Blow-up phenomena for some nonlinear parabolic
- 格式：pdf
- 大小：206.87 KB
- 文档页数：5
∂u + ku = 0 on ∂ Ω × (0, t ∗ ), ∂ν u(x, 0) = g (x) ≥ 0 in Ω ,
(1.3)
✩ The work was supported by the national natural Science Foundation of China (Grants ♯ 10971234, ♯ 11001088, ♯11026227), Excellent Young Fund of Department of Education of Guangdong (Grant ♯ LYM10100), and Guangdong Natural Science Foundation (Grant ♯ S2011040000805). ∗ Corresponding author. E-mail address: yanliu99021324@ (Y. Liu).
|∇ un |2 = n2 u2(n−1) |∇ u|2 ,
and Hölder inequality, we obtain
ϕ ′ (t ) ≤ −
σ (σ − 1)
np+2
∫
Ω
|∇ un |p+2 dx + a1 σ |Ω | σ [ϕ(t )]
1
σ −1 σ
+ a2 σ
∫
Ω
uσ + γ − 1 d x ,
ϕ ′ (t ) = σ
∫
Ω
uσ −1 [((|∇ u|p + 1)u,i ),i + f (u)]dx
≤ −σ [σ − 1]
∫
Ω
u(n−1)(p+2) |∇ u|p+2 dx + σ a1
∫
Ω
uσ −1 dx + σ a2
∫
Ω
uσ −1+γ dx,
(2.5)
where we have used the divergence theorem, the boundary condition on u, and (2.1). Using the fact
article
info
ห้องสมุดไป่ตู้
abstract
This paper deals with the blow-up phenomena of the solutions to some nonlinear parabolic equation under Robin boundary conditions. Lower bounds for blow-up time are determined if the solutions blow up. © 2011 Elsevier Ltd. All rights reserved.
Mathematical and Computer Modelling 54 (2011) 3065–3069
Contents lists available at SciVerse ScienceDirect
Mathematical and Computer Modelling
journal homepage: /locate/mcm
Article history: Received 30 January 2011 Received in revised form 27 July 2011 Accepted 27 July 2011 Keywords: Blow-up Robin boundary conditions Lower bound Parabolic problems
Moreover assume that γ − 1 − p > 0, where γ and a1 , a2 are positive constants. Then the quantity
ϕ(t ) =
∫
Ω
u(n−1)(p+2)+2 dx =
∫
Ω
uσ d x ,
n>1
(2.2)
satisfies the differential inequality
1. Introduction In 2008, Payne et al. in their paper [1] considered the blow-up of solutions of equations of the form ut = div(ρ(|∇ u|2 )grad u) + f (u) (1.1)
p+2 2 (p + 2)2 p v |∇ v |2 , ∇ v 2 = ∫ ∫ 2 2 (p + 2)2 p+ v p |∇ v |2 dx ∇ v 2 dx =
Ω
4
Ω
≤
(p + 2)
4
2
∫
Ω
v
p+2
p ∫ p+ 2
dx
Ω
|∇ v |p+2 dx
2 p+ 2
Blow-up phenomena for some nonlinear parabolic problems under Robin boundary conditions✩
Yuanfei Li a , Yan Liu b,∗ , Shengzhong Xiao c
a b c
Department of Accounting, Huashang College Guangdong University of Business Studies, Guangzhou, 511300, PR China Department of Applied Mathematics, Guangdong University of Finance, Guangzhou 510521, PR China Guangdong AIB College, Guangzhou, Guangdong, 510507, PR China
∫
Ω
v p+2+ n dx,
β
(2.6)
where β = γ − 1 − p > 0. We now begin to consider the first term on the right of (2.6). Since 4 it follows by Hölder inequality that
0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.07.034
3066
Y. Li et al. / Mathematical and Computer Modelling 54 (2011) 3065–3069
where Ω is a smooth bounded domain in R3 with the boundary ∂ Ω , ∇ is the gradient operator, t ∗ is the possible blow-up time and f and g are positive functions. We have by the parabolic maximum principles [3,4] that u is nonnegative in x for t ∈ [0, t ∗ ). Clearly, the function ρ = |∇ u|p + 1 in (1.3) does not satisfy the condition (1.2). In fact the question of blow-up has received a great deal of attention in the recent literature (see [5–16,1] and the references cited therein). A variety of methods are used to study the question of blow-up in many cases, these methods lead to an upper bound on blow-up time when blow-up does occur. In the present paper, we use a differential inequality technique to derive the lower bound for the blow-up time when the blow-up occurs. 2. A lower bound for the blow-up time In this section we seek the lower bound for the blow-up time t ∗ under certain geometrical constraints on the region Ω . Also, in the process of deriving the lower bound, we make some assumption on the data of problem (1.3). We establish the following theorem: Theorem 1. Let u(x, t ) be the classical solution of problem (1.3) in a bounded star-shaped domain Ω assumed to be convex. Assume that the function f (x) satisfying the condition 0 < f (s) ≤ a1 + a2 sγ , s > 0. (2.1)
