Abstract
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.
1. 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 . 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 If , where is a function and is the unit matrix, then (4) becomes and (6) has the form In this paper, we generalized (8) to the following type: and consider the initial-boundary value problem, where is a function, , is a bounded domain with a smooth boundary .
If , , is a constant, (9) is equivalent to the so-called porous medium equation In this case, there exists an abundant literature; one can refer to the survey books [1–6] and the references therein.
If , in one way, (9) can be regarded as a special case of reaction-diffusion equation 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 then the uniqueness of weak solution can be proved only under the entropy condition; one can refer to [7–15]. Moreover, if is degenerate on the boundary, how to impose a suitable boundary value condition to study the well-posedness 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 may not be imposed or be imposed in a weaker sense than the traditional trace. One can refer to [7–12] 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 [16–23] 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 function, and
Definition 1. If a nonnegative function satisfiesand for any function , , , 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).
Theorem 2. If is a function, , , , satisfies (18), then (9) with initial value (13) has a nonnegative solution.
Based on the usual Dirichlet boundary value condition, we have the following.
Theorem 3. If is a function, , ,then the initial-boundary value problems (9), (13) and (14) have a uniqueness solution.
In some cases, we can establish the stability of the weak solutions without any boundary value condition.
Theorem 4. If satisfies (17) and (22) is a solution of (9) with the initial value (13) but without the boundary value condition, satisfiesthen is the unique solution.
At last, we assume thatand probe the stability of weak solutions based on a partial boundary value condition.
Theorem 5. Let be two solutions of (9) with the initial values , respectively, and with a partial boundary value conditionIt is supposed that, for every , either or , satisfies (17) and and satisfyThenHere, if , thenHowever, if , 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 [26–29], the boundary value condition matching up with (32) is with that where is the inner normal vector of . Since (9) is nonlinear, Fichera-Oleinik theory is invalid; whether the partial boundary in (26) can be expressed similar to (34) has become an interesting problem. Theorem 5 partially answers this question. One can see that if 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].
2. The Proof of Theorem 2
Proof of Theorem 2. We suppose that and , and consider the following regularized problem:According to the standard parabolic equation theory, there is a weak solutionandMoreover, by comparison theorem, we clearly havewhich yieldsandIn what follows, we are able to prove that the limit function is a weak solution of (9) with the initial value (13).
Multiplying both sides of the first equation in (36) by , and integrating it over , we haveLet us analyse every term in (42):Here, we have used (41) and the fact In addition, by the factusing the assumption that , we haveaccordinglyand soThen by (41), (42), (43), (44), and (49), we haveThere is a andweakly in . We now prove thatFor any , we haveDenoting that , thenLet in (53). We obtain thatwhich impliesSince , by (40), we haveLetting 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 is a constant; one can refer to [5] for the details. Thus, is a solution of (9) with the initial value (13). If only satisfies (18), by considering the problem of (36) with the initial value which is the mollified function of , then we can get the conclusion by a process of limitation. Certainly, the solution generally is not continuous at , but satisfies (19) and (20). Theorem 2 is proved.
3. The Stability Based on the Dirichlet Boundary Value Condition
Lemma 6. If , then .
Proof. Since ,which yieldsThen Thus 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 , we define the Banach space byand denote by its dual space. In addition, we denote the Banach space byand denote by its dual space. According to [18], we know that
The norm in is defined by
Lemma 7. If is a weak solution of (9) with the initial value (13), then .
Proof. For any and , there holds By Young’s inequality, it follows from (70) that we have
Lemma 8. Suppose that and . For any continuous function , let . For a.e. , there holdsThis lemma can be found in [18].
Theorem 9. Let and let and be two solutions of (9) with the initial values , respectively, and with the same homogeneous boundary value conditionsThen
Proof. For any given positive integer , let , . Then , and we haveandBy the definition of weak solutions, we haveSince , on the boundary, we can choose as the test function. ThenandFor simplism, in what follows, we denote thatand, clearly,We haveSince ,If is with measure, we haveIf the set has a positive measure, then,Therefore, in both cases,At last,Let in (79). By (80), (81), (88), and (89), we have
Corollary 10. Theorem 3 is true.
4. The Stability without the Boundary Value Condition
In this section, we will prove Theorem 4.
Theorem 11. Let be two nonnegative solutions of (9) with the initial values , respectively. If satisfies (17) and and satisfythen
Proof. For all , by the definition of weak solutions, for all , we haveLet be the characteristic function of . By a process of limit, we can chooseas the test function; thenLet us analyse every term in (96). Firstly, we haveSecondly, since ,as .
By thatwhere in the mean value, we haveUsing the Lebesgue dominated convergence theorem, we haveand soas .
Thirdly, since ,Moreover, as in the proof of Theorem 3, we can show thatand, clearly,At last, let in (96). ThenBy the arbitraries of , we have
5. The Stability Based on the Partial Boundary Value Condition
In this section, we will prove Theorem 5. We assume thatwe denote that as before and denote that
Theorem 12. Let be two solutions of (9) with the initial values , respectively, and with a partial boundary value conditionHere, for any given , if ,if ,It is supposed that satisfies (17) and and satisfyThen
Proof. For all , by the definition of weak solutions, for all , we haveFor a small positive constant , letandLet be the characteristic function of . By a process of limit, we can choose as the test function; thenLet us analyse every term in (121):By (114), ,as .as .
This is due to the fact that if is with measure,and by (115)we have where in the mean value. If has a positive measure,and by (115)Moreover, recall thatIf , by the partial boundary value condition (111)If , by the partial boundary value condition (111)As in the proof of Theorem 11 we can show thatClearly,Now, after letting , let in (121). ThenBy the arbitraries of , we have
Proof of Theorem 5. Since we suppose that, for every , either or , by checking the process of the proof of (131) or (132), we can easily obtain the conclusion of Theorem 5.
6. 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.
Acknowledgments
The paper is supported by the Natural Science Foundation of Fujian Province (2019J01858) and the Open Research Fund Program from Fujian Engineering and Research Center of Rural Sewage Treatment and Water Safety, China.