Accepted Paper

Inserted: 16 feb 2006
Last Updated: 18 jan 2007

Year: 2006

Abstract:

We are interested in stabilityinstability of the zero steady state of the superlinear parabolic equation $u_t +\Delta^2u=
u
^{p-1}u$ in ${R}^n\times[0,\infty)$, 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 $t^{-1/(p-1)}$.

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