Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents
We consider the Dirichlet initial boundary value problem , where the exponents , , and are given functions. We assume that is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if , then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when , we get the positivity of solutions for large . In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that and either or a.e. Finally, we give a uniform localization of the support of solutions for all , in the case where a.e. and .
This paper is devoted to studying qualitative properties of nonnegative weak solutions for the following doubly nonlinear parabolic problem with variable exponents
where is a bounded domain of , with smooth boundary , and is defined asThe exponents , , and the coefficient are given measurable functions. It will be assumed throughout the paper that these functions satisfy some specific conditions.
Problems of this form appear in various applications; for instance in models for gas or fluid flow in porous media ([1, 2]) and for the spread of certain biological populations (). Our motivation to study problem with variable exponents is the fact that it is considered as a model of an important class of non-Newtonian fluids which are well known as electrorheological fluids, see (). It appears also as a model in image restoration () and in elasticity ().
It is well known that solutions of problems such as exhibit various qualitative properties, which reflect natural phenomena, according to certain conditions on , , , , and , (see for example [7–13] and the references therein). Among the phenomena that interest us in this work is the finite speed of propagation, which means that if is such that , then , for any , where is a positive function which depends on , (i.e., solutions with compact support). This property has various physical meanings; for instance, in the study of turbulent filtration of gas through porous media, a solution with compact support means that gas will remain confined to a bounded region of space, (see ).
The phenomenon of finite speed of propagation was investigated by Kalashnikov in . He considered, for , the equation in and, under specific conditions, proved that if the initial condition has a compact support, then the condition is necessary and sufficient for solutions to have compact support. This result was extended by Dìaz for , in . Later, in  Dìaz and Hernández considered the doubly nonlinear problem with absorption term , in , where . Under the assumption that has a compact support and , they proved that any solution has a compact support for all . This result was obtained by the construction of a local uniform super-solution. Let us recall that the finite speed of propagation phenomenon has been studied by many authors in the last decades, (see [18–21]).
Besides, extinction and nonextinction are also important properties for solutions of evolution equations that have attracted many authors in the last few decades. Most of them focused on equations with constant exponents of nonlinearity, (see [22–26]). For example, Hong et al., dealt in  with the homogeneous equation , in , where and . They proved that the condition is necessary and sufficient for extinction to occur. Moreover, Zhou and Mu () studied the extinction behavior of weak solutions for the equation with source term , in , where , and . They proved that is a critical extinction exponent.
Otherwise, it is worth noting that problem has been treated by Antontsev and Shamarev in several papers. In [29, 30], they proved the existence of weak and strong solutions. Moreover, under certain regularity hypotheses on , , and under the sign condition a.e, they studied properties of finite speed of propagation and extinction in finite time in [9, 10]. Their results were established by using the local energy method. Here, we shall use the so-called method of sub- and supersolutions to extend some of the results in [9, 10]. To the best of our knowledge, there are few results concerning the study of qualitative properties for parabolic equations with variable exponents by using this method. Furthermore, we shall also extend to the parabolic case some of the results by Zhang et al. in , where radial sub- and supersolutions for some elliptic problems with variable exponents are constructed, and some of the results by Chung and Park in  and by Yuan et al. in , to variable exponents case. In fact, we shall exploit their arguments in our parabolic problem setting with less conditions on the exponents , , and and the coefficient .
The present paper is organized as follows. In Section 2, we introduce some basic facts about the variable exponents spaces. In Section 3, we give assumptions and general definitions; then, we establish a comparison principle which ensures the uniqueness of solutions. In Section 4, we investigate the extinction and nonextinction properties for the solution of . Finally in Section 5, we study the property of finite speed of propagation.
In this section we give some elementary results for the generalized Lebesgue spaces and Sobolev spaces , where is a bounded set of with smooth boundary. For more details, see ([11, 32, 33]). whereFor any , we introduce the variable exponent Lebesgue space as follows: endowed with the Luxemburg norm
Proposition 1 (see [11, 32, 33]). (i)The space is a separable and reflexive Banach space, and its conjugate space is , where . Moreover, for any and , we have (ii)Let be given such that for any then is continuously embedded into .
Now, we define the variable Sobolev space as follows: endowed with the norm We say that satisfies the log-Hölder condition in ifwhere satisfies
Proposition 3 (see [11, 32, 33]). (i) is a separable and reflexive Banach space.(ii)If satisfies the log-Hölder condition (11), then the space is dense in . Moreover, we can define the Sobolev space with zero boundary values, as the completion of , with respect to the .
Next, let and be given functions. For fixed, we denote . Let , we assume that satisfies the following log-Hölder condition in , we havewhere satisfies For every fixed , we introduce the following Banach. endowed with the norm We denote by the following Banach space, endowed with the norm We denote by the dual of .
3. Assumptions and Results
Throughout this paper we assume that the coefficients and the exponents of nonlinearity satisfy the following conditions,and the initial data satisfiesNow, let us state the definition of weak solutions for the problem .
Definition 4. We say that is a super-(sub)solution of on if (1) and .(2)for every nonnegative test function and , we have (3), . A function is a weak solution of if it is simultaneously a supersolution and a subsolution.
The following result concerning the local existence of weak solutions of problem is established in .
Theorem 5. Let , satisfies the log-Hölder condition in (14), and let conditions (20) and (21) be fulfilled. Moreover, we assume that and the exponents , satisfy one of the following conditions (1) is independent of , and in ,(2), , and ,(3), , , and Then, the problem has at least one nonnegative weak solution in , with Moreover, for small the solution satisfies the estimatewith a constant depending only on the data.
The following comparison principle is essential to prove uniqueness and qualitative properties of nonnegative solutions.
Proposition 6. Let (respectively ) be a subsolution (respectively supersolution) of , with the initial datum (respectively ), satisfying (21). We assume that , , and that conditions (20) are fulfilled. If either a.e. in , or , then we have a.e. in .
Remark 7. Note that the comparison principle is true for weak solutions with and recall that in the papers [29, 30], the authors gave some conditions on the data of problem in order to ensure that this class of solutions is nonempty.
Proof. We consider the test function , where and is small. It is easy to see that where , if , and , if . Moreover, we claim that for all the function . Indeed, we observe that for all , . Then, by Proposition 2On the other hand, we have Hence, from Proposition 2 we getTherefore, combining (29) and (31) we deduce the claim. On the other hand, from Definition 4, we obtain Due to a monotonicity argument, we have thenBy Lebesgue’s dominated convergence theorem, we haveNow, we can writeThen, from (34) and (35), by letting , we obtainHence, if a.e. in , it follows that Then, by Gronwall’s lemma we deduce the desired result. Now, we continue the proof without any sign condition on . From (37), by using and the Lebesgue’s dominated convergence theorem it follows that where is depending on the supnorms of and . Hence we deduce from Gronwall’s lemma thatwhich allows us to conclude the result.
Definition 8. We call a strong solution of , if is a weak solution and satisfies
4. Finite Time Extinction and Nonextinction
This section is devoted to studying extinction and positivity properties for nonnegative solutions of problem , without any sign condition on the coefficient , and according to the ranges of , , and . The proof of the results is based on the construction of suitable sub- and supersolutions and on the use of the preceding comparison principle given in Proposition 6.
4.1. Finite Time Extinction
We state and prove our main extinction result.
Theorem 9. Let be a strong solution of . Assume that , , and is small enough. Then, there exists a finite time such that for all
Proof. We consider the following function whereandwhere will be specified later. Our goal is to prove that is a supersolution of and by comparison principle, we can thus deduce the result. Firstly, we shall show that For all and , we have andwhich implies that . Moreover, we havethenand hence . Due to the embedding we get that .
On the other hand, it is clear that , for a.e. , and , for all , . Next, we prove that Since , it suffices to prove thatBy simple calculations, we obtainWe setIf , then . Otherwise, since is small enough, then we can assume that , to deduce that .
Now, we are looking for conditions on to get (54). Thanks to (50) and (55), it is sufficient to haveandAs is small enough, we can assume also that . Then it yields , which implies that . Since and , thus (57) and (58) reduce to