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Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data

Part of: Parabolic equations and systems Partial differential equations Representations of solutions

Published online by Cambridge University Press:  26 January 2019

Yūki Naito
Yūki Naito*
Affiliation:
Department of Mathematics, Ehime University, 2-5 Bunkyo, Matsuyama790-8577, Japan ([email protected])
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Abstract

We consider the Cauchy problem

$$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$
where N > 2, p > 1, and u0 is a bounded continuous non-negative function in RN. We study the case where u0(x) decays at the rate |x|−2/(p−1) as |x| → ∞, and investigate the convergence property of the global solutions to the forward self-similar solutions. We first give the precise description of the relationship between the spatial decay of initial data and the large time behaviour of solutions, and then we show the existence of solutions with a time decay rate slower than the one of self-similar solutions. We also show the existence of solutions that behave in a complicated manner.

Keywords

semilinear heat equationself-similar solutionasymptotic behaviourJoseph-Lundgren critical exponent

MSC classification

Primary: 35K58: Semilinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions 35C06: Self-similar solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 2 , April 2020 , pp. 789 - 811
Copyright
Copyright © Royal Society of Edinburgh 2019

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