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].


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