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: ut=Δu+μ(x)up, u(x,0)=u0(x), where μ(x)≥0 satisfies some condition and the initial data u0(x)(≢0) satisfies ‖μ˜u0‖L∞(RN)<∞ with μ˜=μ1/(p−1). We study weighted solutions μ˜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 μ˜u. We give a sufficient and necessary condition on u0 for a weighted solution to blow up at minimal blow-up time. Moreover, we completely characterize blow-up directions of μ˜u by the profile of the initial data.