The decay of the solutions for the heat equation with a potential

Kazuhiro Ishige, Michinori Ishiwata, Tatsuki Kawakami

Research output: Contribution to journalArticle

24 Citations (Scopus)

Abstract

We study the large time behavior of the solutions for the Cauchy problem, ∂tu = Δu + a(x,t)u in ℝN × (0, ∞u(x,0) = φ(x) inℝN, where φ εL1 (ℝN,(l + |x|K)dx) with K ≥ 0 and ∥a(t)∥ L ∞ (ℝN) = O(t-A) as t → ∞ for some A > 1. In this paper we classify the decay rate of the solutions and give the precise estimates on the difference between the solutions and their asymptotic profiles. Furthermore, as an application, we discuss the large time behavior of the global solutions for the semilinear heat equation, ∂tu = Δu + λ|u|p-1u, where λ ε ℝ. and p > 1.

Original languageEnglish
Pages (from-to)2673-2707
Number of pages35
JournalIndiana University Mathematics Journal
Volume58
Issue number6
DOIs
Publication statusPublished - 2009 Dec 1

Keywords

  • Heat equation
  • Large time behavior
  • Semilinear heat equation

ASJC Scopus subject areas

  • Mathematics(all)

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