PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL

J. Korean Math. Soc. 47 (2010), No. 6, pp. 1317–1328 DOI 10.4134/JKMS.2010.47.6.1317

PROPERTIES OF POSITIVE SOLUTIONS FOR A NONLOCAL REACTION-DIFFUSION EQUATION WITH NONLOCAL NONLINEAR BOUNDARY CONDITION Chunlai Mu, Dengming Liu, and Shouming Zhou Abstract. In this paper, we study the properties of positive solutions for R the reaction-diffusion equation ut = ∆u+ Ω up dx−kuq in Ω×(0, T ) with R nonlocal nonlinear boundary condition u (x, t) = Ω f (x, y) ul (y, t)dy on ∂Ω×(0, T ) and nonnegative initial data u0 (x), where p, q, k, l > 0. Some conditions for the existence and nonexistence of global positive solutions are given.

1. Introduction In this paper, we deal with the existence and nonexistence of positive global solutions for the following nonlocal equation with nonlocal nonlinear boundary condition  R x ∈ Ω, t > 0,  ut = ∆u +R Ω up dx − kuq , u (x, t) = Ω f (x, y) ul (y, t)dy, x ∈ ∂Ω, t > 0, (1.1)  u (x, 0) = u0 (x) , x ∈ Ω, where Ω is a bounded domain in RN for N ≥ 1 with a smooth boundary ∂Ω, p, q and l are positive parameters, the weight function f (x, y) is nonnegative, continuous and defined for x ∈ ∂Ω, y ∈ Ω, while the initial data u0 (x) ∈ L2 (Ω) is function and satisfies the compatibility condition u0 (x) = R a nonnegative l (y) dy for x ∈ ∂Ω. f (x, y)u 0 Ω Many physical phenomena were formulated into nonlocal mathematical models and studied by many authors (see [2, 3, 16, 20]). In the last few years, a lot of works have been devoted to the study of properties of solutions to parabolic problems involving nonlocal terms. Especially, Wang and Wang [17] considered Received April 4, 2009; Revised June 3, 2009. 2000 Mathematics Subject Classification. 35B35, 35K57, 35K60, 35K65. Key words and phrases. reaction-diffusion equation, global existence, blow-up, nonlocal nonlinear boundary condition. This work is supported in part by NSF of China (10771226) and in part by Natural Science Foundation Project of CQ CSTC (2007BB0124). c °2010 The Korean Mathematical Society

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CHUNLAI MU, DENGMING LIU, AND SHOUMING ZHOU

the following reaction-diffusion equation Z (1.2) ut = d∆u + up dx − kuq Ω

with homogeneous Dirichlet boundary condition and positive initial data. They concluded that the blow-up occurs for large initial data if p > q ≥ 1, and that all solutions exist globally if 1 ≤ p < q. In case of p = q, the issue depends on the comparison between |Ω| and k. In [15], Soufi et al. studied the heat equation of the form R  p p 1 x ∈ Ω, t > 0,  ut = ∆u + |u| − |Ω| Ω |u| dx, ∂u (1.3) = 0, x ∈ ∂Ω, t > 0, R  ∂n u (x, 0) = u0 (x) , Ω u0 (x) dx = 0, x ∈ Ω, where 1 < p ≤ 2. Using energy method and Gamma-convergence technique, they concluded that all solutions blow up in a finite time if the energy of u0 is nonpositive. Recently, Jazar and Kiwan [10] generalized the above result, they showed that the solution of (1.3) blow up in a finite time for all p > 1 while the initial energy is nonpositive. On the other hand, parabolic equations with nonlocal boundary conditions are also encountered in other physical applications. For instance, in the study of the heat conduction within linear thermoelastcity, Day [4, 5] investigated a heat equation subject to the following boundary conditions Z L Z L u (−L, t) = f1 (x)u (x, t) dx, u (L, t) = f2 (x)u (x, t) dx. −L

−L

Friedman [8] generalized Day’s result to a general parabolic equation (1.4)

ut = ∆u + g (x, u) , x ∈ Ω, t > 0,

which is subjected to the following nonlocal boundary condition Z (1.5) u (x, t) = f (x, y) u (y, t) dy. Ω

He established the global existence of solution and discussed its monotonic decay property, and then proved that maxΩ |u (x, t)| ≤ ke−ξt under some hypotheses on f (x, y) and g (x, u). In addition, parabolic equations with both nonlocal source and nonlocal boundary condition have been studied as well. Such as, Lin and Liu [12] considered the problem of the form Z (1.6) ut = ∆u + g (u) dx, Ω

which is subjected to boundary condition (1.5) . They established local existence, global existence and nonexistence of solutions and discussed the blow-up properties of solutions. Furthermore, they derived the uniform blow-up estimate for some special g (u).

PROPERTIES OF POSITIVE SOLUTIONS

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In particular, Wang et al. [19] studied problem (1.2) with nonlocal boundary condition (1.5) . They obtained the conditions for existence and nonexistence of global solution. Moreover, they established the precise estimate of the blow-up rate under some suitable hypotheses. However, reaction-diffusion equations coupled with nonlocal nonlinear bounR dary condition, such as u (x, t) = Ω f (x, y) ul (y, t) dy, to our knowledge, has not been well studied. Very recently, Gladkov and Kim [9] considered the following semilinear heat equation  x ∈ Ω, t > 0,  ut = ∆u +R c (x, t) up , u (x, t) = Ω f (x, y, t) ul (y, t) dy, x ∈ ∂Ω, t > 0, (1.7)  u (x, 0) = u0 (x) , x ∈ Ω, where p, l > 0. They obtained some criteria for the existence of global solution as well as for the solution to blow up in a finite time. For other works on nonlocal problems, we refer readers to [13, 14, 18] and references therein. Motivated by those of above works, we will get blow-up criteria for problem (1.1) with nonlocal nonlinear boundary, which are not only different from situations with the null Dirichlet boundary condition, but also different from situations with boundary condition (1.5). We will show that the weight function f (x, y) and the nonlinear term ul (y, t) in the boundary condition of (1.1) play substantial roles in determining blow-up or not of solution. In order to state our results, we introduce some useful symbols. Throughout this paper, we let λ and ϕ (x) be the first eigenvalue and the corresponding normalized eigenfunction of the problem (1.8)

−∆ϕ (x) = λϕ, x ∈ Ω; ϕ (x) = 0, x ∈ ∂Ω,

then

Z λ > 0, ϕ (x) > 0 and

ϕ (x)dx = 1. Ω

For convenience, we denote L = sup ϕ (x) , M1 = inf f (x, y) , M2 = sup f (x, y) . Ω

∂Ω×Ω

∂Ω×Ω

The main results of this paper are stated as follows. Theorem 1.1. Assume that p < q and l ≤ 1. Then, the problem (1.1) has global solutions for any f (x, y) and any nonnegative initial data. 1 1 Theorem 1.2. Assume that max {p, l} > q ≥ 1 and min{ λM L , L } > k. Then for any f (x, y) > 0, the solution R of problem (1.1) blows up in a finite time if the initial data u0 (x) satisfies Ω u0 (x) ϕ (x) dx > 1.

Remark 1.3. In the special case of k = 0 in (1.1) , our results are still true and consistent with those in [9].

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Theorem 1.4. Assume that p = q > 1. Then the problem (1.1) has blow-up solutions in a finite time as well as global solutions. More precisely, (i) if u0 (x) is large enough, then for any f (x, y) ≥ 0, the solution blows up in a finite time. R (ii) if l ≥ 1 and Ω f (x, y)dy < 1, the solution exists globally when u0 (x) ≤ ρψ (x) for some ρ > 0, where ψ (x) is defined in (5.2). R Remark 1.5. If p = 1 or p = q = 1, l > 1 and Ω f (x, y)dy < 1, then there exist positive solutions of (1.1) with sufficiently small initial data, which are globally bounded. Remark 1.6. If q = 1 or p = q = 1, l > 1, then the solution of problem (1.1) blows up in a finite time for any f (x, y) > 0 provided that 1 µ ¶− l−1 Z λM1 u0 (x) ϕ (x) dx > . (λ + k) L Ω R Remark 1.7. If p = q = l = 1, |Ω| > k and Ω f (x, y)dy < 1, it is obvious that the problem has no blow-up solution. In fact, it is easy to verify that v (t) = αeβt is a supersolution of (1.1) if α ≥ maxx∈Ω u0 (x) and β ≥ |Ω| − k. Remark 1.8. When l = 1 in (1.1) , then our results agree with those in [19]. 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 1.4 will be proved in Section 5. 2. Comparison principle and local existence Let ΩT = Ω × (0, T ), ST = ∂Ω × (0, T ) and ΩT = Ω × [0, T ). We begin with the precise definition of a weak solution of problem (1.1) and comparison principle, which will be used in the sequel. ¡ ¢ ¡ ¢ Definition 2.1. A function u ∈ L2 0, T ; H 1 (Ω) , with u0 ∈ L2 0, T ; H −1 (Ω) , is a weak solution of the problem (1.1) if and only if u (x, t)|t=0 = u0 (x) for all x ∈ Ω, and the equality ZZ ZZ ZZ ut φdxdt + ∇u · ∇φdxdt + uφdSdt ΩT ΩT ST µZ ¶ ·Z ¸ ZZ ZZ p q l = φ u dx − ku dxdt + φ f (x, y)u (y, t) dy dSdt ΩT

Ω 2

holds for all test function φ ∈ L

¡

ST 1

¢

Ω

0, T ; H (Ω) .

In a natural way the notion of a subsolution for (1.1) is given by:


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¡ ¢ ¡ ¢ Definition 2.2. A function u ∈ L2 0, T ; H 1 (Ω) , with u0 ∈ L2 0, T ; H −1 (Ω) , is a weak subsolution of the problem (1.1) if and only if u (x, t)|t=0 ≤ u0 (x) for all x ∈ Ω, and the inequality ZZ ZZ ZZ ut φdxdt + ∇u · ∇φdxdt + uφdSdt ΩT ST ΩT µZ ¶ ·Z ¸ ZZ ZZ ≤ φ up dx − kuq dxdt + φ f (x, y)ul (y, t) dy dSdt ΩT

Ω

ST

Ω

¡ ¢ holds for all test function 0 ≤ φ ∈ L2 0, T ; H 1 (Ω) . Similarly, a function u (x, t) is a supersolution of (1.1) if the reversed inequalities hold in Definition (2.2) . A weak solution of (1.1) is a function which is both a subsolution and a supersolution of (1.1). The following comparison principle plays a crucial role in our later proof. Proposition 2.3 (Comparison principle). Let u and u be a positive subsolution and supersolution, respectively, with u (x, 0) ≤ u (x, 0) for x ∈ Ω. Then, u ≤ u in ΩT . Proof. We will modify the method in [1] to prove our result. Step 1. First assume that p, q, l ≥ 1. Let us denote n o c = max kuk ∞ M , kuk ∞ L (ΩT ) L (ΩT ) . Then we have cq−1 ku − uk 2 kuq − uq kL2 (ΩT ) ≤ q M L (ΩT ) ,

(2.1)

and similarly, we have °Z ° ° ° p p ° ° (2.2) ° (u − u ) dx° Ω

L2 (ΩT )

cp−1 ku − uk 2 ≤ p |Ω| M L (ΩT ) .

Consequently

(2.3)

° °Z ° ° ° (up − up ) dx−k (uq − uq )° ° 2 ° Ω L (ΩT ) ³ ´ p−1 q−1 c c ≤ p |Ω| M +qk M ku − ukL2 (ΩT ) ,

where |Ω| denotes the Lebesgue measure of Ω. Let ω (x, t) = u (x, t) − u (x, t) and ω + = max {ω, 0}. Then ¡ ¢ ω + ∈ L2 0, T ; H 1 (Ω) ,

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and since ω (x, 0) ≤ 0, it follows that ω + (x, 0) = 0. Subtracting the defined inequalities for u and u from each other we get ZZ ZZ ZZ ωt φdxdt + ∇ω · ∇φdxdt + ωφdSdt Ω ΩT S ¸ T Z Z T ·Z ≤ φ (up − up ) dx − k (uq − uq ) dxdt (2.4) ΩT Ω ·Z ¸ ZZ ¡ l ¢ l + φ f (x, y) u (y, t) − u (y, t) dy dSdt ST

Ω

¡ ¢ for all 0 ≤ φ ∈ L 0, T ; H 1 (Ω) . Thus inequality (2.4) remains true for any subcylinder of the form Ωτ = Ω×(0, τ ) ⊂ ΩT and corresponding lateral boundary Sτ = ∂Ω × (0, τ ) ⊂ ST . Taking a special test function φ = ω + in (2.4) and applying (2.3) to (2.4) , we find that ° ° °2 ° °2 1° °ω + (x, τ )°2 2 + °∇ω + °L2 (Ω ) + °ω + °L2 (S ) L (Ω) τ τ 2³ ´° ° 2 cp−1 + qk M cq−1 °ω + ° 2 ≤ p |Ω| M (2.5) L (Ωτ ) ¸ ·Z ZZ ¢ ¡ l l + + ω f (x, y) u (y, t) − u (y, t) dy dSdt. 2

Sτ

Ω

Next, our task is to estimate the second term on the right-side of (2.5) . Indeed, we first have Z ¡ ¢ f (x, y) ul (y, t) − ul (y, t) dy ZΩ ¢ ¡ = f (x, y) ul (y, t) − ul (y, t) dy Ω1 Z ¡ ¢ (2.6) + f (x, y) ul (y, t) − ul (y, t) dy Ω2 Z cl−1 ≤ lM2 M ω + (y, t)dy Ω

° ° cl−1 °ω + (x, t)° 2 , ≤ lM2 |Ω| M L (Ω) 1 2

where Ω1 = {y ∈ Ω : u (y, t) < u (y, t)} , Ω2 = {y ∈ Ω : u (y, t) ≥ u (y, t)} . Then by virtue of Young’s inequality, we deduce ·Z ¸ ZZ ¡ ¢ ω+ f (x, y) ul (y, t) − ul (y, t) dy dSdt ST Ω ZZ ° ° + 1 cl−1 °ω (x, t)° 2 ω + dSdt ≤ lM2 |Ω| 2 M L (Ω) (2.7) Z Z Sτ ³ ° °2 ¡ ¢2 ´ 1 cl−1 dSdt C (²) °ω + (x, t)°L2 (Ω) + ² ω + ≤ lM2 |Ω| 2 M Sτ ³ ´ ° ° ° °2 1 cl−1 C (²) |∂Ω| °ω + °2 2 ≤ lM2 |Ω| 2 M + ² °ω + ° 2 L (Ωτ )

L (Sτ )


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for any ² > 0, where C (²) denotes some positive constant depending only on ². Moreover, we see from the trace theorem that ³° ´ ° + °2 ° ° + °2 °ω ° 2 °∇ω + °2 2 °ω ° 2 (2.8) ≤ κ + , L (Sτ ) L (Ωτ ) L (Ωτ ) where κ is a positive constant. Now, let us choose ² sufficiently small such that 1

cp−1 κ² < 1. lM2 |Ω| 2 M From (2.5)-(2.8) , it follows that ° + ° ° °2 °ω (x, τ )°2 2 (2.9) ≤ C °ω + °L2 (Ωτ ) , L (Ω) where C is some positive constant. Now, we write ° °2 (2.10) y (τ ) = °ω + (x, τ )°L2 (Ω) , then, (2.9) implies that Z (2.11)

y (τ ) ≤ C

τ

y (t)dt for a.e. 0 ≤ τ ≤ T. 0

By Gronwall’s inequality, we know that y (τ ) = 0 for any τ ∈ [0, T ] . Thus, ω + = 0, this means that u ≤ u in ΩT as desired. Step 2. Consider now the case that p, q, l < 1, since u and u are positive, there exists a constant µ > 0 such that u ≥ µ > 0, u ≥ µ > 0. Therefore, we have the following estimate °Z ° ° ° ° (up − up ) dx − k (uq − uq )° ° ° 2 Ω (2.12) L (Ω) ¡ ¢ ≤ p |Ω| µp−1 + qkµq−1 ku − ukL2 (Ω) , and

Z

(2.13) Ω

° ° ¡ ¢ 1 f (x, y) ul (y, t) − ul (y, t) dy ≤ lM2 |Ω| 2 µl−1 °ω + (x, t)°L2 (Ω) .

Then, the left arguments are the same as those for the case p, q, l ≥ 1, so we omit them. Step 3. If p < 1, or q < 1, or l < 1. According to Steps 1 and 2, we can obtain our conclusion easily. The proof of Proposition 2.3 is complete. ¤ Remark 2.4. In [11], if u and u are a subsolution and supersolution for the R corresponding problem, respectively. When Ω lf (x, y)χl−1 (y, t) dy ≤ 1 or f (x, y)χl−1 (y, t) ≤ C, where C denotes some positive constant and χ is an intermediate value between u and u, then u (x, 0) ≤ u (x, 0) implies u ≤ u in ΩT . From Proposition 2.3, we know that u (x, 0) ≤ u (x, 0) implies u ≤ u in ΩT and we have no restriction on f (x, y) χl−1 (y, t) here.

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Local in time existence of classical solutions of the problem (1.1) could be obtained by using Schauder’s fixed point theorem, the representation formula and the contraction mapping principle as in [7]. The proof is more or less standard, so is omitted here. From comparison principle, we know that the classical solution is positive when u0 (x) is positive. We assume that u0 (x) > 0 in the rest of this paper. 3. Proof of Theorem 1.1 Proof of Theorem 1.1. Remember that λ and ϕ are the first eigenvalue and the corresponding normalized eigenfunction of −∆ with homogeneous Dirichlet boundary condition. We choose δ to satisfy that for some 0 < ε < 1, Z 1 (3.1) M2 dy ≤ 1. δϕ (y) +ε Ω Let ceγt (3.2) v (x, t) = , δϕ (x) + ε where ( 1 ) ¸ q−p · q Z (δϕ + ε) 1 c = max sup (u0 (x) + 1) (δϕ + ε) , sup , p dx k Ω (δϕ + ε) Ω Ω γ ≥ λ + sup Ω

2δ 2 |∇ϕ| (δϕ + ε)

2

2.

A simple computation shows Z v t − ∆v − v p dx + kv q Ã Ω ! Z (3.3) 2 cp epγt kcq eqγt λδϕ 2δ 2 |∇ϕ| = γv − v − dx + + p q ≥ 0, δϕ + ε (δϕ + ε)2 (δϕ + ε) Ω (δϕ + ε) sup (u0 (x) + 1) (δϕ (x) + ε) 1 ≥ Ω > u0 (x) . δϕ + ε δϕ + ε On the other hand, noticing that v (x, t) > 1 and l < 1, we have on the boundary that Z Z ceγt ceγt v (x, t) = > ceγt ≥ f (x, y) dy = f (x, y) v (y, t) dy ε δϕ (y) + ε Ω Ω Z (3.5) ≥ f (x, y) v l (y, t) dy. (3.4)

v (x, 0) =

Ω

Combining now (3.3)-(3.5), we see that v (x, t) is a supersolution of (1.1) and u (x, t) < v (x, t) by comparison principle, then the problem (1.1) has global solutions. The proof of Theorem 1.1 is complete. ¤


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4. Proof of Theorem 1.2 Proof of Theorem 1.2. The proof is a variant of the eigenfunction method like the one used in [9]. Let u (x, t) be the solution to (1.1). We define the following auxiliary function Z (4.1) J (t) = ϕ (x)u (x, t) dx. Ω

Taking the derivative of J (t) with respect to t, we could obtain ¶ µ Z Z up dx − kuq dx J 0 (t) = ϕ ∆u + Ω ZΩ Z Z Z ∂u = up dx−k ϕuq dx ∇ϕ · ∇udx + ϕdS − ∂Ω ∂ν Ω Ω Ω Z Z Z Z ∂ϕ p = u∆ϕdx − udS + u dx−k ϕuq dx Ω ∂Ω ∂ν Ω Ω µZ ¶ Z Z Z Z ∂ϕ l p = −λ uϕdx− f (x, y) u (y, t) dy dS + u dx−k ϕuq dx. Ω ∂Ω ∂ν Ω Ω Ω R ∂ϕ R Applying the equality ∂Ω ∂ν dS = −λ Ω ϕdx = −λ, we get ¶ Z µ 1 p λM1 l 0 q (4.2) J (t) ≥ −λu + u + u − ku ϕdx. L L Ω Let us first assume that max {l, p} = l. From (4.2) and Jensen’s inequality, it follows that µ ¶ λM1 l λM1 (4.3) J 0 (t) ≥ −λJ + J − kJ q ≥ −λJ + − k J l − k. L L Next, we look for solution J (t) to (4.3) with J (0) > 1 on its interval of exisl 1 tence. Since λM L − k > 0 and the function f (J) = J is convex, there exists η > 1 such that µ ¶ λM1 − k J l ≥ 2 (λJ + k) , ∀J ≥ η. L It follows easily that if J (0) > η, then J (t) is increasing on its interval of existence and 1 (4.4) J 0 (t) ≥ J l . 2 From the above inequality it follows that (4.5)

lim J (t) = +∞,

t→T0−

where

2 . (l − 1) J l−1 (0) Then by assumptions in Theorem 1.2, the solution u (x, t) becomes infinite in a finite time. T0 =

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We next consider the case p > q ≥ 1. Owing to (4.2) and Jensen’s inequality, we get µ ¶ 1 0 (4.6) J (t) ≥ −λJ + − k J l − k. L Then, since the remainder of the proof is similar to the proof in the case of l > q ≥ 1, we omit here. This completes the proof. ¤ 5. Proof of Theorem 1.4 Proof of Theorem 1.4. Firstly, in order to prove our blow-up result, we consider the following well-known nonlocal reaction-diffusion equation Z (5.1) ut = ∆u + up dx − kuq Ω

coupled with zero boundary condition and initial data u0 (x). Let v (x, t) be the solution of this equation. It is obvious that v (x, t) is a subsolution of the problem (1.1). It is known to all that v (x, t) blows up in a finite time if u0 (x) is large enough (see [17, Theorem 3.3]), by Proposition 2.3, we obtain our blow-up result immediately. R Now, we show there exists global solutions if l > 1 and Ω f (x, y)dy < 1. Let ψ (x) be the unique positive solution of the linear elliptic problem Z (5.2) −∆ψ(x) = σ > 0, x ∈ Ω; ψ(x) = f (x, y)dy, x ∈ ∂Ω, Ω R where σ is chosen such that 0 < ψ (x) < 1 (since Ω f (x, y)dy < 1, there exists such a positive constant σ). Let v (x) = ρψ (x) , where ( µ 1 ) ¶ p−1 σ 0 < ρ ≤ min 1, R p . ψ (x) dx − kψ p (x) Ω Calculating directly, we find that

(5.3)

µZ ¶ vt − ∆v = −∆v = σρ > ρp ψ p (x) dx − kψ p (x) Ω Z p = v (x) dx − kv p (x). Ω

For x ∈ ∂Ω, we have that Z Z Z (5.4) v (x) = ρ f (x, y) dy > ρψ (y) f (x, y) dy ≥ v l (y) f (x, y) dy, Ω

Ω

Ω

where the conditions v (x) < 1 and l ≥ 1 are used. By Proposition 2.3, it follows that u (x, t) exists globally provided that u0 (x) < ρψ (x). The proof of Theorem 1.4 is complete. ¤


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Chunlai Mu College of Mathematics and Statistics Chongqing University Chongqing 400044, P. R. China E-mail address: [email protected] Dengming Liu College of Mathematics and Statistics Chongqing University Chongqing 400044, P. R. China E-mail address: [email protected] Shouming Zhou College of Mathematics and Statistics Chongqing University Chongqing 400044, P. R. China E-mail address: [email protected]