Since (9) is nonlinear, Fichera-Oleinik theory is invalid; whether the partial boundary Σ 1 in (26) can be expressed similar to (34) has become an interesting problem. Theorem 5 partially answers this question. One can see that if ( ) = ( ) = dist( , Ω) is the distance function from the boundary, = = , the expression (30) or (31) is similar to (34). In fact, instead of (9), if we consider the equation by a similar method as the proof of Theorem 5, we can show that the partial boundary value condition matching up with (35) has the same expression as (34). Thus, the partial boundary value condition (26) with the expression (30) or (31) is reasonable. At the end of the Introduction section, we would like to suggest that if ( ) = is a constant, then condition (24) in Theorem 4 and condition (28) in Theorem 5 are naturally true. Actually, when ( ) = is a constant, ( ) = ( ); (9) has been studied by the author in [29]. But, one can see that, the results (Theorems 4 and 5) are much better and clearer than the results in [29].

The Proof of Theorem 2
Proof of Theorem 2. We suppose that 0 ∈ ∞ 0 (Ω) and 0 ≤ 0 ≤ , and consider the following regularized problem: According to the standard parabolic equation theory, there is a weak solution ∈ ∞ ( ) , and Moreover, by comparison theorem, we clearly have which yields and | ( , )| ≤ + 1.
In what follows, we are able to prove that the limit function is a weak solution of (9) with the initial value (13).

Discrete Dynamics in Nature and Society
Multiplying both sides of the first equation in (36) by = ( )+1 − (1/ ) ( )+1 , and integrating it over , we have ) . (42) Let us analyse every term in (42): Here, we have used (41) and the fact In addition, by the fact using the assumption that (0) = 0, we have and so Then by (41), (42), (43), (44), and (49), we have There is a → ∈ 2 ( ) and weakly in 2 ( ). We now prove that For any ∀ ∈ ∞ 0 (Ω), we have Denoting that Discrete Dynamics in Nature and Society Let → ∞ in (53). We obtain that which implies Since ∈ 1 (R), by (40), we have Letting → ∞ in (42), by (51), (52), and (62), we know ( , ) satisfies (20). At the same time, the initial value (13) can be proved in a similar way as that when ( ) = − 1 is a constant; one can refer to [5] for the details. Thus, is a solution of (9) with the initial value (13). If 0 only satisfies (18), by considering the problem of (36) with the initial value 0 which is the mollified function of 0 , then we can get the conclusion by a process of limitation. Certainly, the solution ( , ) generally is not continuous at = 0, but satisfies (19) and (20). Theorem 2 is proved.

The Stability Based on the Dirichlet Boundary Value Condition
which yields Then Discrete Dynamics in Nature and Society 7 Thus ( )+1 can be defined by the trace on the boundary in the traditional way. By the definition of the trace, we also know that can be defined by the trace on the boundary in the traditional way. The lemma is proved.
For every fixed ∈ [0, ), we define the Banach space (Ω) by and denote by (Ω) its dual space. In addition, we denote the Banach space W( ) by and denote by W ( ) its dual space. According to [18], we know that The norm in W ( ) is defined by is a weak solution of (9) with the initial value (13), then ∈ W ( ).
Proof. For any V ∈ W( ) and ‖V‖ ( ) = 1, there holds By Young's inequality, it follows from (70) that This lemma can be found in [18].
By the definition of weak solutions, we have and For simplism, in what follows, we denote that and, clearly, We have Since ∫ Ω −1 ( ) < ∞, If the set 0 = { ∈ Ω : | ( )+1 − V ( )+1 | = 0} has a positive measure, then, Discrete Dynamics in Nature and Society 9 Therefore, in both cases,

The Stability without the Boundary Value Condition
In this section, we will prove Theorem 4.
Theorem 11. Let , V be two nonnegative solutions of (9) with the initial values 0 ( ), V 0 ( ), respectively. If ( ) satisfies (17) and and V satisfy Proof. For all 0 ≤ ∈ 1 0 ( ), by the definition of weak solutions, for all 0 ≤ ∈ 1 0 ( ), we have Let , ( ) be the characteristic function of [ , ] ⊂ (0, ). By a process of limit, we can choose as the test function; then 10 Discrete Dynamics in Nature and Society Let us analyse every term in (96). Firstly, we have where ∈ ( , V) in the mean value, we have Using the Lebesgue dominated convergence theorem, we have and so Discrete Dynamics in Nature and Society as → 0.
Thirdly, since |∇ | ∈ Ω = 0, Moreover, as in the proof of Theorem 3, we can show that and, clearly, At last, let → ∞ in (96). Then By the arbitraries of , we have
Proof. For all 0 ≤ ∈ 1 0 ( ), by the definition of weak solutions, for all 0 ≤ ∈ 1 0 ( ), we have For a small positive constant > 0, let Let , ( ) be the characteristic function of [ , ] ⊂ (0, ). By a process of limit, we can choose as the test function; then Let us analyse every term in (121): By (114), as → 0.

Conclusion
The evolutionary equations with variable exponents, especially the so-called electrorheological fluids equations with the form (15), have been brought to the forefront by many scholars since the beginning of this century. There are more or less beyond one's imagination; there are only a few references devoted to the porous medium with variable exponents as (16). So, this paper fills the gaps in the related fields. Moreover, the equation considered in this paper is more general than (16). The most important characteristic lies in that there is a degenerate diffusion coefficient ( ) in the equation. This characteristic may make the usual Dirichlet boundary value condition overdetermined and so a partial boundary value condition is expected. The conclusions in this paper answer the problem partially. In addition, since the equation is with variable exponents, there are many technique difficulties to be overcome. This makes our paper contain many cumbersome calculations, but it is necessary.

Data Availability
There is not any data in this paper.

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