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Stable ergodicity of skew products of one-dimensional hyperbolic flows
Unfocused blow up solutions of semilinear parabolic equations
| 1. | Laboratoire d'Analyse Numérique, Université Pierre et Marie curie, 4, Place Jussieu, 7525 Paris Cedex 05, France |
$u_t - \Delta u = |u|^{p-1} u, \quad x\in\Omega,\quad t\in [0,T]$,
$u(t,x)=0, x\in\partial\Omega, \quad t\in [0,T] $,
$u(0,x) =u_0(x),\quad x\in\Omega $,
around a blow up point other than its centre of symmetry. We assume that $\Omega$ is a ball in $\mathbb R^N$ or $\Omega =\mathbb R^N$, and $p>1$. We show that $u$ behave as of a one-dimensional problem was concerned, that is, the possible asymptotic behaviors and final time profiles around an unfocused blow up point are the ones corresponding to the case of dimesion $N=1$.
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