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Blow up of solutions of supercritical semilinear parabolic equations. (Explosion de solutions d’équations paraboliques semilinéaires supercritiques.) (French. Abridged English version) Zbl 0806.35005

Summary: We consider the equation \[ u_ t = \Delta u + u^ p, \quad x \in \mathbb{R}^ N\;(N \geq 1),\;t>0,\;p>1. \tag{E} \] We show that if \(N \geq 11\) and \(p>(N - 2(N - 1)^{1/2})/((N - 4) - 2(N - 1)^{1/2})\) then there exist radial and positive solutions of (E) which blow up at \(x=0\), \(t=T<\infty\) and such that \(\limsup_{t \uparrow T} (T-t)^{1/(p-1)} u(0,t) = \infty\). Precise asymptotics for these solutions near \(t=T\) are also obtained.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
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