- Email: [email protected]
Uniform blow-up rate for diffusion equations with nonlocal nonlinear source
Nonlinear Analysis 67 (2007) 1947–1957 www.elsevier.com/locate/na
Uniform blow-up rate for diffusion equations with nonlocal nonlinear sourceI Qilin Liu a,∗ , Yuxiang Li b , Hongjun Gao c a Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China b Department of Mathematics, Southeast University, Nanjing 210096, PR China c Department of Mathematics, Nanjing Normal University, Nanjing 210097, PR China
Received 11 March 2006; accepted 15 August 2006
Abstract In this paper, we investigate the blow-up rate of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blow-up and that the rate of blow-up is uniform in all compact subsets of the domain. In each case, the blow-up rate of |u(t)|∞ is precisely determined. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Semi-linear diffusion equations; Uniform blow-up rate; Nonlocal reaction
1. Introduction In this paper, we consider semi-linear parabolic equations with nonlocal nonlinear source. The main questions we address here are the following. · Equations with space integral term of the form Z u t = ∆u + a(x)g f (u(x, t))dx . (1.1) Ω
Some problems involving both local and nonlocal terms of the type Z u t = ∆u + a(x) f (u(x, t))dx + h(u(x, t)),
(1.2)
Ω
will also be considered. · Equations with localized source of the form u t = ∆u + a(x) f (u(0, t)) + h(u(x, t)).
(1.3)
I This Project was Supported by PRC Grant NSFC 10371018, 10571087 and Southeast University Award Program for Outstanding Young Teachers 4007011015. ∗ Corresponding author. E-mail addresses: [email protected] (Q. Liu), [email protected] (Y. Li), [email protected] (H. Gao).
c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.08.030
1948
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
· Equations with space–time integral of the form Z t Z β(x)g(u(x, s))dxds . u t = ∆u + a(x) f 0
(1.4)
Ω
Each equation will be studied in a bounded domain Ω with homogeneous Dirichlet boundary conditions. In the case of a(x) = 1, problems of these types arising in various models in physics and engineering have been studied by a number of authors (for example, see [1–10] and references therein). The aim of the present article is to determine the blow-up rate of solutions for large classes of nonlocal problems of each type above. Namely, we prove that the solutions have global blow-up, and that the blow-up rate is uniform in all compact subsets of the domain. In each case, the blow-up rate of |u(t)|∞ is precisely determined. In the case of a(x) = 1, Souplet (see [1]) obtained the asymptotic blow-up behavior of the solutions for large classes of equations. For the function a(x), it is difficult to get the blow-up rate estimates. Moreover, to our knowledge, there do not seem to be any results in the literature on uniform blow-up rate estimates for problems of these types. In this paper, we use other techniques to prove the global blow-up and to get the blow-up rate for these cases. Throughout this paper, we assume that Ω is the open ball of R N centered at the origin of radius R. Each of the nonlocal nonlinear problems considered below is coupled with the boundary and initial conditions u(x, t) = 0, x ∈ ∂Ω , t > 0, (1.5) u(x, 0) = u 0 (x), x ∈ Ω . As for the functions a(x) and u 0 (x), we assume that (A1) a(x), u 0 (x) ∈ C 2 (Ω ); a(x), u 0 (x) > 0 in Ω and a(x) = u 0 (x) = 0 on ∂Ω . a(x) and u 0 (x) are radially symmetric, i. e., a(x) = a(r ) and u 0 (x) = u 0 (r ) with r = |x|, (A2) a(r ) and u 0 (r ) are non-increasing for r ∈ [0, R]. Also, we denote by λ > 0R the first eigenvalue of the Laplacian in H01 (Ω ) and by ϕ(x) > 0 the corresponding eigenfunction, normalized by Ω a(x)ϕ(x)dx = 1. Moreover, we denote by T ∗ (T ∗ < ∞) the maximal existence time. Then u blows up in L ∞ norm, in the sense that limt→T ∗ |u(t)|∞ = ∞. The outline of the article is as follows. The results are stated in Section 2. The proofs of Theorems 2.1 and 2.2 are presented in Section 3. Section 4 is devoted to the proof of Theorem 2.3. Theorem 2.4 is proved in Section 5. 2. Main results 2.1. Space integral source term We first consider the following problems with space integral nonlocal term: p
u t = ∆u + a(x)|u(t)|r , x ∈ Ω , t > 0, R 1/r where |u(t)|r = Ω |u(x, t)|r dx , 1 ≤ r < ∞ and p > 1.
(2.1)
Theorem 2.1. Assume (A1) and (A2). Let u(x, t) be the blow-up solution of (1.5) and (2.1) and assume that u(x, t) is non-decreasing in time; it holds that lim∗ (T ∗ − t)1/( p−1) u(x, t) = a(x)( p − 1)1/(1− p)
t→T
uniformly in all compact subsets of Ω .
Z Ω
a r (x)dx
p/(r (1− p))
,
(2.2)
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1949
2.2. Space-time integral source terms Next we consider the problems with integral in space and time Z t Z u t = ∆u + a(x) f β(x)u(x, s)dxds , x ∈ Ω , t > 0. 0
(2.3)
Ω
RTheorem 2.2. Assume (A1) and (A2). Let u be the blow-up solution of (1.5) and (2.3) and assume that I = Ω β(x)a(x)dx. (i) if f (s) = s p , p > 1, then lim (T ∗ − t)(1+ p)/( p−1) u(x, t) = a(x)( p − 1)(1+ p)/(1− p) (2 p (1 + p)/I p )1/( p−1) ,
t→T ∗
(2.4)
uniformly in all compact subsets of Ω . (ii) If f (s) = kes − 1, k ≥ 1, then 2 a(x), I uniformly in all compact subsets of Ω . lim (T ∗ − t)u(x, t) =
(2.5)
t→T ∗
2.3. Nonlocal (or localized) source and local term Last we consider the following problems with nonlocal (or localized) source and local term, such as Z u t = ∆u + a(x) u p (x, t)dx − u q (x, t), x ∈ Ω , t > 0,
(2.6)
Ω
and u t = ∆u + u q (x, t) + a(x)u p (0, t),
x ∈ Ω , t > 0,
(2.7)
where p > q ≥ 1. Theorem 2.3. Assume (A1) , (A2) and p > q ≥ 1. Let u be the blow-up solution of (1.5) and (2.6) and assume that u(x, t) is non-decreasing in time; it holds that 1/(1− p) Z ∗ 1/( p−1) p lim∗ (T − t) , (2.8) u(x, t) = a(x) ( p − 1) a (x)dx t→T
Ω
uniformly in all compact subsets of Ω . Theorem 2.4. Assume (A1), (A2) and p > q ≥ 1. Let u be the blow-up solution of (1.5) and (2.7) and assume that u(x, t) is non-decreasing in time; it holds that lim (T ∗ − t)1/( p−1) u(x, t) = a(x)(( p − 1)a p (0))1/(1− p) ,
t→T ∗
(2.9)
uniformly in all compact subsets of Ω . 3. Proof of Theorems 2.1 and 2.2 The problems (2.1) and (2.3) can be written in the form u t = ∆u + a(x)g(t),
x ∈ Ω , t > 0,
u(x, t) = 0, x ∈ ∂Ω , t > 0, u(x, 0) = u 0 (x), x ∈ Ω , where the function g(t) ≥ 0 will depend on the solution u.
(3.1)
1950
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
Rt Through this section, we use the notation G(t) = 0 g(s)ds. The key step in establishing the results of Section 2 is the following theorem. Theorem 3.1. Assume (A1) , (A2) and g(t) non-negative, continuous on (0, T ∗ ), and g(t) non-decreasing in time. Let u be the solution of Eq. (3.1) and limt→T ∗ G(t) = ∞, we then have lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . Proof. Assumption (A2) implies u r ≤ 0(r = |x|) by the maximum principle, which yields maxx∈Ω u(x, t) = u(0, t) and ∆u(0, t) ≤ 0. Clearly, using limt→T ∗ G(t) = ∞, we obtain that lim g(t) = ∞.
(3.2)
t→T ∗
From (3.1), we then get u t (0, t) ≤ a(0)g(t),
0 < t < T ∗,
that is, lim∗ sup
t→T
u(0, t) ≤ a(0). G(t)
(3.3)
Since g(t) is non-decreasing, it follows that for all ε > 0, R T ∗ −ε g(s)ds G(t) 0≤ ≤ 0 + ε, g(t) g(t) and using (3.2), we deduce that limt→T ∗ G(t)/g(t) = 0, so that (3.3) implies limt→T ∗ u(0, t)/g(t) = 0. Set R1 ∈ (0, R), Ω1 = {x ∈ R N : |x| < R1 } and b(x) = 1/a(x), x ∈ Ω1 . By a 0 (r ) ≤ 0, we obtain that b0 (r ) ≥ 0 for 0 ≤ r ≤ R1 . We introduce a function w(x, t) = b(x)u(x, t),
x ∈ Ω1 , 0 < t < T ∗ .
A simple calculation yields that b(x)∆u(x, t) = ∆w(x, t) − u(x, t)∆b(x) − 2 5 u(x, t) 5 b(x),
x ∈ Ω1 , 0 < t < T ∗ .
Since 5u(x, t) 5 b(x) = u r (r, t)b0 (r ) ≤ 0, it follows that b(x)∆u(x, t) ≥ ∆w(x, t) − u(x, t)∆b(x), Set m = maxx∈Ω 1 |∆b(x)|, ε(t) =
x ∈ Ω1 , 0 < t < T ∗ .
mu(0,t) g(t) .
From limt→T ∗ u(0, t)/g(t) = 0, we infer that there exists τ ∈ (0, T ∗ ) such that 0 < ε(t) ≤ Therefore, using (3.1), we obtain wt = b(x)u t = b(x)∆u + g(t) = ∆w − u∆b(x) − 2 5 u 5 b(x) + g(t) ≥ ∆w − u∆b(x) + g(t) ≥ ∆w + (1 − ε(t))g(t) + ε(t)g(t) − mu(0, t) = ∆w + (1 − ε(t))g(t),
x ∈ Ω1 , τ < t < T ∗ . Rt Set g1 (t) = (1 − ε(t))g(t), G 1 (t) = τ g1 (s)ds. We then obtain lim G 1 (t) = ∞
t→T ∗
and lim∗
t→T
G 1 (t) = 1. G(t)
1 2
for τ ≤ t < T ∗ .
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1951
Clearly, w(x, t) is a sup-solution of the following equation: vt = ∆v + g1 (t),
x ∈ Ω1 , τ < t < T ∗ ,
v(x, t) = 0, x ∈ ∂Ω1 , τ < t < T ∗ , v(x, τ ) = v0 (x), x ∈ Ω1 ,
(3.4)
where 0 ≤ v0 (x) ≤ w(x, τ ) in Ω1 and v0 (x) ∈ C 1 (Ω 1 ) with v0 (x)|∂ Ω1 = 0. It is also assumed that v0 (x) is symmetric and non-increasing as a function of |x|(r = |x|). By the maximum principle, we have 0 ≤ v(x, t) ≤ w(x, t) and vr ≤ 0 in Ω1 for τ ≤ t < T ∗ . We denote by λ1 > 0 theR first eigenvalue of the Laplacian in H01 (Ω1 ) and by ϕ1 (x) > 0 the corresponding eigenfunction, normalized by Ω1 ϕ1 dx = 1. Multiplying both sides of Eq. (3.4) by ϕ1 and integrating over Ω1 × (τ, t), we have, for τ < t < T ∗ , Z Z Z tZ vϕ1 dx − v0 ϕ1 dx = −λ1 vϕ1 dxds + G 1 (t). Ω1
Clearly, Z tZ τ
τ
Ω1
vϕ1 dxds ≤
Ω1
t
Z τ
v(0, s)ds ≤
Ω1 t
Z τ
w(0, s)ds,
and lim∗
t→T
w(0, t) 1 u(0, t) w(0, t) = lim∗ = lim∗ = 0, t→T g1 (t) g(t) a(0) t→T g(t)
which imply Rt R lim∗
τ
Ω1
vϕ1 dxds
G 1 (t)
t→T
Rt R = lim∗ t→T
τ
Ω1
vϕ1 dxds
G(t)
= 0.
Therefore, R lim∗
Ω1
vϕ1 dx
G 1 (t)
t→T
= 1,
(3.5)
that is, v(0, t) ≥ 1. t→T G 1 (t) By (3.4), we get lim∗ inf
vt (0, t) ≤ g1 (t),
(3.6)
τ < t < T∗
which implies lim∗ sup
t→T
v(0, t) ≤ 1. G 1 (t)
(3.7)
Combining (3.6) with (3.7) yields that lim∗
t→T
v(0, t) = 1. G 1 (t)
R Using Ω1 ϕ1 dx = 1, vr ≤ 0, (3.5) and (3.8), we deduce that v(x, t) v(x, t) lim∗ = lim∗ = 1, t→T t→T G 1 (t) G(t) uniformly in all compact subsets of Ω1 , that is, lim∗ inf
t→T
u(x, t) ≥ a(x), G(t)
(3.8)
1952
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
uniformly in all compact subsets of Ω1 . By the arbitrariness of Ω1 , we then obtain lim inf
t→T ∗
u(x, t) ≥ a(x), G(t)
(3.9)
uniformly in all compact subsets of Ω , that is, lim inf
t→T ∗
u(0, t) ≥ a(0). G(t)
Using (3.3), we deduce that lim∗
t→T
u(0, t) = a(0). G(t)
(3.10)
Multiplying both sides of the Eq. (3.1) by ϕ and integrating over Ω × (0, t), we have, for 0 < t < T ∗ , Z Z Z tZ uϕdx − u 0 ϕdx = −λ uϕdxds + G(t). Ω
Since
Rt R
Ω
0
uϕdxds ≤ Ω ϕdx Rt R uϕdxds = 0. lim∗ 0 Ω t→T G(t) 0
R
Ω
Rt 0
Ω
u(0, s)ds and limt→T ∗ u(0, t)/g(t) = 0, which imply that
It then follows that R uϕdx = 1. lim∗ Ω t→T G(t)
(3.11)
Now we will show that lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . Assume to the contrary that there exists x0 ∈ Ω \ 0 such that limt→T ∗ sup(u(x0 , t)/G(t)) = c > a(x0 ), that is, there exists a sequence tn → T ∗ such that limtn →T ∗ u(x0 , tn )/G(tn ) = c. Using the continuity of a(x), we deduce that there exists x1 ∈ Ω (|x1 | < |x0 |) such that a(x) < c for |x1 | ≤ |x| ≤ |x0 |. By u r ≤ 0, (3.9) and (3.11), we obtain R Z Z Z u(x, tn ) u(x, tn ) u(x, tn ) Ω uϕdx lim = lim ∗ ϕdx + ϕdx + ϕdx t→T ∗ tn →T G(t) |x0 |<|x| a(x)ϕ(x)dx + a(x)ϕ(x)dx + a(x)ϕ(x)dx = 1. |x0 |<|x|
|x1 |<|x|<|x0 |
|x|<|x1 |
This contradicts (3.11) and we then get the desired result. Rt p Proof of Theorem 2.1. Set g(t) = |u(t)|r , G(t) = 0 g(s)ds. Since u(x, t) is non-decreasing in time, it then follows by (A2) that g 0 (t) ≥ 0 for t > 0 and u r ≤ 0 on [0, R] for t > 0, which imply limt→T ∗ u(0, t) = ∞. Clearly, u t (0, t) ≤ a(0)g(t),
0 < t < T ∗,
1953
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
which implies lim sup
t→T ∗
u(0, t) ≤ a(0). G(t)
Therefore, lim G(t) = ∞.
t→T ∗
By Theorem 3.1, it follows that lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . A simple calculation yields the desired result. R R p Rt t Proof of Theorem 2.2. (i) Set g(t) = 0 Ω β(x)u(x, s)dxds , G(t) = 0 g(s)ds. Clearly, g 0 (t) > 0
for t > 0.
By an argument similar to that at the beginning of the proof of Theorem 2.1, it follows that limt→T ∗ G(t) = ∞. From Theorem 3.1, we obtain lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . A simple calculation yields the desired result. Case (ii) is handled similarly.
4. Proof of Theorem 2.3 We set the following notation: Z Z t p g(t) = u (x, t)dx, G(t) = g(s)ds. Ω
0
First we proceed with some preliminary estimates. Lemma 4.1. Under the assumption of Theorem 2.3, it holds that lim∗
t→T
u q (0, t) = 0. g(t)
Proof. Since u r ≤ 0 and u t ≥ 0, it follows that limt→T ∗ u(0, t) = ∞ and ∆u(0, t) ≤ 0 for t > 0, which imply that u t (0, t) ≤ a(0)|Ω |u p (0, t) − u q (0, t), u t (0, t) ≤ a(0)g(t), 0 < t < T ∗ .
0 < t < T ∗,
A simple computation yields that lim inf(T ∗ − t)1/( p−1) u(0, t) ≥ (a(0)|Ω |( p − 1))1/(1− p) ,
t→T ∗
(using p > q)
and lim sup
t→T ∗
u(0, t) ≤ a(0), G(t)
(4.1)
which imply lim (T ∗ − t)1/( p−1) G(t) ≥ (a p (0)|Ω |( p − 1))1/(1− p) .
t→T ∗
(4.2)
1954
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
Since limt→T ∗ G(t)/g(t) = 0 and (4.1), we deduce that lim∗
t→T
u(0, t) = 0. g(t)
(4.3)
Multiplying both sides of Eq. (2.6) by ϕ and integrating over Ω × (0, t), we have, for 0 < t < T ∗ , Z tZ Z tZ Z Z u q ϕdxds. uϕdxds + G(t) − u 0 ϕdx = −λ uϕdx − Ω
Ω
Ω
0
0
Ω
By the H¨older inequality, we have 1−(q/ p) q/ p Z Z 1−(q/ p) q/ p Z Z Z p q/ p p q , ϕdx u dx ≤ ϕ(0) ϕdx u ϕdx u ϕdx ≤ which implies Rt R lim∗
0
Rt R
uϕdxds =0 G(t)
Ω
t→T
and
lim∗
Ω
Ω
Ω
Ω
Ω
0
t→T
u q ϕdxds = 0. G(t)
Ω
That is, R
uϕdx = 1. lim∗ Ω t→T G(t) R R 1/ p R 1−(1/ p) , it then follows that Since Ω uϕdx ≤ Ω u p ϕdx Ω ϕdx R u p ϕdx 1 lim∗ inf Ω p ≥ R p−1 , t→T G (t) ϕdx
(4.4)
Ω
which implies u p dx 1 ≥ lim∗ inf Ω p R p−1 . t→T G (t) ϕ(0) Ω ϕdx R
(4.5)
Combining (4.2) with (4.5) yields that lim inf(T ∗ − t) p/( p−1) g(t) ≥ d,
(4.6)
t→T ∗
R 1− p p where d = ϕ −1 (0) Ω ϕdx (a (0)|Ω |( p − 1)) p/(1− p) . 0 Using G (t) = g(t), we get lim∗ sup(T ∗ − t)1/( p−1) G(t) ≤ (ϕ(0)/( p − 1))1/( p−1)
t→T
Z Ω
ϕ(x)dx.
(4.7)
From (4.1) and (4.7), we obtain by (4.5) lim∗ sup(T ∗ − t)1/( p−1) u(0, t) ≤ a(0)(ϕ(0)/( p − 1))1/( p−1)
t→T
Z Ω
ϕ(x)dx.
(4.8)
Using p > q, (4.6) and (4.8), it follows that lim∗
t→T
u q (0, t) = 0. g(t)
u q (0,t) a(R2 )g(t) . ε(t) ≤ 12 for
Proof of Theorem 2.3. Set R2 ∈ (0, R), Ω2 = x ∈ R N : |x| < R2 and ε(t) = Using Lemma 4.1, it then follows that there exists τ ∈ (0, T ∗ ) such that 0 < Set Z T∗ g1 (t) = (1 − ε(t))g(t) and G 1 (t) = g1 (s)ds. τ
τ ≤ t < T ∗.
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1955
Clearly, lim G 1 (t) = ∞,
t→T ∗
and lim
t→T ∗
G 1 (t) = 1. G(t)
On the other hand, Z
u p dx − u q
u t = ∆u + a(x) Ω
= ∆u + a(x)(1 − ε(t))g(t) + a(x)ε(t)g(t) − u q (x, t) ≥ ∆u + a(x)g1 (t) + a(R2 )ε(t)g(t) − u q (0, t) = ∆u + a(x)g1 (t),
x ∈ Ω2 , τ < t < T ∗ .
Clearly, u(x, t) is a sup-solution of the following equation: vt = ∆v + a(x)g1 (t),
x ∈ Ω2 , τ < t < T ∗ ,
v(x, t) = 0, x ∈ ∂Ω2 , τ < t < T ∗ , v(x, τ ) = u(x, τ ) − u(R2 , τ ), x ∈ Ω2 .
(4.9)
By the maximum principle, u(x, t) ≥ v(x, t) and vr ≤ 0 in Ω2 × (τ, T ∗ ). Using Theorem 3.1, we have lim∗
t→T
v(x, t) v(x, t) = lim∗ = a(x), t→T G 1 (t) G(t)
uniformly in any compact subsets of Ω2 , that is, lim inf
t→T ∗
u(x, t) ≥ a(x), G(t)
uniformly in any compact subsets of Ω2 . By the arbitrariness of Ω2 , we get lim∗ inf
t→T
u(x, t) ≥ a(x), G(t)
(4.10)
uniformly in any compact Rsubsets of Ω . Since u t ≤ ∆u + a(x) Ω u p dx, it then follows that u(x, t) is a sub-solution of the following equation: v1t = ∆v1 + a(x)g(t),
x ∈ Ω , 0 < t < T ∗,
v1 (x, t) = 0, x ∈ ∂Ω , 0 < t < T ∗ , v1 (x, 0) = u 0 (x), x ∈ Ω .
(4.11)
By the maximum principle, u(x, t) ≤ v1 (x, t) and v1r ≤ 0 in Ω × (0, T ∗ ). Using Theorem 3.1, it holds that lim∗
t→T
v1 (x, t) = a(x), G(t)
uniformly in all compact subsets of Ω , that is, lim∗ sup
t→T
u(x, t) ≤ a(x), G(t)
uniformly in all compact subsets of Ω . Combining (4.10) with (4.12) yields that lim∗
t→T
u(x, t) = a(x), G(t)
(4.12)
1956
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
uniformly in all compact subsets of Ω . A simple calculation yields the desired result. The proof is completed.
5. Proof of Theorem 2.4 Set g(t) = u p (0, t), G(t) =
Rt 0
g(s)ds.
Proof of Theorem 2.4. Since u t ≥ 0 and u r ≤ 0, it then follows that ∆u(0, t) ≤ 0 on (0, T ∗ ) and limt→T ∗ u(0, t) = ∞. Clearly, u t (0, t) ≤ u q (0, t) + a(0)u p (0, t),
0 < t < T ∗,
which implies lim∗ sup
t→T
u(0, t) ≤ a(0), G(t)
(5.1)
that is, lim G(t) = ∞.
t→T ∗
Since u t ≥ ∆u + a(x)u p (0, t), it then follows that u(x, t) is a sup-solution of the following equation: vt = ∆v + a(x)g(t),
x ∈ Ω , 0 < t < T ∗,
v(x, t) = 0, x ∈ ∂Ω , 0 < t < T ∗ , v(x, 0) = u 0 (x), x ∈ Ω .
(5.2)
By the maximum principle, u(x, t) ≥ v(x, t) and vr ≤ 0 in Ω × (0, T ∗ ). Using Theorem 3.1, we obtain lim∗
t→T
v(x, t) = a(x), G(t)
uniformly in any compact subsets of Ω , that is, lim∗ inf
t→T
u(x, t) ≥ a(x), G(t)
(5.3)
uniformly in any compact subsets of Ω , which implies lim∗ inf
t→T
u(0, t) ≥ a(0). G(t)
(5.4)
Combining (5.1) with (5.4) yields that lim∗
t→T
u(0, t) = a(0). G(t)
(5.5)
Multiplying both sides of the Eq. (2.7) by ϕ and integrating over Ω × (0, t), we have, for 0 < t < T ∗ , Z Z Z tZ Z tZ uϕdx − u 0 ϕdx = −λ uϕdxds + u q ϕdxds + G(t). Ω
Ω
0
Rt R
u q ϕdxds
Since p > q ≥ 1 and 0 Ω Rt R uϕdxds lim∗ 0 Ω =0 t→T G(t)
and
Ω
0
Rt
≤ Ω ϕdx 0 Rt R R
lim∗
t→T
0
Ω
u q (0, s)ds,
so we have that
u q ϕdxds = 0, G(t)
Ω
which imply R
uϕdx lim∗ Ω = 1. t→T G(t)
(5.6)
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1957
By an argument similar to that in Theorem 3.1, we obtain lim
t→T ∗
u(x, t) = a(x), G(t)
uniformly in any compact subsets of Ω . A simple calculation yields the desired result. The proof is completed.
References [1] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999) 374–406. [2] J. Bebernes, A. Bressan, A. Lacey, Total blow-up versus single-point blow-up, J. Differential Equations 73 (1988) 30–44. [3] K. Deng, M.K. Kwang, H.A. Levine, The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers equation, Quart. Appl. Math. 5 (6) (1992) 1237–1245. [4] J.M. Chadam, A. Peirce, H.-M. Yin, The blow-up property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992) 313–328. [5] M. Wang, Y. Wang, Properties of positive solutions for non-local reaction–diffusion problems, Math. Methods Appl. Sci. 19 (1996) 1141–1156. [6] J.R. Cannon, H.-M. Yin, A class of non-linear non-classical parabolic equations, J. Differential Equations 79 (1989) 266–288. [7] L. Wang, Q. Chen, The asymptotic behavior of blow-up solution of localized nonlinear equation, J. Math. Anal. Appl. 200 (1996) 315–321. [8] J.-S. Guo, H.-W. Su, The blow-up behavior of the solution of an integro-differential equation, Differential Integral Equations 5 (6) (1992) 1237–1245. [9] A. Okada, I. Fukuda, Total versus single point blow-up of solutions of a semi-linear parabolic equation with localized reaction, J. Math. Anal. Appl. 281 (2003) 485–500. [10] Q. Liu, Y. Li, H. Guo, Uniform blow-up rate for diffusion equations with localized nonlinear source, J. Math. Anal. Appl. 320 (2) (2006) 771–778.
Uniform blow-up rate for diffusion equations with nonlocal nonlinear sourceI Qilin Liu a,∗ , Yuxiang Li b , Hongjun Gao c a Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, PR China b Department of Mathematics, Southeast University, Nanjing 210096, PR China c Department of Mathematics, Nanjing Normal University, Nanjing 210097, PR China
Received 11 March 2006; accepted 15 August 2006
Abstract In this paper, we investigate the blow-up rate of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blow-up and that the rate of blow-up is uniform in all compact subsets of the domain. In each case, the blow-up rate of |u(t)|∞ is precisely determined. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Semi-linear diffusion equations; Uniform blow-up rate; Nonlocal reaction
1. Introduction In this paper, we consider semi-linear parabolic equations with nonlocal nonlinear source. The main questions we address here are the following. · Equations with space integral term of the form Z u t = ∆u + a(x)g f (u(x, t))dx . (1.1) Ω
Some problems involving both local and nonlocal terms of the type Z u t = ∆u + a(x) f (u(x, t))dx + h(u(x, t)),
(1.2)
Ω
will also be considered. · Equations with localized source of the form u t = ∆u + a(x) f (u(0, t)) + h(u(x, t)).
(1.3)
I This Project was Supported by PRC Grant NSFC 10371018, 10571087 and Southeast University Award Program for Outstanding Young Teachers 4007011015. ∗ Corresponding author. E-mail addresses: [email protected] (Q. Liu), [email protected] (Y. Li), [email protected] (H. Gao).
c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.08.030
1948
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
· Equations with space–time integral of the form Z t Z β(x)g(u(x, s))dxds . u t = ∆u + a(x) f 0
(1.4)
Ω
Each equation will be studied in a bounded domain Ω with homogeneous Dirichlet boundary conditions. In the case of a(x) = 1, problems of these types arising in various models in physics and engineering have been studied by a number of authors (for example, see [1–10] and references therein). The aim of the present article is to determine the blow-up rate of solutions for large classes of nonlocal problems of each type above. Namely, we prove that the solutions have global blow-up, and that the blow-up rate is uniform in all compact subsets of the domain. In each case, the blow-up rate of |u(t)|∞ is precisely determined. In the case of a(x) = 1, Souplet (see [1]) obtained the asymptotic blow-up behavior of the solutions for large classes of equations. For the function a(x), it is difficult to get the blow-up rate estimates. Moreover, to our knowledge, there do not seem to be any results in the literature on uniform blow-up rate estimates for problems of these types. In this paper, we use other techniques to prove the global blow-up and to get the blow-up rate for these cases. Throughout this paper, we assume that Ω is the open ball of R N centered at the origin of radius R. Each of the nonlocal nonlinear problems considered below is coupled with the boundary and initial conditions u(x, t) = 0, x ∈ ∂Ω , t > 0, (1.5) u(x, 0) = u 0 (x), x ∈ Ω . As for the functions a(x) and u 0 (x), we assume that (A1) a(x), u 0 (x) ∈ C 2 (Ω ); a(x), u 0 (x) > 0 in Ω and a(x) = u 0 (x) = 0 on ∂Ω . a(x) and u 0 (x) are radially symmetric, i. e., a(x) = a(r ) and u 0 (x) = u 0 (r ) with r = |x|, (A2) a(r ) and u 0 (r ) are non-increasing for r ∈ [0, R]. Also, we denote by λ > 0R the first eigenvalue of the Laplacian in H01 (Ω ) and by ϕ(x) > 0 the corresponding eigenfunction, normalized by Ω a(x)ϕ(x)dx = 1. Moreover, we denote by T ∗ (T ∗ < ∞) the maximal existence time. Then u blows up in L ∞ norm, in the sense that limt→T ∗ |u(t)|∞ = ∞. The outline of the article is as follows. The results are stated in Section 2. The proofs of Theorems 2.1 and 2.2 are presented in Section 3. Section 4 is devoted to the proof of Theorem 2.3. Theorem 2.4 is proved in Section 5. 2. Main results 2.1. Space integral source term We first consider the following problems with space integral nonlocal term: p
u t = ∆u + a(x)|u(t)|r , x ∈ Ω , t > 0, R 1/r where |u(t)|r = Ω |u(x, t)|r dx , 1 ≤ r < ∞ and p > 1.
(2.1)
Theorem 2.1. Assume (A1) and (A2). Let u(x, t) be the blow-up solution of (1.5) and (2.1) and assume that u(x, t) is non-decreasing in time; it holds that lim∗ (T ∗ − t)1/( p−1) u(x, t) = a(x)( p − 1)1/(1− p)
t→T
uniformly in all compact subsets of Ω .
Z Ω
a r (x)dx
p/(r (1− p))
,
(2.2)
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1949
2.2. Space-time integral source terms Next we consider the problems with integral in space and time Z t Z u t = ∆u + a(x) f β(x)u(x, s)dxds , x ∈ Ω , t > 0. 0
(2.3)
Ω
RTheorem 2.2. Assume (A1) and (A2). Let u be the blow-up solution of (1.5) and (2.3) and assume that I = Ω β(x)a(x)dx. (i) if f (s) = s p , p > 1, then lim (T ∗ − t)(1+ p)/( p−1) u(x, t) = a(x)( p − 1)(1+ p)/(1− p) (2 p (1 + p)/I p )1/( p−1) ,
t→T ∗
(2.4)
uniformly in all compact subsets of Ω . (ii) If f (s) = kes − 1, k ≥ 1, then 2 a(x), I uniformly in all compact subsets of Ω . lim (T ∗ − t)u(x, t) =
(2.5)
t→T ∗
2.3. Nonlocal (or localized) source and local term Last we consider the following problems with nonlocal (or localized) source and local term, such as Z u t = ∆u + a(x) u p (x, t)dx − u q (x, t), x ∈ Ω , t > 0,
(2.6)
Ω
and u t = ∆u + u q (x, t) + a(x)u p (0, t),
x ∈ Ω , t > 0,
(2.7)
where p > q ≥ 1. Theorem 2.3. Assume (A1) , (A2) and p > q ≥ 1. Let u be the blow-up solution of (1.5) and (2.6) and assume that u(x, t) is non-decreasing in time; it holds that 1/(1− p) Z ∗ 1/( p−1) p lim∗ (T − t) , (2.8) u(x, t) = a(x) ( p − 1) a (x)dx t→T
Ω
uniformly in all compact subsets of Ω . Theorem 2.4. Assume (A1), (A2) and p > q ≥ 1. Let u be the blow-up solution of (1.5) and (2.7) and assume that u(x, t) is non-decreasing in time; it holds that lim (T ∗ − t)1/( p−1) u(x, t) = a(x)(( p − 1)a p (0))1/(1− p) ,
t→T ∗
(2.9)
uniformly in all compact subsets of Ω . 3. Proof of Theorems 2.1 and 2.2 The problems (2.1) and (2.3) can be written in the form u t = ∆u + a(x)g(t),
x ∈ Ω , t > 0,
u(x, t) = 0, x ∈ ∂Ω , t > 0, u(x, 0) = u 0 (x), x ∈ Ω , where the function g(t) ≥ 0 will depend on the solution u.
(3.1)
1950
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
Rt Through this section, we use the notation G(t) = 0 g(s)ds. The key step in establishing the results of Section 2 is the following theorem. Theorem 3.1. Assume (A1) , (A2) and g(t) non-negative, continuous on (0, T ∗ ), and g(t) non-decreasing in time. Let u be the solution of Eq. (3.1) and limt→T ∗ G(t) = ∞, we then have lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . Proof. Assumption (A2) implies u r ≤ 0(r = |x|) by the maximum principle, which yields maxx∈Ω u(x, t) = u(0, t) and ∆u(0, t) ≤ 0. Clearly, using limt→T ∗ G(t) = ∞, we obtain that lim g(t) = ∞.
(3.2)
t→T ∗
From (3.1), we then get u t (0, t) ≤ a(0)g(t),
0 < t < T ∗,
that is, lim∗ sup
t→T
u(0, t) ≤ a(0). G(t)
(3.3)
Since g(t) is non-decreasing, it follows that for all ε > 0, R T ∗ −ε g(s)ds G(t) 0≤ ≤ 0 + ε, g(t) g(t) and using (3.2), we deduce that limt→T ∗ G(t)/g(t) = 0, so that (3.3) implies limt→T ∗ u(0, t)/g(t) = 0. Set R1 ∈ (0, R), Ω1 = {x ∈ R N : |x| < R1 } and b(x) = 1/a(x), x ∈ Ω1 . By a 0 (r ) ≤ 0, we obtain that b0 (r ) ≥ 0 for 0 ≤ r ≤ R1 . We introduce a function w(x, t) = b(x)u(x, t),
x ∈ Ω1 , 0 < t < T ∗ .
A simple calculation yields that b(x)∆u(x, t) = ∆w(x, t) − u(x, t)∆b(x) − 2 5 u(x, t) 5 b(x),
x ∈ Ω1 , 0 < t < T ∗ .
Since 5u(x, t) 5 b(x) = u r (r, t)b0 (r ) ≤ 0, it follows that b(x)∆u(x, t) ≥ ∆w(x, t) − u(x, t)∆b(x), Set m = maxx∈Ω 1 |∆b(x)|, ε(t) =
x ∈ Ω1 , 0 < t < T ∗ .
mu(0,t) g(t) .
From limt→T ∗ u(0, t)/g(t) = 0, we infer that there exists τ ∈ (0, T ∗ ) such that 0 < ε(t) ≤ Therefore, using (3.1), we obtain wt = b(x)u t = b(x)∆u + g(t) = ∆w − u∆b(x) − 2 5 u 5 b(x) + g(t) ≥ ∆w − u∆b(x) + g(t) ≥ ∆w + (1 − ε(t))g(t) + ε(t)g(t) − mu(0, t) = ∆w + (1 − ε(t))g(t),
x ∈ Ω1 , τ < t < T ∗ . Rt Set g1 (t) = (1 − ε(t))g(t), G 1 (t) = τ g1 (s)ds. We then obtain lim G 1 (t) = ∞
t→T ∗
and lim∗
t→T
G 1 (t) = 1. G(t)
1 2
for τ ≤ t < T ∗ .
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1951
Clearly, w(x, t) is a sup-solution of the following equation: vt = ∆v + g1 (t),
x ∈ Ω1 , τ < t < T ∗ ,
v(x, t) = 0, x ∈ ∂Ω1 , τ < t < T ∗ , v(x, τ ) = v0 (x), x ∈ Ω1 ,
(3.4)
where 0 ≤ v0 (x) ≤ w(x, τ ) in Ω1 and v0 (x) ∈ C 1 (Ω 1 ) with v0 (x)|∂ Ω1 = 0. It is also assumed that v0 (x) is symmetric and non-increasing as a function of |x|(r = |x|). By the maximum principle, we have 0 ≤ v(x, t) ≤ w(x, t) and vr ≤ 0 in Ω1 for τ ≤ t < T ∗ . We denote by λ1 > 0 theR first eigenvalue of the Laplacian in H01 (Ω1 ) and by ϕ1 (x) > 0 the corresponding eigenfunction, normalized by Ω1 ϕ1 dx = 1. Multiplying both sides of Eq. (3.4) by ϕ1 and integrating over Ω1 × (τ, t), we have, for τ < t < T ∗ , Z Z Z tZ vϕ1 dx − v0 ϕ1 dx = −λ1 vϕ1 dxds + G 1 (t). Ω1
Clearly, Z tZ τ
τ
Ω1
vϕ1 dxds ≤
Ω1
t
Z τ
v(0, s)ds ≤
Ω1 t
Z τ
w(0, s)ds,
and lim∗
t→T
w(0, t) 1 u(0, t) w(0, t) = lim∗ = lim∗ = 0, t→T g1 (t) g(t) a(0) t→T g(t)
which imply Rt R lim∗
τ
Ω1
vϕ1 dxds
G 1 (t)
t→T
Rt R = lim∗ t→T
τ
Ω1
vϕ1 dxds
G(t)
= 0.
Therefore, R lim∗
Ω1
vϕ1 dx
G 1 (t)
t→T
= 1,
(3.5)
that is, v(0, t) ≥ 1. t→T G 1 (t) By (3.4), we get lim∗ inf
vt (0, t) ≤ g1 (t),
(3.6)
τ < t < T∗
which implies lim∗ sup
t→T
v(0, t) ≤ 1. G 1 (t)
(3.7)
Combining (3.6) with (3.7) yields that lim∗
t→T
v(0, t) = 1. G 1 (t)
R Using Ω1 ϕ1 dx = 1, vr ≤ 0, (3.5) and (3.8), we deduce that v(x, t) v(x, t) lim∗ = lim∗ = 1, t→T t→T G 1 (t) G(t) uniformly in all compact subsets of Ω1 , that is, lim∗ inf
t→T
u(x, t) ≥ a(x), G(t)
(3.8)
1952
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
uniformly in all compact subsets of Ω1 . By the arbitrariness of Ω1 , we then obtain lim inf
t→T ∗
u(x, t) ≥ a(x), G(t)
(3.9)
uniformly in all compact subsets of Ω , that is, lim inf
t→T ∗
u(0, t) ≥ a(0). G(t)
Using (3.3), we deduce that lim∗
t→T
u(0, t) = a(0). G(t)
(3.10)
Multiplying both sides of the Eq. (3.1) by ϕ and integrating over Ω × (0, t), we have, for 0 < t < T ∗ , Z Z Z tZ uϕdx − u 0 ϕdx = −λ uϕdxds + G(t). Ω
Since
Rt R
Ω
0
uϕdxds ≤ Ω ϕdx Rt R uϕdxds = 0. lim∗ 0 Ω t→T G(t) 0
R
Ω
Rt 0
Ω
u(0, s)ds and limt→T ∗ u(0, t)/g(t) = 0, which imply that
It then follows that R uϕdx = 1. lim∗ Ω t→T G(t)
(3.11)
Now we will show that lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . Assume to the contrary that there exists x0 ∈ Ω \ 0 such that limt→T ∗ sup(u(x0 , t)/G(t)) = c > a(x0 ), that is, there exists a sequence tn → T ∗ such that limtn →T ∗ u(x0 , tn )/G(tn ) = c. Using the continuity of a(x), we deduce that there exists x1 ∈ Ω (|x1 | < |x0 |) such that a(x) < c for |x1 | ≤ |x| ≤ |x0 |. By u r ≤ 0, (3.9) and (3.11), we obtain R Z Z Z u(x, tn ) u(x, tn ) u(x, tn ) Ω uϕdx lim = lim ∗ ϕdx + ϕdx + ϕdx t→T ∗ tn →T G(t) |x0 |<|x|
|x1 |<|x|<|x0 |
|x|<|x1 |
This contradicts (3.11) and we then get the desired result. Rt p Proof of Theorem 2.1. Set g(t) = |u(t)|r , G(t) = 0 g(s)ds. Since u(x, t) is non-decreasing in time, it then follows by (A2) that g 0 (t) ≥ 0 for t > 0 and u r ≤ 0 on [0, R] for t > 0, which imply limt→T ∗ u(0, t) = ∞. Clearly, u t (0, t) ≤ a(0)g(t),
0 < t < T ∗,
1953
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
which implies lim sup
t→T ∗
u(0, t) ≤ a(0). G(t)
Therefore, lim G(t) = ∞.
t→T ∗
By Theorem 3.1, it follows that lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . A simple calculation yields the desired result. R R p Rt t Proof of Theorem 2.2. (i) Set g(t) = 0 Ω β(x)u(x, s)dxds , G(t) = 0 g(s)ds. Clearly, g 0 (t) > 0
for t > 0.
By an argument similar to that at the beginning of the proof of Theorem 2.1, it follows that limt→T ∗ G(t) = ∞. From Theorem 3.1, we obtain lim∗
t→T
u(x, t) = a(x), G(t)
uniformly in all compact subsets of Ω . A simple calculation yields the desired result. Case (ii) is handled similarly.
4. Proof of Theorem 2.3 We set the following notation: Z Z t p g(t) = u (x, t)dx, G(t) = g(s)ds. Ω
0
First we proceed with some preliminary estimates. Lemma 4.1. Under the assumption of Theorem 2.3, it holds that lim∗
t→T
u q (0, t) = 0. g(t)
Proof. Since u r ≤ 0 and u t ≥ 0, it follows that limt→T ∗ u(0, t) = ∞ and ∆u(0, t) ≤ 0 for t > 0, which imply that u t (0, t) ≤ a(0)|Ω |u p (0, t) − u q (0, t), u t (0, t) ≤ a(0)g(t), 0 < t < T ∗ .
0 < t < T ∗,
A simple computation yields that lim inf(T ∗ − t)1/( p−1) u(0, t) ≥ (a(0)|Ω |( p − 1))1/(1− p) ,
t→T ∗
(using p > q)
and lim sup
t→T ∗
u(0, t) ≤ a(0), G(t)
(4.1)
which imply lim (T ∗ − t)1/( p−1) G(t) ≥ (a p (0)|Ω |( p − 1))1/(1− p) .
t→T ∗
(4.2)
1954
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
Since limt→T ∗ G(t)/g(t) = 0 and (4.1), we deduce that lim∗
t→T
u(0, t) = 0. g(t)
(4.3)
Multiplying both sides of Eq. (2.6) by ϕ and integrating over Ω × (0, t), we have, for 0 < t < T ∗ , Z tZ Z tZ Z Z u q ϕdxds. uϕdxds + G(t) − u 0 ϕdx = −λ uϕdx − Ω
Ω
Ω
0
0
Ω
By the H¨older inequality, we have 1−(q/ p) q/ p Z Z 1−(q/ p) q/ p Z Z Z p q/ p p q , ϕdx u dx ≤ ϕ(0) ϕdx u ϕdx u ϕdx ≤ which implies Rt R lim∗
0
Rt R
uϕdxds =0 G(t)
Ω
t→T
and
lim∗
Ω
Ω
Ω
Ω
Ω
0
t→T
u q ϕdxds = 0. G(t)
Ω
That is, R
uϕdx = 1. lim∗ Ω t→T G(t) R R 1/ p R 1−(1/ p) , it then follows that Since Ω uϕdx ≤ Ω u p ϕdx Ω ϕdx R u p ϕdx 1 lim∗ inf Ω p ≥ R p−1 , t→T G (t) ϕdx
(4.4)
Ω
which implies u p dx 1 ≥ lim∗ inf Ω p R p−1 . t→T G (t) ϕ(0) Ω ϕdx R
(4.5)
Combining (4.2) with (4.5) yields that lim inf(T ∗ − t) p/( p−1) g(t) ≥ d,
(4.6)
t→T ∗
R 1− p p where d = ϕ −1 (0) Ω ϕdx (a (0)|Ω |( p − 1)) p/(1− p) . 0 Using G (t) = g(t), we get lim∗ sup(T ∗ − t)1/( p−1) G(t) ≤ (ϕ(0)/( p − 1))1/( p−1)
t→T
Z Ω
ϕ(x)dx.
(4.7)
From (4.1) and (4.7), we obtain by (4.5) lim∗ sup(T ∗ − t)1/( p−1) u(0, t) ≤ a(0)(ϕ(0)/( p − 1))1/( p−1)
t→T
Z Ω
ϕ(x)dx.
(4.8)
Using p > q, (4.6) and (4.8), it follows that lim∗
t→T
u q (0, t) = 0. g(t)
u q (0,t) a(R2 )g(t) . ε(t) ≤ 12 for
Proof of Theorem 2.3. Set R2 ∈ (0, R), Ω2 = x ∈ R N : |x| < R2 and ε(t) = Using Lemma 4.1, it then follows that there exists τ ∈ (0, T ∗ ) such that 0 < Set Z T∗ g1 (t) = (1 − ε(t))g(t) and G 1 (t) = g1 (s)ds. τ
τ ≤ t < T ∗.
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1955
Clearly, lim G 1 (t) = ∞,
t→T ∗
and lim
t→T ∗
G 1 (t) = 1. G(t)
On the other hand, Z
u p dx − u q
u t = ∆u + a(x) Ω
= ∆u + a(x)(1 − ε(t))g(t) + a(x)ε(t)g(t) − u q (x, t) ≥ ∆u + a(x)g1 (t) + a(R2 )ε(t)g(t) − u q (0, t) = ∆u + a(x)g1 (t),
x ∈ Ω2 , τ < t < T ∗ .
Clearly, u(x, t) is a sup-solution of the following equation: vt = ∆v + a(x)g1 (t),
x ∈ Ω2 , τ < t < T ∗ ,
v(x, t) = 0, x ∈ ∂Ω2 , τ < t < T ∗ , v(x, τ ) = u(x, τ ) − u(R2 , τ ), x ∈ Ω2 .
(4.9)
By the maximum principle, u(x, t) ≥ v(x, t) and vr ≤ 0 in Ω2 × (τ, T ∗ ). Using Theorem 3.1, we have lim∗
t→T
v(x, t) v(x, t) = lim∗ = a(x), t→T G 1 (t) G(t)
uniformly in any compact subsets of Ω2 , that is, lim inf
t→T ∗
u(x, t) ≥ a(x), G(t)
uniformly in any compact subsets of Ω2 . By the arbitrariness of Ω2 , we get lim∗ inf
t→T
u(x, t) ≥ a(x), G(t)
(4.10)
uniformly in any compact Rsubsets of Ω . Since u t ≤ ∆u + a(x) Ω u p dx, it then follows that u(x, t) is a sub-solution of the following equation: v1t = ∆v1 + a(x)g(t),
x ∈ Ω , 0 < t < T ∗,
v1 (x, t) = 0, x ∈ ∂Ω , 0 < t < T ∗ , v1 (x, 0) = u 0 (x), x ∈ Ω .
(4.11)
By the maximum principle, u(x, t) ≤ v1 (x, t) and v1r ≤ 0 in Ω × (0, T ∗ ). Using Theorem 3.1, it holds that lim∗
t→T
v1 (x, t) = a(x), G(t)
uniformly in all compact subsets of Ω , that is, lim∗ sup
t→T
u(x, t) ≤ a(x), G(t)
uniformly in all compact subsets of Ω . Combining (4.10) with (4.12) yields that lim∗
t→T
u(x, t) = a(x), G(t)
(4.12)
1956
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
uniformly in all compact subsets of Ω . A simple calculation yields the desired result. The proof is completed.
5. Proof of Theorem 2.4 Set g(t) = u p (0, t), G(t) =
Rt 0
g(s)ds.
Proof of Theorem 2.4. Since u t ≥ 0 and u r ≤ 0, it then follows that ∆u(0, t) ≤ 0 on (0, T ∗ ) and limt→T ∗ u(0, t) = ∞. Clearly, u t (0, t) ≤ u q (0, t) + a(0)u p (0, t),
0 < t < T ∗,
which implies lim∗ sup
t→T
u(0, t) ≤ a(0), G(t)
(5.1)
that is, lim G(t) = ∞.
t→T ∗
Since u t ≥ ∆u + a(x)u p (0, t), it then follows that u(x, t) is a sup-solution of the following equation: vt = ∆v + a(x)g(t),
x ∈ Ω , 0 < t < T ∗,
v(x, t) = 0, x ∈ ∂Ω , 0 < t < T ∗ , v(x, 0) = u 0 (x), x ∈ Ω .
(5.2)
By the maximum principle, u(x, t) ≥ v(x, t) and vr ≤ 0 in Ω × (0, T ∗ ). Using Theorem 3.1, we obtain lim∗
t→T
v(x, t) = a(x), G(t)
uniformly in any compact subsets of Ω , that is, lim∗ inf
t→T
u(x, t) ≥ a(x), G(t)
(5.3)
uniformly in any compact subsets of Ω , which implies lim∗ inf
t→T
u(0, t) ≥ a(0). G(t)
(5.4)
Combining (5.1) with (5.4) yields that lim∗
t→T
u(0, t) = a(0). G(t)
(5.5)
Multiplying both sides of the Eq. (2.7) by ϕ and integrating over Ω × (0, t), we have, for 0 < t < T ∗ , Z Z Z tZ Z tZ uϕdx − u 0 ϕdx = −λ uϕdxds + u q ϕdxds + G(t). Ω
Ω
0
Rt R
u q ϕdxds
Since p > q ≥ 1 and 0 Ω Rt R uϕdxds lim∗ 0 Ω =0 t→T G(t)
and
Ω
0
Rt
≤ Ω ϕdx 0 Rt R R
lim∗
t→T
0
Ω
u q (0, s)ds,
so we have that
u q ϕdxds = 0, G(t)
Ω
which imply R
uϕdx lim∗ Ω = 1. t→T G(t)
(5.6)
Q. Liu et al. / Nonlinear Analysis 67 (2007) 1947–1957
1957
By an argument similar to that in Theorem 3.1, we obtain lim
t→T ∗
u(x, t) = a(x), G(t)
uniformly in any compact subsets of Ω . A simple calculation yields the desired result. The proof is completed.
References [1] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations 153 (1999) 374–406. [2] J. Bebernes, A. Bressan, A. Lacey, Total blow-up versus single-point blow-up, J. Differential Equations 73 (1988) 30–44. [3] K. Deng, M.K. Kwang, H.A. Levine, The influence of nonlocal nonlinearities on the long time behavior of solutions of Burgers equation, Quart. Appl. Math. 5 (6) (1992) 1237–1245. [4] J.M. Chadam, A. Peirce, H.-M. Yin, The blow-up property of solutions to some diffusion equations with localized nonlinear reactions, J. Math. Anal. Appl. 169 (1992) 313–328. [5] M. Wang, Y. Wang, Properties of positive solutions for non-local reaction–diffusion problems, Math. Methods Appl. Sci. 19 (1996) 1141–1156. [6] J.R. Cannon, H.-M. Yin, A class of non-linear non-classical parabolic equations, J. Differential Equations 79 (1989) 266–288. [7] L. Wang, Q. Chen, The asymptotic behavior of blow-up solution of localized nonlinear equation, J. Math. Anal. Appl. 200 (1996) 315–321. [8] J.-S. Guo, H.-W. Su, The blow-up behavior of the solution of an integro-differential equation, Differential Integral Equations 5 (6) (1992) 1237–1245. [9] A. Okada, I. Fukuda, Total versus single point blow-up of solutions of a semi-linear parabolic equation with localized reaction, J. Math. Anal. Appl. 281 (2003) 485–500. [10] Q. Liu, Y. Li, H. Guo, Uniform blow-up rate for diffusion equations with localized nonlinear source, J. Math. Anal. Appl. 320 (2) (2006) 771–778.