Abstract and Applied Analysis
Volume 2018 (2018), Article ID 3821217, 14 pages
https://doi.org/10.1155/2018/3821217

Research Article

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

Zakariya Chaouai and Abderrahmane El Hachimi

Laboratory of Mathematical Analysis and Applications, Mohammed V University, Avenue Ibn Battouta, BP 1014, Rabat, Morocco

Correspondence should be addressed to Abderrahmane El Hachimi; aelhachi@yahoo.fr

Received 18 July 2018; Revised 3 October 2018; Accepted 11 October 2018; Published 8 November 2018

Academic Editor: Changbum Chun

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.

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 as

The 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]).

where

For 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]). Let

Then, 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 if

where 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 have

where 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 satisfies

Now, 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 estimate

with 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 2

On the other hand, we have

Hence, from Proposition 2 we get

Therefore, combining (29) and (31) we deduce the claim. On the other hand, from Definition 4, we obtain

Due to a monotonicity argument, we have

then

By Lebesgue’s dominated convergence theorem, we have

Now, we can write

Then, from (34) and (35), by letting , we obtain

Hence, 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 that

which 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

where

and

where 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

and

which implies that . Moreover, we have

then

and 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 that

By simple calculations, we obtain

We set

If , 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 have

and

As is small enough, we can assume also that . Then it yields , which implies that . Since and , thus (57) and (58) reduce to

and

which are satisfied if

and

By 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

where

and

We 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. Then

for 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:

where

and

where and are positive constants small and large enough, respectively, is a positive constant such that

and , , are positive constants and will be determined later. By direct calculations, we get

and

We can see easily that a.e. in . Then, by using (88) we obtain

and

Moreover, 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 that

Using (89), we obtain by simple calculations

We set , , and

To get (96), by (90) and (97) it suffices to have

and

Now, 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 get

Then, it yields

Therefore, we have

so (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

where

we 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 follows

where ,

and

with sufficiently small .
Firstly, we denote . It is clear that and by direct calculations, we have

Thus, it follows that

Moreover, from the definition of we have

Next, we will show that

Since , it suffices to show that

Using the hypothesis , we have for all . Then, from (119) and (116) we obtain

Due to the last inequality, it remains to prove that

We have

and

We set

Then

where

for 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 that

Set . Now, to get (125) it suffices to have

Since is small enough then, from the value of and , we obtain that

and

which 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 that

where 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 that

where 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 , and

where 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 that

We have

and

Then

with

By 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 that

To do so, we just need to get

Due 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.

References

S. N. Antontsev and S. I. Shmarev, “A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions,” Nonlinear Analysis. Theory, Methods and Applications. An International Multidisciplinary Journal, vol. 60, no. 3, pp. 515–545, 2005.

D. G. Aronson, The Porous Medium Equation, F. Primicerio, Ed., vol. 1224 of Nonlinear Diffusion Problems, Springer, Berlin, Germany, 1986.

M. E. Gurtin and R. C. MacCamy, “On the diffusion of biological populations,” Mathematical Biosciences, vol. 33, no. 1-2, pp. 35–49, 1977.

M. Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, vol. 1748 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2000.

Y. M. Chen, S. Levine, and M. Rao, “Variable exponent, linear growth functionals in image restoration,” SIAM Journal on Applied Mathematics, vol. 66, no. 4, pp. 1383–1406, 2006.

V. V. Zhikov, “Density of smooth functions in sobolev-orlicz spaces,” Journal of Mathematical Sciences, vol. 132, no. 3, pp. 285–294, 2006.

G. Akagi, “Doubly nonlinear parabolic equations involving variable exponents,” Discrete and Continuous Dynamical Systems - Series S, vol. 7, no. 1, pp. 1–16, 2014.

G. Akagi and K. Matsuura, “Nonlinear diffusion equations driven by the p(.)-Laplacian,” Nonlinear Differential Equations and Applications NoDEA, vol. 20, no. 1, pp. 37–64, 2013.

S. N. Antontsev and S. I. Shmarev, “Doubly degenerate parabolic equations with variable nonlinearity II: Blow-up and extinction in a finite time,” Nonlinear Analysis, vol. 95, pp. 483–498, 2014.

S. Antontsev and S. Shmarev, “On the localization of solutions of doubly nonlinear parabolic equations with nonstandard growth in filtration theory,” Applicable Analysis: An International Journal, vol. 95, no. 10, pp. 2162–2180, 2016.

S. Antontsev and S. Shmarev, Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-Up, vol. 4 of Atlantis Studies in Differential Equations, Atlantis Press, 2015.

B. Guo and W. Gao, “Study of weak solutions for parabolic equations with nonstandard growth conditions,” Journal of Mathematical Analysis and Applications, vol. 374, no. 2, pp. 374–384, 2011.

B. Guo and W. Gao, “Existence and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity,” Acta Mathematica Scientia B, vol. 32, no. 3, pp. 1053–1062, 2012.

M. Bertsch, R. Kersner, and L. A. Peletier, “Positivity versus localization in degenerate diffusion equations,” Nonlinear Analysis Journal, vol. 9, no. 9, pp. 987–1008, 1985.

A. S. Kalashnikov, “Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations,” Uspekhi Matematicheskikh Nauk, vol. 254, no. 2, pp. 135–176, 1987.

J. I. Díaz, “Solutions with compact support for some degenerate parabolic problems,” Nonlinear Analysis, vol. 3, no. 6, pp. 831–847, 1979.

J. I. Díaz and J. Hernandez, “Qualitative properties of free boundaries for some nonlinear degenerate parabolic equations,” in Nonlinear Parabolic Equations: Qualitative Properties of Solutions, L. Boccardo and A. Teseieds, Eds., vol. 149 of Pitman Research Notes, pp. 85–93, Longman, 1987.

S. Antontsev, J. I. Díaz, and S. Shmarev, Energy Methods for Free Boundary Problems: Applications to Non-Linear PDEs and Fluid Mechanics, vol. 48 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, Massachusetts, Mass, USA, 2002.

J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries, vol. 106 of Pitman (Advanced Publishing Program), London, UK, 1985.

J. I. Díaz and M. A. Herrero, “Proprietés de support compact pour certaines équations elliptiques et paraboliques non linéaires,” Comptes Rendus de l'Académie des Sciences, vol. 286, no. 19, pp. 815–817, 1978.

J. I. Díaz and M. A. Herrero, “Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems,” Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, vol. 89, no. 3-4, pp. 249–258, 1981.

S.-Y. Chung and J.-H. Park, “A complete characterization of nonlinear absorption for the evolution p-Laplacian equations to have positive or extinctive solutions,” Computers and Mathematics with Applications, vol. 71, no. 8, pp. 1624–1635, 2016.

J. I. Díaz, “Qualitative study of nonlinear parabolic equations: an introduction,” Extracta Mathematicae, vol. 16, no. 3, pp. 303–341, 2001.

Z. B. Fang and G. Li, “Extinction and decay estimates of solutions for a class of doubly degenerate equations,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1795–1802, 2012.

C. Jin, J. Yin, and Y. Ke, “Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources,” Proceedings of the Edinburgh Mathematical Society, vol. 52, no. 2, pp. 419–444, 2009.

Y. G. Gu, “Necessary and sufficient conditions for extinction of solutions to parabolic equations,” Acta Mathematica Sinica, vol. 37, no. 1, pp. 73–79, 1994.

H. J. Yuan, S. Z. Lian, C. L. Cao, W. J. Gao, and X. J. Xu, “Extinction and positivity for a doubly nonlinear degenerate parabolic equation,” Acta Mathematica Sinica, vol. 23, no. 10, pp. 1751–1756, 2007.

J. Zhou and C. Mu, “Critical blow-up and extinction exponents for non-Newton polytropic filtration equation with source,” Bulletin of the Korean Mathematical Society, vol. 46, no. 6, pp. 1159–1173, 2009.

S. Antontsev and S. Shmarev, “Existence and uniqueness for doubly nonlinear parabolic equations with nonstandard growth conditions,” Differential Equations and Applications, vol. 4, no. 1, pp. 67–94, 2012.

S. Antontsev and S. Shmarev, “Doubly degenerate parabolic equations with variable nonlinearity I: Existence of bounded strong solutions,” Advances in Differential Equations, vol. 17, no. 11-12, pp. 1181–1212, 2012.

Q. Zhang, X. Liu, and Z. Qiu, “On the boundary blow-up solutions of -Laplacian equations with singular coefficient,” Nonlinear Analysis, vol. 70, no. 11, pp. 4053–4070, 2009.

L. Diening, P. Harjulehto, P. Hästö, and M. Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Berlin, Germany, 2011.

X. L. Fan and Q.-H. Zhang, “On the spaces and ,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 424–446, 2001.