Asymptotic expansions of the solutions of the Cauchy problem for nonlinear parabolic equations

Abstract

Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $${\partial }_{t}u=\Delta u+F(x,t,u,\nabla u)in{\mathbf{R}}^N\times (0,\infty ),u(x,0)=\phi (x)in{\mathbf{R}}^N,$$ and assume that the solution $u$ behaves like the Gauss kernel as $t\to \infty$. In this paper, under suitable assumptions of the reaction term $F$ and the initial function $\phi$, we establish the method of obtaining higher order asymptotic expansions of the solution $u$ as $t\to \infty$. This paper is a generalization of our previous paper, and our arguments are applicable to the large class of nonlinear parabolic equations.

References in corpus (1)

• Spikes and diffusion waves in one-dimensional model of chemotaxis

Cited by in corpus (2)

• Asymptotics for a nonlinear integral equation with a generalized heat kernel
• Asymptotic expansions of solutions of fractional diffusion equations