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Unfocused blow up solutions of semilinear parabolic equations

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  • The aim of this paper is to study the blow up behavior of a radially symmetric solution $u$ of the semilinear parabolic equation

    $u_t - \Delta u = |u|^{p-1} u, \quad x\in\Omega,\quad t\in [0,T]$,

    $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 $N=1$.

    Mathematics Subject Classification: 35B40, 35K55, 35K57.


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