Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we consider a semilinear parabolic equation ut = �u + u q Z t 0 u p (x, s)ds, x ∈ , t > 0

From a physical point of view, (1.1) represents the slow-diffusion equations with memory.Problem (1.1) with p = q = 1 and ϕ ≡ 0 appears in the theory of nuclear reactor dynamics (see [13] and the references therein, where a more detailed physical background can be found).Parabolic equations with nonlinear memory and homogeneous Dirichlet boundary conditions have been studied by several authors (see [6], [20], [24], [25], [26] and the references therein).For instance, in [1], Bellout considered the following equation with null Dirichlet boundary condition, where g (x) ≥ 0 is a smooth function and λ > 0.
In [27], Yamada investigated the stability properties of the global solutions of the following nonlocal Volterra diffusion equation Recently, in [14], Li and Xie considered the following equation u t = ∆u + u q t 0 u p ds, x ∈ Ω, t > 0, u (x, t) = 0, x ∈ ∂Ω, t > 0, u (x, 0) = u 0 (x) , x ∈ Ω, (1.4) where p, q ≥ 0. They established the conditions for global and non-global solutions.Moreover, under some appropriate hypotheses, they obtained the blow-up rate estimate for the special case q = 0. On the other hand, parabolic equations with nonlocal boundary conditions are also encountered in other physical applications.For example, in the study of the heat conduction within linear thermoelastcity, in [3], [4], Day investigated a heat equation which is subjected to the following boundary conditions Friedman [7] generalized Day's result to the following general parabolic equation in n dimensions u t = ∆u + g (x, u) , x ∈ Ω, t > 0, (1.5) which is subjected to the following nonlocal boundary condition EJQTDE, 2010 No. 51, p. 2 and studied the global existence of solution and its monotonic decay property under some hypotheses on ϕ (x, y) and g (x, u).
In addition, parabolic equations with both space-integral source terms and nonlocal boundary conditions have been studied as well (see [2], [5], [19], [23] and the references therein).For example, Lin and Liu [16] considered the problem of the form which is subjected to boundary condition (1.6).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).However, to the authors' best knowledge, there is little literature on the study of the global existence and blow-up properties for the reaction-diffusion equations coupled with nonlocal nonlinear boundary condition.Recently, Gladkov and Kim [9] considered the following semilinear heat equation where p, l > 0. They obtained some criteria for the existence of global solution as well as for the solution to blow-up in finite time.
Motivated by those of works above, our main objectives of this paper are to investigate conditions for the occurrence of the blow-up in finite time or global existence and to estimate the blow-up rate of the blow-up solution.Due to the appearance of the nonlocal nonlinear boundary condition, the approaches used in [14] can not be extended to handle our problem (1.1).Meanwhile, our method is very different from those previously used in [16] because the space-integral source term Ω g (u) dx is replaced by time-integral term t 0 u p ds.By a modification of the methods used in [9], we show that the nonlinear memory term t 0 u p (x, s) ds, the weight function ϕ (x, y) and the nonlinear term u l (y, t) in the boundary condition play substantial roles in determining blow-up or not of the solution.
In order to state our 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.
The main results of this paper are stated as follows.
Theorem 1.1.Assume that p + q ≤ 1 and l ≤ 1, then the solution of problem (1.1) exists globally for any nonnegative ϕ and initial data u 0 .
Theorem 1.3.Assume that l > 1, then for any positive ϕ, the solution of problem (1.1) blows up in finite time provided that the initial data u 0 satisfies Ω u 0 (x) φ (x) dx ≥ ̺ > 1 for some ̺, where φ is given by (1.9).
Theorem 1.4.Assume that p + q > 1.If q ≥ 1, then the solution of problem (1.1) blows up in finite time for sufficiently large initial data u 0 .If q < 1, then the solution of problem (1.1) blows up in finite time for any nonnegative initial data u 0 .
Consider problem (1.1) with q = 0 and l = 1.In order to obtain the blow-up rate, we need to add the following assumption on initial data u 0 (assume T * is the blow-up time of the blow-up solution u (x, t) to problem (1.1)): (H1) There exists a constant t 0 ∈ (0, T * ) such that u t (x, t 0 ) ≥ 0 for all x ∈ Ω.
Remark 1.8.In [14], the authors proved the blow-up rate under the additional assumptions Ω = B R and u 0 is radially symmetric decreasing.Motivated by the idea of Souplet in [22], we have no restriction on Ω and u 0 here.
The rest of this paper is organized as follows.In Section 2, we shall establish the comparison principle for problem (1.1).In Section 3, we shall discuss the global existence of the solution and prove Theorems 1.1 and 1.2.In Section 4, we shall discuss the blow-up results of the solution and prove Theorems 1.3 and 1.4.Finally, we shall estimate the blow-up rate and give the proof of Theorem 1.6 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 subsolutions and supersolutions of problem (1.1).EJQTDE, 2010 No. 51, p. 4 (2.1) A supersolution u (x, t) is defined analogously by the above inequalities with "≤" replaced by "≥".We say that u (x, t) is a solution of the problem (1.1) in Ω T if it is both a subsolution and a supersolution of problem First of all, we give some hypotheses on g i (x, t) and χ (x, y) as follows, which will be used in the sequel.
Then h ≥ 0 on Ωt, and there exists at least one point (x, t) such that h (x, t) = 0.If (x, t) ∈ Ωt, by virtue of the first inequality of (2.8) and the strong maximum principle,we conclude that h (x, t) ≡ 0 in Ωt, a contradiction.If (x, t) ∈ St, by (H2), this also results in a contradiction, that This proves h > 0, and in turn g 1 (x, t) ≥ g 2 (x, t) in Ω T .The proof of Lemma 2.2 is complete.
Lemma 2.3.Let the hypotheses of Lemma 2.1, with (H2) replaced by (H3), be satisfied.Then Then from (2.2), we have where is a uniformly elliptic operator.By (H3), it is easy to see that 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).
Proof.It is easy to check that u, u and ϕ satisfy hypotheses (H3).

Global existence of the solution
In this section, we investigate the global existence of the solution to problem (1.1).
Proof of Theorem 1.1.Let T be any positive number.In order to prove our conclusion, according to Proposition 2.4, we only need to construct a suitable gloabl supersolution of problem (1.1) in Ω T .Remember that λ and φ are the first eigenvalue and the corresponding normalized eigenfunction of −∆ with homogeneous Dirichlet boundary condition.We choose ζ to satisfy that for some 0 Now, let v(x, t) be defined as η q e qκt (ζφ + ǫ) q t 0 η p e pκs (ζφ + ǫ) p ds η p+q e (p+q)κt κp (ζφ + ǫ) p+q + η p+q e (p+q)κt κp (ζφ + ǫ) p+q .
Combining now from (3.3) to (3.6), we know that v (x, t) is a supersolution of (1.1) in Ω T and the solution u (x, t) ≤ v (x, t) by comparison principle, therefore the problem (1.1) has global solutions.The proof of Theorem 1.1 is complete.
Proof of Theorem 1.2.Let Ω 1 be a bounded domain in R N such that Ω ⊂⊂ Ω 1 , let λ 1 be the first eigenvalue of the following eigenvalue problem and φ 1 the corresponding eigenfunction.Denote sup EJQTDE, 2010 No. 51, p. 9 Let where 0 < ξ ≤ µ − l l−1 is a constant.Then we can know that sup Furthermore, it is easy to verify that φ 2 satisfies (3.7).Then, from (3.9), it follows immediately that inf For q > 1, let where α > 0 and A > 1 are constants to be determined later.Then after a simple computation, we have .
Since that q > 1, we can choose α to satisfy Then we have that P v ≥ 0 with A large enough.
On the other hand, since Ω ϕ (x, y) dy < 1 and l > 1, we have on the boundary that Thus, by comparison principle, we know that the solution of problem (1.1) exists globally provided that x) e τ t , where β < 1 and τ > 0 are two constants to be determined later.Computing directly, we obtain EJQTDE, 2010 No. 51, p. 10 If τ < λ 1 and β is sufficiently small, then we can conclude that P v ≥ 0. On the other hand, since 0 < ξ ≤ µ − l l−1 , we have on the boundary that where the conditions Ω ϕ (x, y) dy < 1 are used.Therefore, v (x, t) is a supersolution of problem (1.1) if u 0 (x) ≤ βφ 2 (x).The proof of Theorem 1.2 is complete.

Blow-up of the solution
In this section, we will discuss the blow-up property of the solution to problem (1.1), and give the proofs of Theorems 1.3 and 1.4.
Proof of Theorem 1.3.We employ a variant of Kaplan's method to prove our blow-up result of the case l > 1.Let u (x, t) be the unique solution to (1.1) and Taking the derivative of J (t) with respect to t, and using Green's formula we could obtain  Next, we look for the solution J (t) to (4.3) with J (0) > 1 on its interval of existence.Since the function f (J) = J l is convex, then there exists ̺ > 1 such that It follows easily that if J (0) > ̺, then J (t) is increasing on its interval of existence and From the above inequality it follows that lim where .
Then by assumptions in Theorem 1.3, the solution u (x, t) becomes infinite in a finite time.
The proof of Theorem 1.3 is complete.
Proof of Theorem 1.4.Consider the following equation and let v (x, t) be the solution to problem (4.6).It is obvious that v (x, t) is a subsolution of problem (1.1).For the case q ≥ 1, from Theorem 3.1 in [14], we know that v(x, t) blows up in finite time for sufficiently large u 0 (x).For the case q < 1, it is well-known that v (x, t) blows up in a finite time for any nonnegative u 0 (x) (see [14], Theorem 3.3).By Proposition (2.4), we obtain our blow-up result immediately.The proof of Theorem 1.4 is complete.

Blow-up rate estimate
In this section, we consider problem (1.1) with q = 0 and l = 1, i.e., where p > 1.By Theorem 1.4, for any nonnegative nontrivial initial data u 0 , u blows up in a finite time T * < ∞.We first give the upper bounder of the blow-up rate near the blow-up time.
EJQTDE, 2010 No. 51, p. 12 Lemma 5.1.Suppose that Ω ϕ(x, y)dy ≤ 1 and assumptions (H1) hold, then for any t 1 ∈ (t 0 , T * ), the blow-up solution u (x, t) to problem (5.1) satisfies where C 1 is a positive constant. Proof.Let where δ is a sufficiently small positive constant.After straightforward computation, we then obtain which proves the lower estimate.Combining (5.12) with Lemma 5.1, we obtain the blow-up estimate.The proof of the Theorem 1.6 is complete.