一类带非局部项的抛物型方程爆破时间的下界问题*

2016-02-07王忠谦宋明亮

湘潭大学自然科学学报 2016年3期

关键词：下界南京师范大学抛物

王忠谦，宋明亮,2*

(1.江苏第二师范学院数学与信息技术学院，江苏南京 210013；2.南京师范大学数学科学学院，江苏南京 210046)

一类带非局部项的抛物型方程爆破时间的下界问题*

王忠谦1，宋明亮1,2*

(1.江苏第二师范学院数学与信息技术学院，江苏南京 210013；2.南京师范大学数学科学学院，江苏南京 210046)

爆破时间；Dirichlet边界条件；Sobolev不等式

在[1]中，Payne和Schaefer得到了半线性抛物方程ut=Δu+f(u)在齐次Dirichlet边界条件下解的爆破时间的下界，这里，f满足某些适当的假设.从那以后，带Dirichlet边界条件的各类抛物方程解的爆破时间的下界被广泛研究(见[2-6]及其参考文献).在[7]中，作者研究了问题

(1)

u(x,t)=0,x∈∂Ω,t>0,

(2)

u(x,0)=u0(x),x∈Ω,

(3)

受到以上工作的启发，本文中对于问题(1)～(3)，在1

我们有如下主要结论：

为证明定理1，需要先证明如下引理.

‖v(u)‖.特别地，‖v(u)‖.

‖v(u)‖.

(4)

特别地，当q=n-1时，(4)可写作‖v(u)‖≤C(Ω ,n).

下面我们估计爆破时间t*的下界和爆破率的下界.

定理1的证明：设φ(t):=∫Ωumdx，则利用分部积分公式和引理2,

(5)

(6)

再由Sobolev不等式(见[12])

(7)

(8)

(9)

(10)

由于Y(φ)关于φ单调递减，故Y存在反函数Y-1且Y-1仍为单调递减函数.由(10)可得爆破率的下估计，φ(t)≥Y-1(t*-t).

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责任编辑：龙顺潮

Lower Bound for a Nonlocal Equation of Parabolic Type

WANGZhong-qian1,SONGMing-liang1,2*

(1. Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing 210013；2. School of Mathematical Sciences,Nanjing Normal University, Nanjing 210046 China)