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

Abstract

Let u be a solution of the Cauchy problem for the nonlinear parabolic equationand assume that the solution u behaves like the Gauss kernel as t → ∞. In this paper, under suitable assumptions of the reaction term F and the initial function ϕ, we establish the method of obtaining higher order asymptotic expansions of the solution u as t → ∞.

Global solutions for a nonlinear integral equation with a generalized heat kernel

“…Then, by the same argument as in the proof of Theorem 1.1, we see that there exists a solution u of (1.1) satisfying (1.10). Furthermore, applying a similar argument as in the proof of Theorem 3.1 in [17] with (1.10), we obtain assertions (a) and (b).…”

Large time behavior of ODE type solutions to nonlinear diffusion equations

“…In [9], developing the arguments in [7] and [8], the second author of this paper and Kawakami studied the Cauchy problem to nonlinear diffusion equations of the form…”

Higher Order Asymptotic Expansion for the Heat Equation with a Nonlinear Boundary Condition

“…The argument in [9] is applicable to the large class of the nonlinear parabolic equations in the whole space (see [8]). However it is not applicable directly to nonlinear boundary value problem (1.1) because the integral equation has a di¤erent kernel and the nonlinear term appears on the boundary (see Definition 2.1).…”

“…In this paper, as an extension of [9], under condition (1.2) and p > 1 þ 1=N, we consider the initial-boundary value problem (1.1), and study the large time behavior of the solutions satisfying (1.3). In particular, modifying the arguments in [8] and [9], we give higher order asymptotic expansions of the solution u of (1.1). Throughout this paper we write…”