Blow-Up Phenomena for Certain Nonlocal Evolution Equations and Systems

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 , , andfor 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:orThen (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 takewhere is large enough, , and is given by From the definition of , clearly we have which yieldsWriting and applying Lemma 1, we obtainfor some , where . On the other hand, where Using the change of variable and , we obtainThe above equality with (20) yieldsfor 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 haveThe property (J3) and the inequality yieldFrom this, we have Writing using Hölder’s inequality and Lemma 1, we obtain for some . We getfor 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 , , andfor every regular test function with .

We have the following result.

Theorem 6. Let . Suppose thatThen (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 haveWriting and using Hölder’s inequality, we obtain where . On the other hand, from (23), we have which yieldsfor some constant . Using (33), we obtain which yieldsfor some constant . As consequence, from (38), (45), and (47), it follows thatwhere . Similarly, we havefor some constant . Combining (48) with (49), we obtainfor 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 , , , , andfor every regular test function with .

We have the following result.

Theorem 9. Let . Ifwhere 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 haveFrom (47), we haveUsing (53), (57), and (58), we getSimilarly, using (54), (57), and (58), we getHere, , , , are some positive constants. Set we obtain from (59) and (60) the following system: Using Lemma 2, we obtainwhere Similarly. we havewhere 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).