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Asymptotically self-similar behaviour of global solutions for semilinear heat equations with algebraically decaying initial data. (English) Zbl 1439.35305

Summary: We consider the Cauchy problem \[\begin{cases} u_t = \Delta u + u^p,\quad & x\in\mathbb{R}^N, t \leq 0, \\ u(x,0) = u_0(x),\quad & x\in \mathbb{R}^N,\end{cases}\] where \(N > 2, p > 1\), and \(u_0\) is a bounded continuous non-negative function in \(\mathbb{R}^N\). We study the case where \(u_0(x)\) decays at the rate |\(x|^{-2/(p -1)}\) as |\(x| \rightarrow \infty \), 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.

MSC:

35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35C06 Self-similar solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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