On directional blow-up for a semilinear heat equation with space-dependent reaction

Abstract

We consider nonnegative solutions u of the Cauchy problem for a semilinear heat equation with space-dependent reaction: $u_t=\Delta u+\mu \left(x\right)u^p$, $u(x,0)=u_0\left(x\right)$, where $\mu \left(x\right)\ge 0$ satisfies some condition and the initial data $u_0\left(x\right)(\not\equiv 0)$ satisfies ${\Vert \tilde{\mu }{u}_0\Vert }_{L^{\infty }\left(R^N\right)}<\infty$ with $\tilde{\mu }={\mu }^{1/(p-1)}$. We study weighted solutions $\tilde{\mu }u$ which blow up at minimal blow-up time. Such a weighted solution blows up at space infinity in some direction (directional blow-up). We call this direction a blow-up direction of $\tilde{\mu }u$. We give a sufficient and necessary condition on $u_0$ for a weighted solution to blow up at minimal blow-up time. Moreover, we completely characterize blow-up directions of $\tilde{\mu }u$ by the profile of the initial data.