Instantaneous and finite time blow-up of solutions to a reaction-diffusion equation with Hardy-type singular potential

Abstract

We deal with radially symmetric solutions to the reaction-diffusion equation with Hardy-type singular potentialut=Δum+K|x|2um, posed in RN×(0,T), in dimension N≥3, where m>1 and 0<K<(N−2)2/4. We prove that, in dependence of the initial condition u0∈L∞(RN)∩C(RN), its solutions may either blow up instantaneously or blow up in finite time at the origin, thus developing a singularity at x=0, but they can be continued globally in weak sense. The instantaneous blow-up occurs for example for any data u0 such that u0(0)>0. The proofs are based on a transformation mapping solutions to our equation into solutions to a non-homogeneous porous medium equation.