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

Authors: Mauricio Bogoya; Julio D. Rossi
Journal: Abstract and Applied Analysis
 (2018)

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## 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.


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## Body

## 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(1)utx,t=∫ΩJx-yuy,t-ux,tdy+vpx,t,x∈Ω,t>0,vtx,t=∫RNJx-yvy,t-vx,tdy+uqx,t,x∈Ω,t>0,vx,t=0x∉Ω,t>0,ux,0=u0x,vx,0=v0x,x∈Ω.Here p,q>0, u0(x), v0(x)∈C(Ω¯) are nonnegative and nontrivial functions, J:Rn→R is a nonnegative, smooth, symmetric radially and strictly decreasing kernel, with ∫RnJ(x)dx=1 and supported in the unitary ball, and Ω⊂RN (N≥1) is a bounded connected and smooth domain.In the first equation, we are imposing that the diffusion takes place only inΩ; 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 RN but we impose that u vanishes outside Ω. 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(2)utx,t=Δu+vpx,t,x∈Ω,t>0vtx,t=∫RNJx-yvy,t-vx,tdy+uqx,t,x∈Ω,t>0∂u∂ηx,t=0,x,t∈∂Ω×0,T;vx,t=0,x∉Ω,t>0ux,0=u0x,vx,0=v0x,x∈Ω.Again, we assume that p,q>0, u0(x),v0(x)∈C(Ω¯) are nontrivial functions, J is as before, and Ω⊂RN (N≥1) is a bounded connected and smooth domain with η 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(u,v) blows up in finite time if and only if a finite time T>0 exists such that (3)limt↗T⁡sup⁡ux,tL∞Ω+vx,tL∞Ω=∞.If solutions are well defined and finite for every t∈(0,T) with T=∞ the solution (u,v) is global, i.e., the solution exists for all t≥0.For both problems our results can be summarized as follows.Theorem 1.
For(u0,v0)∈C(Ω¯)×C(Ω¯) positive real functions, there exists a unique positive solution (u,v). Moreover, (u,v) can be extended to a maximal interval [0,T) with T≤∞.
A comparison principle between super- and subsolutions holds.
Ifpq>1, the solution (u,v) blows up in finite time T, while for pq≤1 the solution (u,v) exists globally.
For solutions that attain their maximum atx=0 for every t (for example, for radially symmetric nondecreasing solutions), the blow-up rates are give by (4)u0,t~T-t-p+1/pq-1v0,t~T-t-q+1/pq-1.


### 1.3. Previous Results

Nonlocal diffusion equations of the form(5)utx,t=J∗u-ux,t=∫RnJx-yuy,tdy-ux,t,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 u at a point x and time t depends not only on u(x,t), but also on all the values of u in a neighborhood of x through the convolution term J∗u. This equation shares many properties with the classical heat equation ut=Δu such as the fact that bounded stationary solutions are constant, a maximum principle holds for both of them, and even if J 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(6)utx,t=∫RNJx-yuy,t-ux,tdy,x∈Ω,t>0,ux,t=0,x∉Ω,t>0,ux,0=u0x,x∈Ω,In this model they have that diffusion takes place in the whole RN but impose that u vanishes outside Ω. 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(7)utx,t=∫ΩJx-yuy,t-ux,tdy,x∈Ω,t>0ux,0=u0x,x∈Ω.Since we are now integrating in Ω, the diffusion takes place only in Ω. The individuals may not enter nor leave Ω. 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(8)utx,t=∫ΩJx-yuy,t-ux,tdy+upx,t,x∈Ω,t>0,ux,0=u0x,x∈Ω.They prove that nonnegative and nontrivial solutions blow up in finite time if and only if p>1. Moreover, they find that the blow-up rate is the same as the one that holds for the ODE ut=up, that is, limt→T-⁡T-t1/(p-1)u(·,t)∞=(1/(p-1))1/(p-1).For systems, Bogoya in [9] studied the analogous system to (1) with Neumann boundary conditions.The paper is organized as follows: In Section2, 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.


## 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(1)utx,t=∫ΩJx-yuy,t-ux,tdy+vpx,t,x∈Ω,t>0,vtx,t=∫RNJx-yvy,t-vx,tdy+uqx,t,x∈Ω,t>0,vx,t=0x∉Ω,t>0,ux,0=u0x,vx,0=v0x,x∈Ω.Here p,q>0, u0(x), v0(x)∈C(Ω¯) are nonnegative and nontrivial functions, J:Rn→R is a nonnegative, smooth, symmetric radially and strictly decreasing kernel, with ∫RnJ(x)dx=1 and supported in the unitary ball, and Ω⊂RN (N≥1) is a bounded connected and smooth domain.In the first equation, we are imposing that the diffusion takes place only inΩ; 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 RN but we impose that u vanishes outside Ω. 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(2)utx,t=Δu+vpx,t,x∈Ω,t>0vtx,t=∫RNJx-yvy,t-vx,tdy+uqx,t,x∈Ω,t>0∂u∂ηx,t=0,x,t∈∂Ω×0,T;vx,t=0,x∉Ω,t>0ux,0=u0x,vx,0=v0x,x∈Ω.Again, we assume that p,q>0, u0(x),v0(x)∈C(Ω¯) are nontrivial functions, J is as before, and Ω⊂RN (N≥1) is a bounded connected and smooth domain with η 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(u,v) blows up in finite time if and only if a finite time T>0 exists such that (3)limt↗T⁡sup⁡ux,tL∞Ω+vx,tL∞Ω=∞.If solutions are well defined and finite for every t∈(0,T) with T=∞ the solution (u,v) is global, i.e., the solution exists for all t≥0.For both problems our results can be summarized as follows.Theorem 1.
For(u0,v0)∈C(Ω¯)×C(Ω¯) positive real functions, there exists a unique positive solution (u,v). Moreover, (u,v) can be extended to a maximal interval [0,T) with T≤∞.
A comparison principle between super- and subsolutions holds.
Ifpq>1, the solution (u,v) blows up in finite time T, while for pq≤1 the solution (u,v) exists globally.
For solutions that attain their maximum atx=0 for every t (for example, for radially symmetric nondecreasing solutions), the blow-up rates are give by (4)u0,t~T-t-p+1/pq-1v0,t~T-t-q+1/pq-1.


## 1.3. Previous Results

Nonlocal diffusion equations of the form(5)utx,t=J∗u-ux,t=∫RnJx-yuy,tdy-ux,t,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 u at a point x and time t depends not only on u(x,t), but also on all the values of u in a neighborhood of x through the convolution term J∗u. This equation shares many properties with the classical heat equation ut=Δu such as the fact that bounded stationary solutions are constant, a maximum principle holds for both of them, and even if J 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(6)utx,t=∫RNJx-yuy,t-ux,tdy,x∈Ω,t>0,ux,t=0,x∉Ω,t>0,ux,0=u0x,x∈Ω,In this model they have that diffusion takes place in the whole RN but impose that u vanishes outside Ω. 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(7)utx,t=∫ΩJx-yuy,t-ux,tdy,x∈Ω,t>0ux,0=u0x,x∈Ω.Since we are now integrating in Ω, the diffusion takes place only in Ω. The individuals may not enter nor leave Ω. 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(8)utx,t=∫ΩJx-yuy,t-ux,tdy+upx,t,x∈Ω,t>0,ux,0=u0x,x∈Ω.They prove that nonnegative and nontrivial solutions blow up in finite time if and only if p>1. Moreover, they find that the blow-up rate is the same as the one that holds for the ODE ut=up, that is, limt→T-⁡T-t1/(p-1)u(·,t)∞=(1/(p-1))1/(p-1).For systems, Bogoya in [9] studied the analogous system to (1) with Neumann boundary conditions.The paper is organized as follows: In Section2, 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(u,v) of (1).We will deal with this issue for a slightly more general system in which the reaction termsuq, vp are replaced by f(v),g(u), two Lipschitz nonnegative functions. Existence and uniqueness will be obtained via Banach’s fixed point theorem. Let t0>0 be fixed and consider (9)Xt0=C0,t0:CΩ¯×CΩ¯=u,v:0,t0→CΩ¯×CΩ¯:u,viscontinuosinspaceandtime,that is, a Banach space with the norm (10)u,v=max0≤t≤t0⁡u·,t,v·,tI,I=L∞Ω¯×L∞Ω¯and(11)u·,t,v·,tI=maxx∈Ω¯⁡ux,t+maxx∈Ω¯⁡vx,t.Let (12)Pt0=u,v∈Xt0:u≥0,v≥0,that is, a closed subspace of Xt0. In what follows, we will use the notation (13)uκ=max0≤t≤t0⁡u·,tL∞Ω¯.We define the operator(14)ψu0,v0:Pt0→Pt0,asψu0,v0u,v=Tu0u,Sv0v,where(15)Tu0ux,t=∫0t∫ΩJx-yuy,s-ux,sdyds+∫0tfvx,sds+u0xSv0vx,t=∫0t∫RNJx-yvy,s-vx,sdyds+∫0tgux,sds+v0xx∈Ω0x∉Ω.The following Lemma is just the first step in order to show thatψ(u0,v0) has a fixed point in Pt0.Lemma 2.
Letf and g be Lipschitz functions with Lipschitz constants K1 and K2 and (u0,v0) and (w0,z0) two pairs of initial conditions in C(Ω¯)×C(Ω¯) and (u,v),(w,z)∈Pt0. Then, there exists a positive constant C=C(K1,K2,Ω,J) such that(16)ψu0,v0u,v-ψw0,z0w,z≤Ct0u,v-w,z+u0,v0-w0,z0IProof.
For any(x,t)∈Ω¯×[0,t0] we have (17)Tu0ux,t-Tw0wx,t≤u0x-w0x+∫0t∫ΩJx-yuy,s-wy,sdyds+∫0t∫ΩJx-yux,s-wx,sdyds+∫0tfvx,s-fzx,sds≤u0-w0L∞Ω¯+2∫0tu·,s-w·,sL∞Ω¯ds∫ΩJx-ydy+K1∫0tv·,s-z·,s)L∞Ω¯ds≤u0-w0L∞Ω¯+2KΩ+K1tu-wκ, where K=J∞ and K1 is the Lipschitz constant of f.
Analogously, taking into account the fact thatv is zero outside Ω and denoting by K2 the Lipschitz constant for g, we have that (18)Sv0vx,t-Sz0zx,t≤v0-z0L∞Ω¯+2KΩ+K2tv-zκ. Therefore, we have obtained(19)ψu0,v0u,v-ψw0,z0w,z≤2KΩ+K3t0u,v-w,z+u0,v0-w0,z0I, where K3=max{K1,K2}. Then, if C=2KΩ+K3 we have that(20)ψu0,v0u,v-ψw0,z0w,z≤Ct0u,v-w,z+u0,v0-w0,z0I.Next, we show the existence and uniqueness of the solution for functionsf and g that are locally Lipschitz.Theorem 3.
Letf and g be locally Lipschitz functions and (u0,v0)∈C(Ω¯)×C(Ω¯) be nonnegative real functions; then there exists a unique solution (u,v) such that (u,v)∈Pt0. Moreover, (u,v) can be extended to a maximal interval [0,T) with T≤∞.Proof.
We check first thatψ(u0,v0):Pt0→Pt0. Taking (w,z)=(w0,z0)=(0,0) in Lemma 2 we get that ψ(u0,v0)(u,v)∈Pt0. Choose t0 such that Ct0<1. Now taking (w0,z0)=(u0,v0) in Lemma 2 we get that (u0,v0) is a strict contraction in Pt0 and the existence and uniqueness part of the theorem follow from Banach’s fixed point theorem in the interval [0,t0]. To extend the solution to [0,T) we may take as initial conditions u(x,t0),v(x,t0)∈C(Ω) and obtain a solution up to [0,2t0]. Iterating this procedure, we get a solution defined in [0,T).Now, we use the same ideas of the previous analysis to obtain the following results.Corollary 4.
The solution(u,v) depends continuously on the initial data. In fact, let f and g be locally Lipschitz functions and if (u,v) and (w,z) are solutions with initial data (u0,v0) and (w0,z0), respectively, then there exists a constant C~ such that (21)u,v-w,z≤C~u0,v0-w0,z0I.Corollary 5.
( u ( x , t ) , v ( x , t ) ) ∈ P t 0 is a solution of (1) if and only if(22)ux,t=∫0t∫ΩJx-yuy,s-ux,sdyds+∫0tfvx,sds+u0xvx,t=∫0t∫ΩJx-yvy,s-vx,sdyds+∫0tgux,sds+v0xx∈Ω0x∉Ω.Now, we look for existence and uniqueness of solutions to (1).Theorem 6.
Ifp,q≥1 and (u0,v0)∈C(Ω¯)×C(Ω¯) are nonnegative real functions, then there exists a unique solution (u,v) of (1) such that (u,v)∈Pt0.Proof.
The functionsf(v)=vp,g(u)=uq are locally Lipschitz; then by Theorem 3, system (1) has a unique solution (u,v)∈Pt0.For exponents less than one, we have the following existence result.Theorem 7.
Letp,q>0 and (u0,v0)∈C(Ω¯)×C(Ω¯) be nonnegative and bounded functions; then there exists a solution (u,v)∈Pt0 to (1).Proof.
The existence of the solution of (1) is obtained with an approximation procedure. We assume that 0<p<1≤q (the other possibilities are left to the reader). Let (fn)n be a sequence of locally Lipschitz functions such that, for n fixed fn(s)=0, if s≤0, fn is nondecreasing and limn→∞⁡fn(s)=sp for s≥0. Consider the system with fn(v) and uq. From Lemma 2 and Theorem 3, we have that there is a unique solution (un(x,t),vn(x,t)). Now, (un(x,t))nand(vn(x,t))n are nondecreasing and bounded sequences. Therefore, passing to the limit as n→∞ using Corollary 5, we get the existence of (u(x,t),v(x,t)), a solution to (1).Remark 8.
For future reference, let us analyze the solution to the ODE system(23)u′t=vpt,v′t=uqtfort>0u0=a≥0,v0=b≥0.
Ifpq>1, the solution to (23) is given by(24)ut=C1pq-1p+1/pq-1p+1q+1p1/pq-1apq-1/p+1-t-p+1/pq-1,vt=C2pq-1q+1/pq-1p+1qq+11/pq-1bpq-1/q+1-t-q+1/pq-1,with (25)C1=p+1q+1ppq-1p+11/pq-1andC2=p+1qq+1pq-1q+11/pq-1.
Ifpq<1, the solution of (23) is(26)ut=a1-pq/p+1+C1tp+1/1-pq,vt=b1-pq/q+1+C2tq+1/1-pq,with (27)C1=1-pqp+1p+1q+1p/p+1andC2=1-pqq+1q+1p+1q/q+1.
Ifpq=1, the solution of (23) is(28)ut=aeC1t,C1=p+1q+1p/p+1;vt=beC2t,C2=q+1p+1q/q+1.
We will use the notation(a,b)≥(c,d) to indicate that a≥c and b≥d.Definition 9.
Letu¯,v¯∈C1([0,T);C(Ω¯)). (u¯,v¯) is called a supersolution of (1) if(29)u¯tx,t≥∫ΩJx-yu¯y,t-u¯x,tdy+v¯px,t,x∈Ω,t>0,v¯tx,t≥∫RNJx-yv¯y,t-v¯x,tdy+u¯qx,t,x∈Ω,t>0,vx,t≥0,x∉Ω,t>0,u¯x,0≥u0x,v¯x,0≥v0x,x∈Ω.Analogously (u_,v_)∈Pt0 is called a subsolution of (1) if it satisfies the opposite inequalities.Lemma 10 (comparison principle).
Let(u_,v_) and (u¯,v¯) be a subsolution and supersolution of (1), respectively. If (u_0(x),v_0(x))≤(u¯0(x),v¯0(x)) for all x∈Ω, then (30)u_x,t,v_x,t≤u¯x,t,v¯x,tfor all (x,t)∈Ω×(0,T).Proof.
Letw(x,t)=u¯(x,t)-u_(x,t), z(x,t)=v¯(x,t)-v_(x,t). Assume first that w(x,0),z(x,0)>0 for x∈Ω. We observe that w and z verify (31)wtx,t≥∫ΩJx-ywy,t-wx,tdy+v¯px,t-v_px,tv¯x,t-v_x,tzx,t,ztx,t≥∫ΩJx-yzy,t-zx,tdy+u¯qx,t-u_qx,tu¯x,t-u_x,twx,t. Now, set 0<δ=min⁡wx,0,zx,0 and suppose that the conclusion of the Lemma does not hold. Thus, let t1 be the first time such that (32)δ2=min⁡wx,t1,zx,t1.We can assume that w attains the previous minimum. At that time, there must be a point x1∈Ω such that w(x1,t1)=δ/2. But, on the one hand wt(x1,t1)≤0 and, on the other hand (33)wtx1,t1≥∫ΩJx1-ywy,t-wx1,t1dy+v¯px1,t1-v_px1,t1v¯x1,t1-v_x1,t1zx1,t1≥pηp-1x1,t1zx1,t1>0,where v_(x1,t1)<η(x1,t1)<v¯(x1,t1). 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(u¯,v¯)∈Pt0 be a supersolution of (1). Then, if (u0,v0)≥(0,0) for x∈Ω we have that (u¯(x,t),v¯(x,t))≥(0,0) for all (x,t)∈Ω×(0,T) and moreover, a strict inequality holds if (u0,v0) is positive.The following lemma gives us the existence of a maximal solution of (1) when pq<1. Its proof is analogous to those given in [10]; therefore we omit the details here.Lemma 12.
Let0<pq<1. Then there exists (uM(x,t),vM(x,t)), a maximal solution of (1), in the sense that if (u(x,t),v(x,t)) is any other solution there holds(34)ux,t≤uMx,t,vx,t≤vMx,tinΩ×0,∞.Moreover, uM(x,t)>0, vM(x,t)>0 for all (x,t)∈Ω×(0,∞).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(u0,v0)=(0,0).(1)
Ifpq≥1, then the unique solution of (1) is (u,v)=(0,0).(2)
Ifpq<1, then there exists exactly one solution (u,v) of (1) such that u and v are strictly positives.Theorem 14.
If(u0,v0)≠(0,0), 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.
Letpq>1 and u0,v0∈C(Ω¯) be nonnegative and nontrivial functions. Then, the solution (u,v) of (1) blows up in finite time T.Proof.
Letpq>1 and u0(x),v0(x)∈C(Ω¯) be nonnegative and nontrivial functions and (u,v) solution of (1). We define the functions for t≥0(35)Mt=∫Ωux,tdx,Nt=∫RNϕ1xvx,tdx,where C1≤ϕ1≤C2 is eigenfunction of the nonlocal operator, which is positive in Ω¯; see [7].
Asp≥1, by Holder’s inequality we have that(36)Npt=∫RNϕ1xvx,tdxp≤C2p∫Ωvx,tdxp≤C2pΩp/p′∫Ωvpx,tdx≤C∫Ωvpx,tdxwith 1/p+1/p′=1. Now, differentiating M with respect to t and using (1) we obtain that(37)M′t=∫Ωutx,tdx=∫Ωvpx,tdx≥CNpt. As q≥1, by Holder’s inequality we have that(38)∫Ωϕ1xuqx,tdx≥C1∫Ωuqx,tdx≥C1Ωq/q′∫Ωux,tdxq=CMqt.Now, taking the derivative of N with respect to t we have that(39)N′t=∫RNϕ1xvtx,tdx=∫RNϕ1x∫RNJx-yvy,t-vx,tdy+∫RNϕ1xuqx,tdx=∫RNvx,tϕ1x∫RNJx-yϕ1y-ϕ1xdy+∫RNϕ1xuqx,tdx=-λ1∫RNϕ1xvx,tdx+∫Ωϕ1xuqx,tdx≥-λ1Nt+CMqt. Summarizing, we get that(40)M′t≥CNpt,N′t≥-λ1Nt+CMqt. If u0,v0 are large, then M,N blow up in finite time; therefore (u,v) blow up in finite time.
In general,(u,v) blow up in finite time for any initial data u0, v0. In fact, if we consider u_(x,t)=a(t),v_(x,t)=b(t)ϕ1(x) for x∈Ω, t≥0, with a(t), b(t) positive functions, then(41)a′t=c1bpt,b′t=-λ1bt+c2aqt.Then, it can be obtained that (a(t),b(t)) (and then (u,v)) blows up in finite time.Theorem 16.
Letpq≤1 and u0(x),v0(x)∈C(Ω¯) be nonnegative and nontrivial functions. Then, the solution (u,v) of (1) exists globally.Proof.
Considera,b such that (u0,v0)≤(a,b). Since pq≤1, by Remark 8, we have that (u¯(t),v¯(t)); the solution of (23) with a≥u0,b≥v0 is globally defined. Notice that (u¯(t),v¯(t)) is a supersolution of (1). If (u(x,t),v(x,t)) is a solution of (1) with initial conditions (u0(x),v0(x)), then, by Lemma 10 (comparison principle), we have that u(x,t)≤u¯(t) and v(x,t)≤v¯(t) in Ω×(0,T). Therefore, any solution of (1) can be continued for all times in the case pq≤1.Next, we analyze the blow-up rate of the solutions of (1). We assume that x=0∈Ω and note that for smooth radially symmetric and nondecreasing initial conditions (that is, when u0(r),v0(r) are C1 such that u0′(r)≤0, v0′(r)≤0) the solutions are also radially symmetric and radially nondecreasing (that is, it holds that ur(r,t)≤0 and vr(r,t)≤0). Hence, for every t∈(0,T), the maximum of both components is attained at x=0. We state this result as follows.Lemma 17.
IfΩ=B(0,R) is a ball and (u0,v0) are smooth, radially symmetric, and nondecreasing initial conditions (i.e. u0(r),v0(r) are C1 such that u0′(r)≤0, v0′(r)≤0), then both components of the solution are radially symmetric and radially nondecreasing (they verify ur(r,t)≤0 and vr(r,t)≤0 for every r∈[0,R) and every t>0).Proof.
For a proof we refer to Lemma 4.1 in [8].Theorem 18.
Forpq>1, let (u,v) be a positive solution to (1) such that the maximum is attained at x=0 for every t∈(0,T). Then, there exist C1, C2, C3, C4 positive constants such that(42)C1T-t-p+1/pq-1≤u0,t≤C2T-t-p+1/pq-1C3T-t-q+1/pq-1≤v0,t≤C4T-t-q+1/pq-1,where T stands for the blow-up time of the solution.Proof.
Aspq>1, we have that (u,v), the solution of (1), blows up in finite time (that we called T). We assumed that (43)u0,t=maxx∈Ω¯⁡ux,tandv0,t=maxx∈Ω¯⁡vx,t.By (1), we have that(44)ut0,t=∫ΩJ0-yuy,t-u0,tdy+vp0,t≤vp0,tvt0,t=∫RNJ0-yvy,t-v0,tdy+uq0,t≤uq0,t,As (45)1=∫RNJζdζ≥∫ΩJζdζ,we have that(46)ut0,t=∫ΩJ0-yuy,t-u0,tdy+vp0,t≥-u0,t+vp0,tvt0,t=∫RNJ0-yvy,t-v0,tdy+uq0,t≥-v0,t+uq0,t.Therefore, we get that for all 0<t<T(47)-u0,t+vp0,t≤ut0,t≤vp0,tand(48)-v0,t+uq0,t≤vt0,t≤uq0,t.Multiplying the second inequality of (47) by uq(0,t) and the first inequality of (48) by vp(0,t), we obtain (49)ut0,tuq0,t≤vt0,tvp0,t+vp+10,t,which is equivalent to (50)uq+1q+1t0,t≤vp+1p+1t0,t+vp+10,t.Multiplying the inequality by (p+1)e(p+1)t and integrating in [0,t] with t<T, we have that(51)uq0,t≤q+1ep+1tvp+10,t+Cq/q+1≤q+1ep+1Tvp+10,t+Cq/q+1≤Cv0,tp+1q/q+1.Replacing the second inequality of (48) by inequality (51), we get (52)vt0,t≤Cv0,tp+1q/q+1. Integrating this inequality on [t,T), we finally obtain (53)v0,t≥C3T-t-β,withβ=q+1pq-1.In an analogous way we obtain that (54)u0,t≥C1T-t-α,withα=p+1pq-1. With an analysis similar to the one developed above, we obtain that there exists a constant C>0 such that for 0<t<T(55)Cv0,tp+1q/q+1≤uq0,t.Replacing the first inequality of (48) by inequality (55) and as pq>1 we have that (p+1)q/(q+1)>1 and (56)Cv0,tp+1q/q+1≤-v0,t+Cv0,tp+1q/q+1≤vt0,t. Integrating the inequality from above on [t,T), we obtain that (57)v0,t≤C4T-t-β. In an analogous way we get (58)u0,t≤C2T-t-α.


## 4. Local and Nonlocal Diffusion

In this section, we analyze the existence and uniqueness of nonnegative solutions(u,v) of (2).As in Section2, we will initially study the system with g(u) and f(v) replacing uq, vp, with f, g being nonnegative Lipschtiz functions. Existence and uniqueness will be again obtained via Banach’s fixed point theorem. Let t0>0 be fixed; we consider the Banach space Xt0 with the norm (59)u,v=max0≤t≤t0⁡u·,t,v·,tI,and Pt0 a closed subspace of Xt0 already considered in Section 2.We define the operatorϖ(u0,v0):Pt0→Pt0, by (60)ϖu0,v0u,v=φu0u,ϕv0v,where(61)φu0ux,t=Stu0+∫0tSt-sfvx,sdsϕv0vx,t=∫0t∫RNJx-yvy,s-vx,sdyds+∫0tgux,sds+v0xx∈Ω0x∉Ω,where for any function h∈L1(Ω)(62)Sth=∫ΩGx,y,thydywhere G is Green’s function. That is, S(t)h denote the solution of (see [11])(63)ut-Δu=0x,t∈Ω×0,T∂u∂ηx,t=0x,t∈∂Ω×0,T;ux,0=hx,x∈Ω,Now, we need the following lemma.Lemma 19.
Let(u0,v0),(w0,z0)∈C(Ω¯)×C(Ω¯) and (u,v),(w,z)∈Pt0. Then, there exists a positive constant C=C(K1,K2,Ω,J) such that(64)ϖu0,v0u,v-ϖw0,z0w,z≤Ct0u,v-w,z+u0,v0-w0,z0I.Proof.
For any(x,t)∈Ω¯)×[0,t0] we have(65)φu0ux,t-φw0wx,t≤Stu0-w0+∫0tSt-s(fvx,s-fzx,sds≤C1u0-w0L∞Ω¯+C2K1tu-wκ, where K1 is the Lipschitz constant of f.
Taking into account thatv is zero outside of Ω, we have that (66)ϕv0vx,t-ϕz0zx,t≤u0-w0L∞Ω¯+2KΩ+K2tu-wκ,where K=J∞ and K2 is the Lipschitz constant of g.
Therefore,(67)ϖu0,v0u,v-ϖw0,z0w,z≤Ct0u,v-w,z+u0,v0-w0,z0I.In the following theorem, we analyze the existence and uniqueness of the solution by considering that the functionsf and g are locally Lipschitz. The proof is analogous to the one for Theorem 3; hence we omit it.Theorem 20.
Letf and g be locally Lipschitz functions and (u0,v0)∈C(Ω¯)×C(Ω¯) are nonnegative real functions; then there exists a unique solution (u,v)∈Pt0. Moreover, (u,v) can be extended to a maximal interval [0,T) with T≤∞.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 Section2; hence we omit them.Theorem 21.
Let(u0,v0)∈C(Ω¯)×C(Ω¯) be nonnegative real functions. (i)
Ifpq≥1 then there exists a unique solution (u,v)∈Pt0.(ii)
Ifpq<1 then there exists a solution (u,v)∈Pt0.Remark 22.
The stationary problem(68)0=Δu+vp,x∈Ω,0=∫RNJx-yvy-vxdy+uq,x∈Ω∂u∂ηx=0x∈Ωvx=0x∉Ωhas a solution u(x)=v(x)=0,x∈Ω.Definition 23.
Letu¯,v¯∈C1([0,T);C(Ω¯)). (u¯,v¯) is called a supersolution of (2) if(69)u¯tx,t≥Δu¯+v¯px,t,x∈Ω,t>0v¯tx,t≥∫RNJx-yv¯y,t-v¯x,tdy+u¯qx,t,x∈Ω,t>0∂u¯∂ηx,t≥δ>0x∈∂Ω,t>0v¯x,t≥0x∉Ω,t>0u¯x,0≥u0x,v¯x,0≥v0x,x∈Ω.Analogously (u_,v_)∈Pt0 is called a subsolution of (2) if it satisfies the opposite inequalities.Lemma 24 (comparison principle).
Let(u_,v_) and (u¯,v¯) be a subsolution and supersolution of (2), respectively. If (u_0(x),v_0(x))≤(u¯0(x),v¯0(x)) for all x∈Ω, then (u_(x,t),v_(x,t))≤(u¯(x,t),v¯(x,t)) for all (x,t)∈Ω×(0,T).


## 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 Theorems15 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.
Letu0,v0∈C(Ω¯) be nonnegative and nontrivial functions.(1)
Ifpq>1, then the solution (u,v) of (2) blows up in finite time T.(2)
Ifpq≤1, then the solution (u,v) of (2) exists globally.Next, we analyze the blow-up rate of the solutions of (2). We assume that x=0∈Ω.Theorem 26.
Letpq>1 and u0(x), v0(x)∈C(Ω¯) be nonnegative and nontrivial functions. Let (u,v) be the solution such that the maximum is reached at x=0 for every t∈(0,T). Then, there exist C1,C2,C3,C4 positive constants such that(70)C1T-t-p+1/pq-1≤u0,t≤C2T-t-p+1/pq-1C3T-t-q+1/pq-1≤v0,t≤C4T-t-q+1/pq-1.Proof.
Aspq>1, we have that (u,v), the solution of (1), blow up in finite time T. Let u(0,t)=maxx∈Ω¯u(x,t) and v(0,t)=maxx∈Ω¯v(x,t). By (1), we have that(71)ut0,t=Δu0,t+vp0,t≤vp0,tvt0,t=∫RNJ0-yvy,t-v0,tdy+uq0,t≤uq0,t,and we have that(72)ut0,t≥δvp0,tvt0,t=∫RNJ0-yvy,t-v0,tdy+uq0,t≥-v0,t+uq0,t.Therefore, we have that for all 0<t<T(73)δvp0,t≤ut0,t≤vp0,tand(74)-v0,t+uq0,t≤vt0,t≤uq0,t.Multiplying the second inequality of (73) by uq(0,t) and the first inequality of (74) by vp(0,t), we have that (75)ut0,tuq0,t≤vt0,tvp0,t+vp+10,t, which is equivalent to (76)uq+10,tq+1t≤vp+10,tp+1t+vp+10,t. From this point the proof follows analogously to what was done in the proof of Theorem 18.



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*Source: 3906431-2018-10-01.xml*