## 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