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2006-03

Global solutions for superlinear parabolic equations

by Gazzola, F., Grunau, H.-Ch..


Series: 2006-03, Preprints

MSC:
35K55 Nonlinear parabolic equations

Abstract:
We are interested in stability/instability of the zero steady state of the superlinear
parabolic equation $u_t +\Delta^2u=|u|^{p-1}u$ in $\mathbb{R}^n\times[0,\infty)$,
where the exponent is considered in the ``super-Fujita'' range $p>1+4/n$.
We determine the corresponding limiting growth at infinity for the initial data
giving rise to global bounded solutions.
In the supercritical case $p>(n+4)/(n-4)$ this is related to the asymptotic behaviour of positive steady states,
which the authors have recently studied.
Moreover, it is shown that the solutions found for the parabolic problem decay to $0$ at rate $t^{-1/(p-1)}$.

Keywords:
super-Fujita, stability, decay rate, global strong solution

This paper was published in:
Calc. Var. 30, 389-415 (2007).