Refined asymptotic profiles for a semilinear heat equation

Abstract

We study the large time behavior of the solutions of the Cauchy problem for a semilinear heat equation,whereAssume that u is a solution of (P) satisfyingfor some constants C > 0 and A > 1. Then it is well known that the solution u behaves like the heat kernel. In this paper we give the ([K ] + 2)th order asymptotic expansion of the solution u, and reveal the relationship between the asymptotic profile of the solution u and the nonlinear term F. Here [K ] is the integer satisfying K − 1 < [K ] ≤ K .

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

“…Before treating our main theorem, modifying [9], we introduce the functions U n ¼ U n ðx; tÞ defined inductively by …”

“…Furthermore, combining (1.5), and (2.1)-(2.3) we have uðx; tÞ ¼ ðSðtÞjÞðxÞ þ 2k The end of this subsection we introduce some operators P K ðtÞ and Q K ðtÞ. Following [6] and [9], for any k b 0 and t > 0, we introduce a linear operator P k ðtÞ on L for all t > 0 (see [6]). This operators are key of our proof, in particular (2.8) and (2.9) are crucial properties in our analysis.…”

“…Furthermore, as far as we know, there are no results treating higher order asymptotic expansions of the solution of (1.1) even if j A C with k A R and p > 1 þ 2=N, in [9], improving the argument in [6], the author of this paper and Ishige established the useful method which derives higher order asymptotic expansions of the solutions for the Cauchy problem (1.10). The argument in [9] is applicable to the large class of the nonlinear parabolic equations in the whole space (see [8]).…”

“…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).…”

“…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).…”