Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents

Chaouai, Zakariya; El Hachimi, Abderrahmane

doi:https://doi.org/10.1155/2018/3821217

Abstract and Applied Analysis

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Abstract Introduction Preliminaries Results Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 3821217 | https://doi.org/10.1155/2018/3821217

Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents

Zakariya Chaouai¹and Abderrahmane El Hachimi¹

Academic Editor: Changbum Chun

Received18 Jul 2018

Revised03 Oct 2018

Accepted11 Oct 2018

Published08 Nov 2018

Abstract

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 .

1. Introduction

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 ([3]). 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 ([4]). It appears also as a model in image restoration ([5]) and in elasticity ([6]).

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 [14]).

The phenomenon of finite speed of propagation was investigated by Kalashnikov in [15]. 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 [16]. Later, in [17] 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 [27] with the homogeneous equation , in , where and . They proved that the condition is necessary and sufficient for extinction to occur. Moreover, Zhou and Mu ([28]) 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 [31], where radial sub- and supersolutions for some elliptic problems with variable exponents are constructed, and some of the results by Chung and Park in [22] and by Yuan et al. in [27], 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.

2. Preliminaries

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 .

Proposition 2 (see [11, 32, 33]). LetThen, we have(i)(ii)(iii).

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 [29].

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 toandwhich are satisfied if andBy setting and , we can choose such that Therefore, we get the desired result.

Next, we will mention an extinction result where there is no condition between the ranges of and .

Proposition 10. Let be a strong solution of . Assume that and , and is small enough. Then, there exists a finite time such that for all

Proof. We consider the same supersolution as in the proof of Theorem 9 but we choose here , which means whereandWe have already shown in Theorem 9 that . We claim that by using the same lines as in the proof of Theorem 9. Since , it is therefore sufficient to have which is satisfied if we choose Consequently, by the comparison principle we deduce the extinction of solution in finite time.

4.2. Nonextinction of Solutions

The following theorem deals with the positivity of weak solutions.

Theorem 11. Let be a strong solution of and not identically zero. Assume that , , and . Then, there exists a finite time such that for all

The method of proof is inspired from [27], where the constant exponents case is treated. However, some difficulties arise in the construction of subsolutions due to the fact that the exponents are variable. The proof of this theorem is divided into two lemmas. In the first lemma, we show by using a comparison function that the support of weak solution is nondecreasing with respect to time. In the second lemma, we show that the solution is positive locally in ; then, by a finite covering argument, we deduce the result.

Lemma 12. Let be a strong solution of . Assume that and the initial condition is nontrivial. Thenfor all .

The proof of Lemma 12 follows the same lines as that of lemma in [22], where the constant exponents case is studied. For completeness, we shall give it here.

Proof. The argument used here is based on a comparison function with which we show that the support of solution is increasing. For that we consider an arbitrary set which is a nonzero measure subset of such that . We divide the proof in two cases, firstly we treat the case where and then the case where . If , we consider the following function:where . It is clear that , and it is easy to verify that On the other hand, by direct calculations we get Hence Since and for all , , it follows that Moreover, from the definition of , we have , almost everywhere in , and for all, , . Thus, by comparison principle we conclude that for any arbitrary where , the weak solution of satisfies and the result follows in this case.
If , we consider the following function:where . By the same argument used previously we obtain that , and . Moreover, by direct calculations we get since and for all , . Hence Therefore, we have Thus, by the same argument used previously we deduce our result.

Lemma 13. Under the same assumptions of Theorem 11, let the initial condition satisfies , for some . Then, there exists , such that for any , where .

Proof. We consider the following function: whereandwhere and are positive constants small and large enough, respectively, is a positive constant such thatand , , are positive constants and will be determined later. By direct calculations, we getandWe can see easily that a.e. in . Then, by using (88) we obtain andMoreover, from the definition of we have Since for all , then on , which implies that on . Now, let us show that To do so, it suffices to show thatUsing (89), we obtain by simple calculationsWe set , , and To get (96), by (90) and (97) it suffices to haveandNow, our goal is to choose the constants , , and in order to verify each of the inequalities above. Firstly, let us show the inequality (99). Since is large enough and , we have from which we getThen, it yields Therefore, we haveso (99) is satisfied. Next, to get (100) and (101), we use the fact that and a.e. in . By setting and using large enough, so that wherewe obtain the inequalities (100) and (101). Finally, to get the inequality (102), it suffices to have which reduces to Since is small enough, the last inequality holds, which means that the inequality (102) is satisfied and allows us to deduce inequality (96). Then, we obtain by comparison principle, for each , Therefore, the result follows from Lemma 12.

Proof of Theorem 11. The proof is similar to that of heorem 1.2, in [27], and we omit the details here.

5. Finite Speed of Propagation Property

In this section we shall give precise estimates for the of support of the solution of , depending on the size of the support of . Let us emphasize that each estimation is obtained under a sign condition on and depending on the range of the exponents , , and . As in [21], the proof is based on the construction of local supersolutions and on the use of the comparison principle.

Concerning the construction of supersolutions, we shall proceed as in [31].

Note that under some conditions on the data, if is positive, then the solutions will blow up in finite time (see [10]). For that it requires to construct a supersolution defined locally in time, which means in for any , where is the maximal existence time. We denote .

Theorem 14. Let be a strong solution of . Let be such that . We assume , , and . Then, for any , there exists a unique compactly supported solution of such that where will be specified in the proof below.

Proof. The idea of the proof is to construct a suitable supersolution with compact support which is not necessarily defined in the whole . Then, by the comparison principle, we deduce directly that , and the result follows. For all , we define as followswhere , andwith sufficiently small .
Firstly, we denote . It is clear that and by direct calculations, we haveThus, it follows that Moreover, from the definition of we have Next, we will show that Since , it suffices to show thatUsing the hypothesis , we have for all . Then, from (119) and (116) we obtainDue to the last inequality, it remains to prove thatWe haveandWe setThenwherefor all . By straightforward considerations, (boundedness of different functions), there exist positive constants such that, for all and , Thus In this case we have , and from (118), we can choose small enough, so that is also small for all , whence we have for sufficiently small , which implies thatSet . Now, to get (125) it suffices to have Since is small enough then, from the value of and , we obtain that andwhich implies (125). Therefore we deduce the desired result.

Theorem 15. Let be a strong solution of . Let be such that . We assume , , and . Then, for any , there exists a unique compactly supported solution of such thatwhere is specified in the proof above.

Proof. We consider the same supersolution as in the proof of Theorem 14 and we just need to prove that where . Since , it is sufficient to prove that Combining the same lines as in the previous theorem and the comparison principle in Proposition 6, we conclude the result.

Finally, we state the following result on uniform localization of the support of solution.

Theorem 16. Let be a strong solution of . Let be such that . We assume , , and . Then for any , there exists a unique compactly supported solution of such thatwhere is positive constants determined in the proof below.

Proof. In order to get the desired estimation of the support of the solution, we define a suitable local supersolution associated with the stationary problem related to . Let . We define as follows:where , andwhere and is a constant that will be determined hereafter. It is easy to verify that for any , where . Concerning initial condition, we have . Now, let us show thatWe have andThenwithBy straightforward considerations, there exists a constant such that, for every with , we have Hence, we get Now, in order to obtain the inequality (144), we need to show thatTo do so, we just need to getDue to the fact that , (152) holds if Then, it is suffices to take to complete the proof.

Remark 17. As in [20, 21], we can assume in this section that is a set of , not necessarily bounded.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is a part of the Ph.D. thesis of the first author which is in preparation under the supervision of the second author at the Laboratory of Mathematical Analysis and Applications at the Mohammed V University in Rabat, Morocco.

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Copyright © 2018 Zakariya Chaouai and Abderrahmane El Hachimi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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