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

TL;DR

This paper develops a method for deriving higher order asymptotic expansions of solutions to nonlinear parabolic equations, generalizing previous results and applicable to a broad class of such equations as time approaches infinity.

Contribution

It introduces a systematic approach for obtaining detailed asymptotic expansions of solutions to nonlinear parabolic equations under broad conditions.

Findings

Established higher order asymptotic expansions for solutions

Applicable to a large class of nonlinear parabolic equations

Generalized previous results in the field

Abstract

Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$ \partial_t u=\Delta u+F(x,t,u,\nabla u) \quad in \quad{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad in \quad{\bf R}^N, $$ 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 $\varphi$, 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.