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Asymptotics for a nonlinear integral equation with a generalized heat kernel

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Abstract

This paper is concerned with a nonlinear integral equation

$$(P)\qquad u(x, t)=\int_{{\bf R}^N}G(x-y, t)\varphi(y){d}y+\int_0^t \int_{{\bf R}^N}G(x-y, t-s)f(y, s:u){d}y{d}s, \quad $$

where N ≥  1, \({\varphi \in L^\infty({\bf R}^N) \cap L^1({\bf R}^N,(1+|x|^K){d}x)}\) for some K ≥  0. Here, G = G(x,t) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P) behaving like a multiple of the integral kernel G as \({t \to \infty}\) .

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

    Kazuhiro Ishige & Kanako Kobayashi

  2. Department of Mathematical Sciences, Osaka Prefecture University, Sakai, 599-8531, Japan

    Tatsuki Kawakami

Authors
  1. Kazuhiro Ishige
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  2. Tatsuki Kawakami
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  3. Kanako Kobayashi
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Correspondence to Tatsuki Kawakami.

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Ishige, K., Kawakami, T. & Kobayashi, K. Asymptotics for a nonlinear integral equation with a generalized heat kernel. J. Evol. Equ. 14, 749–777 (2014). https://doi.org/10.1007/s00028-014-0237-3

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  • DOI: https://doi.org/10.1007/s00028-014-0237-3

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