非线性扩散方程静态解的稳定性

Stability of Stationary State Solution for a Reaction Density-Dependent Diffusion Equation

在本文中我们考虑下列非线性扩散方程在时间充分长时的性态u_t=((?)(u))_(xx)+(?)(u),(x∈R,t∈R^+=(0,+∞))其中函数(?)(u)和(?)(u)允许此方程具有行波解.首先我们给出该方程柯西问题的广义解的存在性、唯一性和一些比较原理.然后给定(?)(u)的某些条件,我们证明了一些阀值效应.由这些结果我们可以看到在这些假设条件下,静态解u=a稳定的,而u=0或u=1是不稳定的,等等.

In this paper we are interested in the large time behavior of the nonlinear diffusion equation We consider functions and which allow the equation to possess traveling wave solutions. We first present an existence and uniqueness as well as some comparison principle result of generalized solutions to the Cauchy problem. Then we give for some threshold results, from which we can see that u=a is stable, u=0 or u=1 is unstable under some assumptions, etc.