一类抛物型Kirchhoff方程解的爆破性质

Blow-Up Properties of Solutions to a Class of Parabolic Type Kirchhoff Equations

该文研究一类抛物型Kirchhoff方程的初边值问题解的爆破性质.主要结果包含两个部分.第一部分中考虑了具一般扩散系数M(‖▽u‖22)和一般非线性项f(u)的抛物型Kirchhoff方程.确立了新的有限时刻爆破准则,同时给出了爆破时刻的上下界估计.第二部分中研究了当M(‖▽u‖22)=a+b‖▽u‖22且f(u)=|u|^(q-1)u的情形,Han和Li就初始能量为次临界,临界和超临界的情形分别给出了当q>3时该问题解全局存在和有限时刻爆破的阈值结果^([11]).该文补充了他们的结果,证明了q=3在某种意义上是该问题存在有限时刻爆破解的临界指数.

In this paper,blow-up properties of solutions to an initial-boundary value problem for a parabolic type Kirchhoff equation are studied.The main results contain two parts.In the first part,we consider this problem with a general diffusion coefficient M(||▽u||22)and general nonlinearity f(u).A new finite time blow-up criterion is established,and the upper and lower bounds for the blow-up time are also derived.In the second part,we deal with the case that M(||▽u||22)=a+b||▽u||22 and f(u)=|u|^(q-1)u,which was considered in[Computers and Mathematics with Applications,2018,75:3283-3297]with q>3,where global existence and finite time blow-up of solutions were obtained for subcritical,critical and supercritical initial energy.Their results are complemented in this paper in the sense that q=3 will be shown to be critical for the existence of finite time blow-up solutions to this problem.