The blow-up problem for a semilinear parabolic equation with a potential

Abstract

Let $Ω$ be a bounded smooth domain in $\RR^N$. We consider the problem $u_t= Δu + V(x) u^p$ in $Ω\times [0,T)$, with Dirichlet boundary conditions $u=0$ on $\partial Ω\times [0,T)$ and initial datum $u(x,0)= M ϕ(x)$ where $M \geq 0$, $ϕ$ is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of $M$. As a consequence of these estimates we find that, for $M$ large, the blow up set concentrates near the points where $ϕ^{p-1}V$ attains its maximum.