带梯度项的非线性抛物方程正解的爆破

Blow-up of Positive Solutions for a Family of Nonlinear Parabolic Equations with a Gradient Term

研究了RN中一般区域上的一族带非线性梯度项的非线性退缩抛物方程解的blow-up性质.通过构造适当的辅助函数,利用特征函数法和不等式技巧,给出了其齐次Dirichlet边值问题的正解产生blow-up的充分条件;利用能量方法,证明了其Cauchy问题非平凡整体解的不存在性.本文的方法也适用于研究其它带非线性源的退缩非线性抛物方程解的blow-up问题.

The blow-up behavior of positive solutions for a family of nonlinear degenerate parabolic equations with nonlinear gradient term in general domain in R^N was Studied. By constructing an auxiliary function and using the eigenfunction method and inequality technic on it, the sufficient conditions for blow-up positive solutions of the equations with homogeneous Dirichlet boundary conditions were obtained,and then the nonexistence of global solutions of the Cauchy problem was proved by using the energy method. The methods in here＇could be applied to a wide class of nonlinear degenerate parabolic equations with nonlinear sources.