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

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

2012-02-06

arxiv.org/abs/1202.1037v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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