Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

and Applied Analysis 3 1, 2, 6–9 and the references cited therein . For blowup results for other parabolic systems, we refer the readers to 10–13 and the references cited therein. Moreover, in recent years, many authors see studies such as those in 14, 15 and the references cited therein considered semilinear reaction-diffusion systems with nonlocal Dirichlet boundary conditions of the form ut Δu f u, v , vt Δv g u, v , x ∈ Ω, t > 0, u ∫ Ω φ ( x, y ) u ( y, t ) dy, v ∫ Ω ψ ( x, y ) v ( y, t ) dy, x ∈ ∂Ω, t > 0, u x, 0 u0 x , v x, 0 v0 x , x ∈ Ω. 1.5 They studied how the weight functions φ x, y and ψ x, y in the nonlocal boundary conditions affect the blowup properties of the solutions of 1.5 . However, reaction-diffusion problems coupled with nonlocal nonlinear boundary conditions, to our knowledge, have not been well studied. Recently, Gladkov and Kim 16 considered the following problem for a single semilinear heat equation: ut Δu c x, t u, x ∈ Ω, t > 0, u x, t ∫ Ω f ( x, y, t ) u ( y, t ) dy, x ∈ ∂Ω, t > 0, u x, 0 u0 x , x ∈ Ω, 1.6 where p, l > 0. They obtained some criteria for the existence of the global solution as well as for blowup of the solution in finite time. The main purpose of this paper is to understand how the reaction terms, the weight functions and the nonlinear terms in the boundary conditions affect the blowup properties for problem 1.1 .We will show that the weight functions φ x, y , ψ x, y and the nonlinear terms u y, t , v y, t in the boundary conditions of 1.1 play substantial roles in determining blowup or not of solutions. Before starting the main results, we introduce some useful symbols. Throughout this paper, we let λ be the first eigenvalue of the eigenvalue problem −Δφ x λφ, x ∈ Ω; φ x 0, x ∈ ∂Ω, 1.7 and φ x the corresponding eigenfunction with ∫ Ω φ x dx 1, φ x > 0 in Ω. In addition, for convenience, we denote that L supΩφ x and M1 sup ∂Ω×Ω φ ( x, y ) , M2 inf ∂Ω×Ω φ ( x, y ) , K1 sup ∂Ω×Ω ψ ( x, y ) , K2 inf ∂Ω×Ω ψ ( x, y ) . 1.8 4 Abstract and Applied Analysis The main results of this paper are stated as follows. Theorem 1.1. Assume that 0 < pq ≤ 1 and m,n ≤ 1. Then the solution of problem 1.1 exists globally for any positive initial data. Theorem 1.2. Assume that pq > 1 or m,n > 1. Then for any φ x, y , ψ x, y > 0, the solution of problem 1.1 blows up in finite time for sufficiently large initial data. Theorem 1.3. Assume that p > 1, q > 1, m > 1, and n > 1. Then for any nonnegative continuous φ x, y and ψ x, y , the solution of problem 1.1 exists globally for sufficiently small initial data. Remark 1.4. When p q, m n, φ x, y ψ x, y , and u0 x v0 x , system 1.1 is then reduced to a single equation ut Δu u with nonlocal nonlinear boundary condition u x, t ∫ Ω φ x, y u m y, t dy. In this case, our above results are still true and consistent with those in 16 . The rest of this paper is organized as follows. In Section 2, we establish the comparison principle for problem 1.1 . In Sections 3 and 4, we will give the proofs of Theorems 1.1 and 1.2, respectively. Finally, Theorem 1.3 will be proved in Section 5. 2. Preliminaries In this section, we will give a suitable comparison principle for problem 1.1 . Let ΩT Ω × 0, T , ST ∂Ω × 0, T , and ΩT Ω × 0, T . We begin with the precise definitions of a subsolution and supersolution of problem 1.1 . Definition 2.1. A pair of functions u, v ∈ C2,1 ΩT ∩ C ΩT × C2,1 ΩT ∩ C ΩT is called a subsolution of problem 1.1 in ΩT if ut ≤ Δu v, vt ≤ Δv u, x, t ∈ ΩT , u x, t ≤ ∫ Ω φ ( x, y ) u ( y, t ) dy, x, t ∈ ST , v x, t ≤ ∫ Ω ψ ( x, y ) v ( y, t ) dy, x, t ∈ ST , u x, 0 ≤ u0 x , v x, 0 ≤ v0 x , x ∈ Ω. 2.1 Similarly, a pair of functions u, v ∈ C2,1 ΩT ∩ C ΩT × C2,1 ΩT ∩ C ΩT is a supersolution of system 1.1 if the reversed inequalities hold in 2.1 . We say that u, v is a solution of system 1.1 in ΩT if it is both a subsolution and a supersolution of problem 1.1 in ΩT . Let gi x, t ,hi x, t ∈ C2,1 ΩT ∩ C ΩT , χi x, y ≥ 0 on ∂Ω × Ω, i 1, 2. We first give some hypotheses as follows, which will be used in the sequel. H1 For x ∈ ∂Ω, y ∈ Ω, t > 0, χ1 x, y gm−1 1 y, t , χ1 x, y hm−1 1 y, t , χ2 x, y gn−1 2 y, t , and χ2 x, y hn−1 2 y, t are nonnegative. Further, ∫ Ωmχ1 x, y g m−1 1 y, t dy ≤ 1, Abstract and Applied Analysis 5 ∫ Ωmχ1 x, y h m−1 1 y, t dy ≤ 1, ∫ Ω nχ2 x, y g n−1 2 y, t dy ≤ 1, and ∫ Ω nχ2 x, y h n−1 2 y, t dy ≤ 1. H2 For x ∈ ∂Ω, y ∈ Ω, t > 0, there exists M > 0 such that 0 ≤ mχ1 x, y gm−1 1 y, t ≤ M, 0 ≤ mχ1 x, y hm−1 1 y, t ≤ M, 0 ≤ nχ2 x, y gn−1 2 y, t ≤ M, and 0 ≤ nχ2 x, y hn−1 2 y, t ≤M. Lemma 2.2. Let (H1) hold, and cij dij i, j 1, 2 be bounded inΩT and let cij dij ≥ 0 i / j, i, j 1, 2 . Further, assume that wi x, t ≥ si x, t i 1, 2 . If χi x, y ≥ 0 on ∂Ω × Ω; and gi,hi ∈ C2,1 ΩT ∩ C ΩT i 1, 2 satisfyand Applied Analysis 5 ∫ Ωmχ1 x, y h m−1 1 y, t dy ≤ 1, ∫ Ω nχ2 x, y g n−1 2 y, t dy ≤ 1, and ∫ Ω nχ2 x, y h n−1 2 y, t dy ≤ 1. H2 For x ∈ ∂Ω, y ∈ Ω, t > 0, there exists M > 0 such that 0 ≤ mχ1 x, y gm−1 1 y, t ≤ M, 0 ≤ mχ1 x, y hm−1 1 y, t ≤ M, 0 ≤ nχ2 x, y gn−1 2 y, t ≤ M, and 0 ≤ nχ2 x, y hn−1 2 y, t ≤M. Lemma 2.2. Let (H1) hold, and cij dij i, j 1, 2 be bounded inΩT and let cij dij ≥ 0 i / j, i, j 1, 2 . Further, assume that wi x, t ≥ si x, t i 1, 2 . If χi x, y ≥ 0 on ∂Ω × Ω; and gi,hi ∈ C2,1 ΩT ∩ C ΩT i 1, 2 satisfy g1t − ( n ∑ k,l 1 a 1 k,l ∂g1 ∂xk∂xl n ∑ k 1 b 1 k ∂g1 ∂xk ) ≥ 2 ∑ i 1 c1igi − 2 ∑ i 1 d1ihi w1 x, t , x, t ∈ ΩT , g2t − ( n ∑ k,l 1 a 2 k,l ∂g2 ∂xk∂xl n ∑ k 1 b 2 k ∂g2 ∂xk ) ≥ 2 ∑ i 1 c2igi − 2 ∑ i 1 d2ihi w2 x, t , x, t ∈ ΩT , h1t − ( n ∑ k,l 1 a 1 k,l ∂h1 ∂xk∂xl n ∑ k 1 b 1 k ∂h1 ∂xk ) ≤ 2 ∑ i 1 c1ihi − 2 ∑ i 1 d1igi s1 x, t , x, t ∈ ΩT , h2t − ( n ∑ k,l 1 a 2 k,l ∂h2 ∂xk∂xl n ∑ k 1 b 2 k ∂h2 ∂xk ) ≤ 2 ∑ i 1 c2ihi − 2 ∑ i 1 d2igi s2 x, t , x, t ∈ ΩT , g1 x, t ≥ ∫ Ω χ1 ( x, y ) g 1 ( y, t ) dy, g2 x, t ≥ ∫ Ω χ2 ( x, y ) g 2 ( y, t ) dy, x, t ∈ ST , h1 x, t ≤ ∫ Ω χ1 ( x, y ) hm1 ( y, t ) dy, h2 x, t ≤ ∫ Ω χ2 ( x, y ) hn2 ( y, t ) dy, x, t ∈ ST , g1 x, 0 ≥ h1 x, 0 , g2 x, 0 ≥ h2 x, 0 , x ∈ Ω. 2.2 Then g1 x, t , g2 x, t ≥ h1 x, t , h2 x, t in ΩT . Proof. For any given ε > 0, define g̃i gi εe, h̃i hi − εe, i 1, 2, 2.3


Introduction
In this paper, we deal with the following semilinear reaction-diffusion system with nonlinear nonlocal boundary conditions and nontrivial nonnegative continuous initial data: u t Δu v p , v t Δv u q , x ∈ Ω, t > 0, u x, t Ω ϕ x, y u m y, t dy, x ∈ ∂Ω, t > 0, v x, t Ω ψ x, y v n y, t dy, x ∈ ∂Ω, t > 0, where Ω is a bounded domain in R N for N ≥ 1 with a smooth boundary ∂Ω,p,q,m,n > 0, the weight functions ϕ x, y and ψ x, y are nonnegative continuous defined in ∂Ω × Ω, and Ω ϕ x, y dy, Ω ψ x, y dy > 0 on ∂Ω.Moreover, for x ∈ ∂Ω, the initial data u 0 x , v 0 x satisfy 2 Abstract and Applied Analysis the compatibility conditions u 0 x Ω ϕ x, y u m 0 y dy and v 0 x Ω ψ x, y v n 0 y dy, respectively.System 1.1 has been formulated from physical models arising in various fields of applied sciences.For example, it can be interpreted as a heat conduction problem with nonlocal nonlinear sources on the boundary of the material body see 1, 2 .In this case, u x, t and v x, t represent the temperatures of the interacting components in the evolution processes.
The local in time existence of classical solutions of system 1.1 can be derived easily by standard parabolic theory.We say that the solution u x, t , v x, t of problem 1.1 blows up in finite time if there exists a positive constant T < ∞ such that In this case, T is called the blowup time.We say that the solution u x, t , v x, t exists globally if In the last few years, a lot of efforts have been devoted to the study of properties of solutions to the semilinear parabolic equation u t Δu u p with homogeneous Dirichlet boundary condition see, e.g., the classical works in 3, 4 and to the heat equation u t Δu with Neumann boundary condition ∂u/∂ν u p see, e.g., 5 .
Blowup properties for the problem of systems have been studied very extensively over past years by many researchers.Here p, q > 0, ν denotes the unit outer normal vector on ∂Ω.They were concerned with the existence, uniqueness, and regularity of solutions.Furthermore, they investigated the global and nonglobal existence, the blowup set, and the blowup rate for the above systems see, e.g., Abstract and Applied Analysis 3 1, 2, 6-9 and the references cited therein .For blowup results for other parabolic systems, we refer the readers to 10-13 and the references cited therein.Moreover, in recent years, many authors see studies such as those in 14, 15 and the references cited therein considered semilinear reaction-diffusion systems with nonlocal Dirichlet boundary conditions of the form

1.5
They studied how the weight functions ϕ x, y and ψ x, y in the nonlocal boundary conditions affect the blowup properties of the solutions of 1.5 .However, reaction-diffusion problems coupled with nonlocal nonlinear boundary conditions, to our knowledge, have not been well studied.Recently, Gladkov and Kim 16 considered the following problem for a single semilinear heat equation: where p, l > 0. They obtained some criteria for the existence of the global solution as well as for blowup of the solution in finite time.
The main purpose of this paper is to understand how the reaction terms, the weight functions and the nonlinear terms in the boundary conditions affect the blowup properties for problem 1.1 .We will show that the weight functions ϕ x, y , ψ x, y and the nonlinear terms u m y, t , v n y, t in the boundary conditions of 1.1 play substantial roles in determining blowup or not of solutions.
Before starting the main results, we introduce some useful symbols.Throughout this paper, we let λ be the first eigenvalue of the eigenvalue problem and φ x the corresponding eigenfunction with Ω φ x dx 1, φ x > 0 in Ω.In addition, for convenience, we denote that L sup Ω φ x and The main results of this paper are stated as follows.
Theorem The rest of this paper is organized as follows.In Section 2, we establish the comparison principle for problem 1.1 .In Sections 3 and 4, we will give the proofs of Theorems 1.1 and 1.2, respectively.Finally, Theorem 1.3 will be proved in Section 5.

Preliminaries
In this section, we will give a suitable comparison principle for problem 1.1 .Let Ω T Ω × 0, T , S T ∂Ω × 0, T , and Ω T Ω × 0, T .We begin with the precise definitions of a subsolution and supersolution of problem 1.1 .

Definition 2.1. A pair of functions
We first give some hypotheses as follows, which will be used in the sequel.
Abstract and Applied Analysis 5

2.4
Then, a direct computation yields

2.5
On the other hand, for x, t ∈ S T , we have

2.8
Abstract and Applied Analysis 7 In addition, it is obvious that and hence, we know that Next, our task is to show that η 1 x, t , η 2 x, t > 0, 0 .

2.12
Actually, if 2.12 is true; then we can immediately get which means that g 1 x, t , g 2 x, t ≥ h 1 x, t , h 2 x, t in Ω T as desired.
In order to prove 2.12 , we set where θ 2 is an intermediate value between g 1 and h 1 , θ 3 is an intermediate value between g 2 and h 2 .

2.16
Then η 1 , η 2 ≥ 0 on Ω t , and at least one of η 1 , η 2 vanishes at x, t for some x ∈ Ω.Without loss of generality, suppose that η 1 x, t 0 inf Ω t η 1 .If x, t ∈ Ω t , by virtue of the first inequality of 2.15 , we find that

2.17
This leads us to conclude that η 1 ≡ 0 in Ω t by the strong maximum principle, a contradiction.If x, t ∈ S t , this also results in a contradiction, that is

2.20
Then from 2.2 , we have

2.21
where is a uniformly elliptic operator.By H2 , it is easy to see that

2.23
Similarly, we have Therefore, in view of Lemma 2.2, we have which implies that

2.26
The proof of Lemma 2.3 is complete.
On the basis of the above lemmas, we obtain the following comparison principle for problem 1.1 .

Proposition 2.4
Comparison principle .Let u, v and u, v be a nonnegative supersolution and a nonnegative subsolution of problem 1.1 in Ω T , respectively.Suppose that u, v > 0, 0 and Proof.It is easy to check that u, v, u, v and ϕ, ψ satisfy hypotheses H2 .
Next, we state the local existence theorem, and its proof is standard; hence we omit it.

2.27
Remark 2.6.From maximum principle, we know that the solution of system 1.1 is positive when u 0 x and v 0 x are positive.Indeed, since u t − Δu − v p ≥ 0 and v t − Δv − u q ≥ 0, the minimum of u, v in Ω T should be obtained at a parabolic boundary point by maximum principle.Furthermore, Ω ϕ x, y dy, Ω ψ x, y dy > 0 on ∂Ω imply that ϕ x, t / ≡ 0 and ψ x, t / ≡ 0, then we have u, v > 0, 0 for x, t ∈ ∂Ω× 0, T .Thus u, v > 0, 0 provided that u 0 x and v 0 x are positive.In the rest of this paper, we assume that u 0 x , v 0 x > 0, 0 .
Remark 2.7.If pq ≥ 1, m ≥ 1, and n ≥ 1, we could obtain the uniqueness of the solution easily by comparison principle.

Proof of Theorem 1.1
In this section, by constructing special supersolution, we will give the sufficient condition for the existence of global solution of problem 1.1 under the hypotheses 0 < pq ≤ 1 and m, n < 1.

3.6
Likewise, we also have for v that

3.7
On the other hand, since α < 1, we have and similarly, Therefore, u, v is a global supersolution of 1.1 ; by Proposition 2.4, the solution of 1.1 exists globally.The proof of Theorem 1.1 is complete.

Proof of Theorem 1.2
In this section, we will establish that the solution of system 1.1 blows up in finite time for the case pq > 1 or m, n > 1.We employ a variant of Kaplan's method see 17 for more details to obtain our blowup conclusion.

4.2
Applying the equality ∂Ω ∂φ/∂ν dS −λ to 4.2 , we find that Symmetrically, we deduce that Case 1.For the case pq > 1; we first prove the assertion under the stronger assumption p, q > 1.
Without loss of generality, we assume that q max{p, q} > 1.Then using Jensen's inequality to 4.3 and 4.4 , we see that In view of the inequality J 1 p J 2 p ≥ 2 1−p J p , we discover that It follows that J blows up in finite time whenever is sufficiently large.Furthermore, the solution of system 1.1 blows up in finite time.
If p < 1 or q < 1, in order to obtain our conclusion, we consider system 1.1 with zero Dirichlet boundary condition; then in light of Theorem 2 in 1 , we obtain our result immediately.
Case 2. Consider now the case that m, n > 1.Since m, n > 1, Jensen's inequality can be applied to 4.3 and 4.4 like Step 1 to get 4.9 Then the left arguments are the same as those for Case 1, we omit the details.The proof of Theorem 1.2 is complete.

Proof of Theorem 1.3
In this section, we will use an idea from Gladkov and Kim 16 to prove Theorem 1.3.
Proof of Theorem 1.3.Let Ω 1 be a bounded domain in R N satisfying the property that Ω Ω 1 and let λ 1 be the first eigenvalue of −Δ on Ω 1 with null Dirichlet boundary condition which satisfies the inequality 0 < λ 1 < λ.Since ϕ x, y and ψ x, y are nonnegative continuous defined in ∂Ω × Ω; then there exist some constants 0 < A, B < ∞ such that Ω ϕ x, y dy ≤ A, Ω ψ x, y dy ≤ B.

5.1
Denote φ an eigenfunction corresponding the eigenvalue λ 1 ; then it is obviously that where δ > 1 is some constant.Choosing any ε which satisfies the inequality

5.6
It is easy to check that f t satisfy the following ordinary differential equation: f t λ 1 f − max δε p−1 , δε q−1 f p 0. 5.7 Observe next that f t < 1, and so f p ≥ f q under the condition q ≥ p > 1.Let u x, t v x, t φ x f t .5.8 A series of computations yields u t − Δu − v q φ f λ 1 f − φ q−1 f q ≥ φ f λ 1 f − max δε p−1 , δε q−1 f p ≥ 0.

5.10
On the other hand, since Ω ϕ x, y dy ≤ A, we have on the boundary that u x, t > εf t ≥ A δε m f t ≥ Case 2. For p > q > 1, set max δε p−1 , δε q−1 e − q−1 λ 1 t λ 1

5.14
We can immediately verify that f t satisfy the following ordinary differential equation: f t λ 1 f − max δε p−1 , δε q−1 f q 0. 5.15 In addition, it is obvious that f t < 1.Then we have that f q ≥ f p under the condition p > q > 1.Let u x, t v x, t φ x f t .

5.16
Similar to the arguments for the case q ≥ p > 1, we can prove that u x, t , v x, t is a global supersolution of problem 1.1 provided that max{u 0 x , v 0 x } ≤ ⎡ ⎢ ⎣1 max δε p−1 , δε q−1 λ 1

1.1. Assume
that 0 < pq ≤ 1 and m,n ≤ 1.Then the solution of problem 1.1 exists globally for any positive initial data.In this case, our above results are still true and consistent with those in 16 .
is defined in 1.7 .Taking the derivative of J 1 t with respect to t, we could obtain