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

Abstract

Let $\Omega$ be a bounded smooth domain in $\RR^N$. We consider the problem $u_t=\Delta u+V(x)u^p$ in $\Omega \times [0,T)$, with Dirichlet boundary conditions $u=0$ on $\partial \Omega \times [0,T)$ and initial datum $u(x,0)=M\varphi (x)$ where $M\ge 0$, $\varphi$ 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 ${\varphi }^{p-1}V$ attains its maximum.

Cited by in corpus (2)

• Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS
• Some Blow-Up Problems for a Semilinear Parabolic Equation with a Potential