Journal of Evolution Equations

, Volume 14, Issue 4–5, pp 749–777 | Cite as

Asymptotics for a nonlinear integral equation with a generalized heat kernel

  • Kazuhiro Ishige
  • Tatsuki KawakamiEmail author
  • Kanako Kobayashi


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 \({t \to \infty}\) .


Asymptotic Expansion Heat Kernel Decay Estimate Nonlinear Integral Equation Nonlinear Parabolic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Basel 2014

Authors and Affiliations

  • Kazuhiro Ishige
    • 1
  • Tatsuki Kawakami
    • 2
    Email author
  • Kanako Kobayashi
    • 1
  1. 1.Mathematical InstituteTohoku UniversityAobaJapan
  2. 2.Department of Mathematical SciencesOsaka Prefecture UniversitySakaiJapan

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