Systems with Local and Nonlocal Diffusions, Mixed Boundary Conditions, and Reaction Terms
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.
1.1. Description of the Problem
Our first goal in this paper is to study the following nonlocal reaction-diffusion system with mixed boundary conditionsHere , , 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 . 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 . 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 localAgain, 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 formand 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  studied the Dirichlet boundary conditions problemIn 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  the Neumann boundary conditions case is studiedSince 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  studied the problemThey 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, .
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 andLet 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 .
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 systemIf , the solution to (23) is given bywith If , the solution of (23) iswith If , the solution of (23) isWe will use the notation to indicate that and .
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.
Lemma 12. Let . Then there exists , a maximal solution of (1), in the sense that if is any other solution there holdsMoreover, , 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 ; hence we omit them.
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 .
As , by Holder’s inequality we have thatwith . Now, differentiating with respect to and using (1) we obtain that As , by Holder’s inequality we have thatNow, 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, thenThen, 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 .
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 thatwhere 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 thatAs we have thatTherefore, we get that for all andMultiplying 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 thatReplacing 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 wherewhere for any function where is Green’s function. That is, denote the solution of (see )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 .
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 problemhas a solution .
Definition 23. Let . is called a supersolution of (2) if