## Initial blow-up for the solutions of a semilinear parabolic equation with source term.(English)Zbl 0914.35055

Équations aux dérivées partielles et applications. Articles dédiés à Jacques-Louis Lions. Gauthier-Villars: Paris. 189-198 (1998).
Summary: We study the initial blow-up of the positive solutions of the semilinear parabolic equation $$u_t=\Delta u+u^q$$ in $$\Omega\times(0,T)$$ with $$q>1$$, for any domain $$\Omega$$ of $$\mathbb{R}^N$$. We show that the Harnack inequality holds, and prove the following a priori estimate in any domain $$\overline \omega\Subset \Omega$$, (or $$\overline\omega =\Omega=\mathbb{R}^N)$$: $u(x,t) \leq Ct^{-1/(q-1)}\quad \text{in }\overline\omega\times(0,T/2),$ with $$C=C (\overline\omega,N,q)$$, whenever $$q<N(N+2)/(N-1)^2$$.
For the entire collection see [Zbl 0899.00020].

### MSC:

 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs

### Keywords:

positive solutions; Harnack inequality