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Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 26 January 2019, pp. 789-811
- Print publication:
- April 2020
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- Article
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We consider the Cauchy problem
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.
$$\left\{ {\matrix{ {u_t = \Delta u + u^p,\quad } \hfill & {x\in {\bf R}^N,\;t \leq 0,} \hfill \cr {u(x,0) = u_0(x),\quad } \hfill & {x\in {\bf R}^N,} \hfill \cr } } \right.$$
