一类带非线性源的拟线性抛物方程解的熄灭问题

Quenching Phenomena for a Quasilinear Parabolic Equation with Nonlinear Source

研究了一类拟线性抛物方程边值问题解的熄灭问题,非线性抛物方程在随时间增加时,不一定存在连续的解,有的问题解是整体存在的,有的问题解就不存在,熄灭是解不存在的一种。形如ut-div[σ(|荦u|2)荦u]=λup的拟线性抛物方程在RN中有界凸空间上的解的熄灭问题,是几何学领域率先提出的,此时方程的主部项不再是一致椭圆的,这是与p-Laplace型方程的主要区别也是难点所在,利用上下解方法及积分估计方法得到两类在有限时间内解熄灭的结果。研究中所利用方法是一种常用方法,可以推广到更一般的拟线性抛物方程的研究。

For many quasilinear differential equations, their origin may be traced in biology and astrophysics, as related with generalized reaction diffusion theory, non-Newtonian fluid theory, non-Newtonian filtration theory and the turbulent flow of a gas in porous medium.Quenching phenomena for a class of quasilinear parabolic equations are studied in this paper.In the time domain, nonlinear parabolic equations not necessarily have a continuous solution.For some equations, solutions exist as a whole, for others solutions may not exist.Quenching phenomena are related with cases where solutions do not exist in some time period.This paper considers the extinction of solutions of an initial boundary value problem of the quasilinear parabolic equation ut-divσ（｜▽u｜2）▽u=λup in a bounded convex domain of RN with N≥2.This problem is first introduced in the field of geometry.At this time, the principal part of the equation is no longer uniformly elliptic.Using upper and lower solutions and the integral estimate method, two results of the extinction of the solution are obtained.The results obtained can be extended to a more general form of quasilinear parabolic equations.