Abstract and Applied Analysis
Volume 2018 (2018), Article ID 3906431, 10 pages
https://doi.org/10.1155/2018/3906431

Research Article

Systems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Terms

Mauricio Bogoya¹ and Julio D. Rossi²

¹Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia
²Departamento de Matemática, FCEyN, Universidad de Buenos Aires, Ciudad Universitaria, Pabellon I (1428), Buenos Aires, Argentina

Correspondence should be addressed to Mauricio Bogoya; mbogoyal@unal.edu.co

Received 12 July 2018; Revised 24 August 2018; Accepted 13 September 2018; Published 1 October 2018

Academic Editor: Jozef Banas

Copyright © 2018 Mauricio Bogoya and Julio D. Rossi. 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 study systems with different diffusions (local and nonlocal), mixed boundary conditions, and reaction terms. We prove existence and uniqueness of the solutions and then analyze global existence vs blow up in finite time. For blowing up solutions, we find asymptotic bounds for the blow-up rate.

1. Introduction

1.1. Description of the Problem

Our first goal in this paper is to study the following nonlocal reaction-diffusion system with mixed boundary conditions

Here , , are nonnegative and nontrivial functions, is a nonnegative, smooth, symmetric radially and strictly decreasing kernel, with and supported in the unitary ball, and () is a bounded connected and smooth domain.

In the first equation, we are imposing that the diffusion takes place only in ; no mass may enter or leave the domain. This corresponds to what is called Neumann boundary conditions in the literature; see [1]. In the second equation, we have that the diffusion takes place in the whole but we impose that vanishes outside . This is the analogous of what is called Dirichlet boundary conditions; see [1]. Hence this system is governed by nonlocal diffusion with mixed boundary conditions.

Our second objective in this paper is the study of the same kind of system when one of the diffusions is local

Again, we assume that , are nontrivial functions, is as before, and () is a bounded connected and smooth domain with denoting its outer normal unit vector field.

Now, we face a system similar to the previous one, in which one of the two components obeys a local classical diffusion (with the usual Laplacian) while the other component has a nonlocal diffusion operator.

1.2. Main Results

For both problems (1) and (2) we analyze the existence and uniqueness of nonnegative solutions. Once we settle this issue, we look for the existence of solutions globally defined in time and solutions blowing up in finite time. Finally, when we have solutions that blow up, we analyze the blow-up rate.

We say that a solution blows up in finite time if and only if a finite time exists such that

If solutions are well defined and finite for every with the solution is global, i.e., the solution exists for all .

For both problems our results can be summarized as follows.

Theorem 1. For positive real functions, there exists a unique positive solution . Moreover, can be extended to a maximal interval with .
A comparison principle between super- and subsolutions holds.
If , the solution blows up in finite time , while for the solution exists globally.
For solutions that attain their maximum at for every (for example, for radially symmetric nondecreasing solutions), the blow-up rates are give by

1.3. Previous Results

Nonlocal diffusion equations of the form

and variations of it, have been widely used in the last decade to model diffusion processes; see [1–7] and references therein. This equation is called nonlocal diffusion equation since the diffusion of the density at a point and time depends not only on , but also on all the values of in a neighborhood of through the convolution term . This equation shares many properties with the classical heat equation such as the fact that bounded stationary solutions are constant, a maximum principle holds for both of them, and even if is compactly supported, perturbations propagate with infinite speed.

Concerning boundary conditions for nonlocal diffusion, Chasseigne et al. in [5] studied the Dirichlet boundary conditions problem

In this model they have that diffusion takes place in the whole but impose that vanishes outside . This is the analogous of what is called Dirichlet boundary conditions for the heat equation.

Also in [5] the Neumann boundary conditions case is studied

Since we are now integrating in , the diffusion takes place only in . The individuals may not enter nor leave . This is the analogous of what is called homogeneous Neumann boundary conditions in the literature.

Concerning reaction terms, Pérez-Llanos and Rossi in [8] studied the problem

They prove that nonnegative and nontrivial solutions blow up in finite time if and only if . Moreover, they find that the blow-up rate is the same as the one that holds for the ODE , that is, .

For systems, Bogoya in [9] studied the analogous system to (1) with Neumann boundary conditions.

The paper is organized as follows: In Section 2, we analyze the existence and uniqueness of solutions to (1). In Section 3, we look for global existence vs blow up of solutions to (1) and its blow-up rate when appropriate. In Section 4, we deal with the existence and uniqueness of solutions to (2). Finally, in Section 5, we analyze the global existence and blow up of solutions of (2) and the corresponding blow-up rate.

2. Existence and Uniqueness for (1)

In this section, we analyze the existence and uniqueness of nonnegative solutions of (1).

We will deal with this issue for a slightly more general system in which the reaction terms , are replaced by , two Lipschitz nonnegative functions. Existence and uniqueness will be obtained via Banach’s fixed point theorem. Let be fixed and consider

that is, a Banach space with the norm

and

Let

that is, a closed subspace of . In what follows, we will use the notation

We define the operator

where

The following Lemma is just the first step in order to show that has a fixed point in .

Lemma 2. Let and be Lipschitz functions with Lipschitz constants and and and two pairs of initial conditions in and . Then, there exists a positive constant such that

Proof. For any we have

where and is the Lipschitz constant of .
Analogously, taking into account the fact that is zero outside and denoting by the Lipschitz constant for , we have that

Therefore, we have obtained

where . Then, if we have that

Next, we show the existence and uniqueness of the solution for functions and that are locally Lipschitz.

Theorem 3. Let and be locally Lipschitz functions and be nonnegative real functions; then there exists a unique solution such that . Moreover, can be extended to a maximal interval with .

Proof. We check first that . Taking in Lemma 2 we get that . Choose such that . Now taking in Lemma 2 we get that is a strict contraction in and the existence and uniqueness part of the theorem follow from Banach’s fixed point theorem in the interval . To extend the solution to we may take as initial conditions and obtain a solution up to . Iterating this procedure, we get a solution defined in .

Now, we use the same ideas of the previous analysis to obtain the following results.

Corollary 4. The solution depends continuously on the initial data. In fact, let and be locally Lipschitz functions and if and are solutions with initial data and , respectively, then there exists a constant such that

Corollary 5. is a solution of (1) if and only if

Now, we look for existence and uniqueness of solutions to (1).

Theorem 6. If and are nonnegative real functions, then there exists a unique solution of (1) such that .

Proof. The functions are locally Lipschitz; then by Theorem 3, system (1) has a unique solution .

For exponents less than one, we have the following existence result.

Theorem 7. Let and be nonnegative and bounded functions; then there exists a solution to (1).

Proof. The existence of the solution of (1) is obtained with an approximation procedure. We assume that (the other possibilities are left to the reader). Let be a sequence of locally Lipschitz functions such that, for fixed , if , is nondecreasing and for . Consider the system with and . From Lemma 2 and Theorem 3, we have that there is a unique solution . Now, are nondecreasing and bounded sequences. Therefore, passing to the limit as using Corollary 5, we get the existence of , a solution to (1).

Remark 8. For future reference, let us analyze the solution to the ODE system

If , the solution to (23) is given by

with

If , the solution of (23) is

with

If , the solution of (23) is

We will use the notation to indicate that and .

Definition 9. Let . is called a supersolution of (1) if

Analogously is called a subsolution of (1) if it satisfies the opposite inequalities.

Lemma 10 (comparison principle). Let and be a subsolution and supersolution of (1), respectively. If for all , then

for all .

Proof. Let , . Assume first that for . We observe that and verify

Now, set and suppose that the conclusion of the Lemma does not hold. Thus, let be the first time such that

We can assume that attains the previous minimum. At that time, there must be a point such that . But, on the one hand and, on the other hand

where . This gives a contradiction. Using the continuity of solutions of (1) with respect to the initial condition and an approximation argument, the result follows for general initial condition.

Corollary 11. Let be a supersolution of (1). Then, if for we have that for all and moreover, a strict inequality holds if is positive.

The following lemma gives us the existence of a maximal solution of (1) when . Its proof is analogous to those given in [10]; therefore we omit the details here.

Lemma 12. Let . Then there exists , a maximal solution of (1), in the sense that if is any other solution there holds

Moreover, , for all .

The following theorems deal with the uniqueness problem for (1) in the cases of identically null initial condition and of nontrivial initial condition, respectively. Their proofs are analogous to those given in [9]; hence we omit them.

Theorem 13. Let .

(1)If , then the unique solution of (1) is .(2)If , then there exists exactly one solution of (1) such that and are strictly positives.

Theorem 14. If , then the solution of problem (1) is unique.

3. Global Existence vs Blow Up for Solutions to (1)

In this section, we look for conditions that ensure that solutions of (1) blow up in finite time or are globally defined.

Theorem 15. Let and be nonnegative and nontrivial functions. Then, the solution of (1) blows up in finite time .

Proof. Let and be nonnegative and nontrivial functions and solution of (1). We define the functions for

where is eigenfunction of the nonlocal operator, which is positive in ; see [7].
As , by Holder’s inequality we have that

with . Now, differentiating with respect to and using (1) we obtain that

As , by Holder’s inequality we have that

Now, taking the derivative of with respect to we have that

Summarizing, we get that

If are large, then blow up in finite time; therefore blow up in finite time.
In general, blow up in finite time for any initial data , . In fact, if we consider for , , with , positive functions, then

Then, it can be obtained that (and then ) blows up in finite time.

Theorem 16. Let and be nonnegative and nontrivial functions. Then, the solution of (1) exists globally.

Proof. Consider such that . Since , by Remark 8, we have that ; the solution of (23) with is globally defined. Notice that is a supersolution of (1). If is a solution of (1) with initial conditions , then, by Lemma 10 (comparison principle), we have that and in . Therefore, any solution of (1) can be continued for all times in the case .

Next, we analyze the blow-up rate of the solutions of (1). We assume that and note that for smooth radially symmetric and nondecreasing initial conditions (that is, when are such that , ) the solutions are also radially symmetric and radially nondecreasing (that is, it holds that and ). Hence, for every , the maximum of both components is attained at . We state this result as follows.

Lemma 17. If is a ball and are smooth, radially symmetric, and nondecreasing initial conditions (i.e. are such that , ), then both components of the solution are radially symmetric and radially nondecreasing (they verify and for every and every ).

Proof. For a proof we refer to Lemma 4.1 in [8].

Theorem 18. For , let be a positive solution to (1) such that the maximum is attained at for every . Then, there exist , , , positive constants such that

where stands for the blow-up time of the solution.

Proof. As , we have that , the solution of (1), blows up in finite time (that we called ). We assumed that

By (1), we have that

As

we have that

Therefore, we get that for all

and

Multiplying the second inequality of (47) by and the first inequality of (48) by , we obtain

which is equivalent to

Multiplying the inequality by and integrating in with , we have that

Replacing the second inequality of (48) by inequality (51), we get

Integrating this inequality on , we finally obtain

In an analogous way we obtain that

With an analysis similar to the one developed above, we obtain that there exists a constant such that for

Replacing the first inequality of (48) by inequality (55) and as we have that and

Integrating the inequality from above on , we obtain that

In an analogous way we get

4. Local and Nonlocal Diffusion

In this section, we analyze the existence and uniqueness of nonnegative solutions of (2).

As in Section 2, we will initially study the system with and replacing , , with , being nonnegative Lipschtiz functions. Existence and uniqueness will be again obtained via Banach’s fixed point theorem. Let be fixed; we consider the Banach space with the norm

and a closed subspace of already considered in Section 2.

We define the operator , by

where

where for any function

where is Green’s function. That is, denote the solution of (see [11])

Now, we need the following lemma.

Lemma 19. Let and . Then, there exists a positive constant such that

Proof. For any we have

where is the Lipschitz constant of .
Taking into account that is zero outside of , we have that

where and is the Lipschitz constant of .
Therefore,

In the following theorem, we analyze the existence and uniqueness of the solution by considering that the functions and are locally Lipschitz. The proof is analogous to the one for Theorem 3; hence we omit it.

Theorem 20. Let and be locally Lipschitz functions and are nonnegative real functions; then there exists a unique solution . Moreover, can be extended to a maximal interval with .

With these ingredients we can show existence and uniqueness of the solution as well as a comparison principle. The proofs are analogous to the ones that we included in Section 2; hence we omit them.

Theorem 21. Let be nonnegative real functions.

(i)If then there exists a unique solution .(ii)If then there exists a solution .

Remark 22. The stationary problem

has a solution .

Definition 23. Let . is called a supersolution of (2) if

Analogously is called a subsolution of (2) if it satisfies the opposite inequalities.

Lemma 24 (comparison principle). Let and be a subsolution and supersolution of (2), respectively. If for all , then for all .

5. Global Existence and Blow Up for the Local/Nonlocal Diffusion System

In this section, we analyze under what conditions the solutions of (2) blow up in finite time or are global.

Reasoning analogously to Theorems 15 and 16, we have the following theorem on the global existence and blow up of the solutions of (2); again we omit their proofs.

Theorem 25. Let be nonnegative and nontrivial functions.

(1)If , then the solution of (2) blows up in finite time .(2)If , then the solution of (2) exists globally.

Next, we analyze the blow-up rate of the solutions of (2). We assume that .

Theorem 26. Let and , be nonnegative and nontrivial functions. Let be the solution such that the maximum is reached at for every . Then, there exist positive constants such that

Proof. As , we have that , the solution of (1), blow up in finite time . Let and . By (1), we have that

and we have that

Therefore, we have that for all

and

Multiplying the second inequality of (73) by and the first inequality of (74) by , we have that

which is equivalent to

From this point the proof follows analogously to what was done in the proof of Theorem 18.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

F. Andreu-Vaillo, J. M. Mazón, J. . Rossi, and J. J. Toledo-Melero, Nonlocal Diffusion Problems, vol. 165 of Mathematical Surveys and Monographs, American Mathematical Society, Real Sociedad Matematica EspaNola, Madrid, Providence, RI, USA, 2010.

P. W. Bates, P. C. Fife, X. Ren, and X. Wang, “Traveling waves in a convolution model for phase transitions,” Archive for Rational Mechanics and Analysis, vol. 138, no. 2, pp. 105–136, 1997.

M. Bogoya, “A nonlocal nonlinear diffusion equation in higher space dimensions,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 601–615, 2008.

C. Cortazar, M. Elgueta, and J. D. Rossi, “A nonlocal diffusion equation whose solutions develop a free boundary,” Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics, vol. 6, no. 2, pp. 269–281, 2005.

E. Chasseigne, M. Chaves, and J. D. Rossi, “Asymptotic behavior for nonlocal diffusion equations,” Journal de Mathématiques Pures et Appliquées, vol. 86, no. 3, pp. 271–291, 2006.

P. Fife, “Some nonclassical trends in parabolic and parabolic-like evolutions,” in Trends in Nonlinear Analysis, pp. 153–191, Springer, Berlin, Germany, 2003.

J. Garcia-Melian and J. D. Rossi, “On the principal eigenvalue of some nonlocal diffusion problems,” Journal of Differential Equations, vol. 246, no. 1, pp. 21–38, 2009.

M. Pérez-Llanos and J. D. Rossi, “Blow-up for a non-local diffusion problem with Neumann boundary conditions and a reaction term,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 70, no. 4, pp. 1629–1640, 2009.

M. Bogoya, “On non-local reaction-diffusion system in a bounded domain,” Boundary Value Problems, Paper No. 38, 16 pages, 2018.

M. Escobedo and M. A. Herrero, “A semilinear parabolic system in a bounded domain,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 165, pp. 315–336, 1993.

C. Bandle and H. Brunner, “Blow-up in diffusion equations,” Journal of Computational and Applied Mathematics, vol. 97, no. 1-2, pp. 3–22, 1998.