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Dynamics of global solutions of a semilinear parabolic equation. (English) Zbl 1194.35069

Du, Yihong (ed.) et al., Recent progress on reaction-diffusion systems and viscosity solutions. Based on the international conference on reaction-diffusion systems and viscosity solutions, Taichung, Taiwan, January 3–6, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-283-473-7/hbk). 300-331 (2009).
The trivial steady state of the parabolic Cauchy problem \[ \begin{cases} u_t=\Delta u+u^p&x\in\mathbb{R}^N,\;t>0,\\ u(x,0)=u_0(x)&x\in\mathbb{R}^N,\\ u\geq0&x\in\mathbb{R}^N, \end{cases}\tag{1} \] where \(p>1\), is not stable in \(C_0\). Moreover, if \(p\) is large enough there is a continuum of steady states of (1). These facts lead to a structurally rich set of global solutions of (1). In the present survey article the author reviews the state of the art concerning this structure. After the definition of various critical values for \(p\) the following aspects are considered: attractivity of steady states, non-convergence and quasi-convergence, grow-up of solutions, convergence to the trivial steady state, convergence to self-similar solutions, and rapidly decaying solutions.
For the entire collection see [Zbl 1165.35002].

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35B09 Positive solutions to PDEs
35B33 Critical exponents in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K58 Semilinear parabolic equations
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