Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 102639, 7 pages
http://dx.doi.org/10.1155/2015/102639

Research Article

Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

Mohamed Jleli and Bessem Samet

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received 12 July 2015; Accepted 15 November 2015

Academic Editor: Yuming  Qin

Copyright © 2015 Mohamed Jleli and Bessem Samet. 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 provide sufficient conditions for the nonexistence of global positive solutions to the nonlocal evolution equation  ∗ , where , , and . Next, we deal with global nonexistence for certain nonlocal evolution systems. Our method of proof is based on a duality argument.

1. Introduction

In [1], García-Melián and Quirós considered the nonlocal diffusion problem:

where is a compactly supported nonnegative function with unit integral, , and . Equation (1) may model a variety of biological, epidemiological, ecological, and physical phenomena involving media with properties varying in space [2, 3]; similar equations appear, for example, in Ising systems with Glauber dynamics [4]. In [1] the authors proved that (1) has a critical exponent:

which is the Fujita exponent for the classical nonlinear heat equation [5]. More precisely, they proved that if , the solution blows up in finite time for any nonnegative and nontrivial initial data ; if , there exist global solutions for small initial data . Very recently, Yang [6] considered the nonlinear coupled nonlocal diffusion system:

where and . Equation (3) can serve as a model for the processes of heat diffusion and combustion in two-component continua with nonlinear heat conduction and volumetric release [7]. In this case, Yang established that the critical Fujita curve is given by

which is also the Fujita curve for the coupled heat system and , obtained by Escobedo and Herrero [8].

In this paper, we are first concerned with the following evolution problem:

where , , and . We provide a sufficient condition for the nonexistence of global positive solutions to (5). Next, we consider the following two systems:

where . For each system, we find a bound on leading to the absence of global nontrivial solutions. Our method of proof is based on a duality argument developed by Mitidieri and Pokhozhaev [9, 10].

2. Main Results

Through this paper, , , and is a continuous function satisfying the following conditions:

(J1) is symmetric; that is, , for every .(J2).(J3).

The following lemmas will be used later.

Lemma 1. Let and . Then

where .

Lemma 2 (see [11]). Let , , , , , and be nonnegative functions and let and , , be positive reals such that , , , and . If

then

for some constant .

2.1. A Nonexistence Result for (5)

The definition of solutions we adopt for (5) is as follows.

Definition 3. Let . We say that is a global weak solution to (5) if , , and

for every regular test function with .

Our first main result is given by the following theorem.

Theorem 4. Suppose that one of the following conditions hold:

or

Then (5) admits no global weak solutions other than the trivial one.

Proof. Suppose that is a nontrivial global weak solution to (5). As a test function, we take

where is large enough,

, and is given by

From the definition of , clearly we have

which yields

Writing

and applying Lemma 1, we obtain

for some , where . On the other hand,

where

Using the change of variable and , we obtain

The above equality with (20) yields

for some constant . Next, we have

Using the symmetry of and Fubini’s theorem, we obtain

Therefore,

Using the property (J1), we obtain

By the property (J2) and the definition of , we have

The property (J3) and the inequality yield

From this, we have

Writing

using Hölder’s inequality and Lemma 1, we obtain

for some . We get

for some constant . Consequently, it follows from (11), (18), (24), and (34) that

For , we obtain

where . Observe that if one of conditions (12) or (13) is satisfied, then . In this case, letting in the above inequality and using the monotone convergence theorem, we obtain

which is a contradiction. The proof is finished.

2.2. A Nonexistence Result for System (6)

The definition of solutions we adopt for (6) is as follows.

Definition 5. Let , . We say that the pair is a global weak solution to (6) if , , and

for every regular test function with .

We have the following result.

Theorem 6. Let . Suppose that

Then (6) admits no global weak solutions other than the trivial one.

Proof. Suppose that is a nontrivial global weak solution to (6). As a test function, we take the function defined by (14). From the definition of , we have

Writing

and using Hölder’s inequality, we obtain

where . On the other hand, from (23), we have

which yields

for some constant . Using (33), we obtain

which yields

for some constant . As consequence, from (38), (45), and (47), it follows that

where . Similarly, we have

for some constant . Combining (48) with (49), we obtain

for some constant , where

Observe that (40) is equivalent to , . Under this condition, letting in (50), we get

which is a contradiction.

Remark 7. Taking and in Theorem 6, we obtain the result given by Theorem 4 for (5).

2.3. A Nonexistence Result for System (7)

The definition of solutions we adopt for (7) is as follows.

Definition 8. Let , . We say that the pair is a global weak solution to (6) if , , , , and

for every regular test function with .

We have the following result.

Theorem 9. Let . If

where

then (7) admits nonglobal weak solutions.

Proof. As before, we argue by contradiction. Suppose that is a nontrivial global weak solution to (7). As a test function, we take the function defined by (14). From (45), we have

From (47), we have

Using (53), (57), and (58), we get

Similarly, using (54), (57), and (58), we get

Here, , , , are some positive constants. Set

we obtain from (59) and (60) the following system:

Using Lemma 2, we obtain

where

Similarly. we have

where

It is not difficult to observe that condition (55) is equivalent to , , or , . In both cases, letting in (63) or in (65), we obtain , which is a contradiction.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors extend their sincere appreciations to the Deanship of Scientific Research at King Saud University for its funding of this Prolific Research group (PRG-1436-10).

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