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

Abstract

We deal with radially symmetric solutions to the reaction-diffusion equation with Hardy-type singular potential ut = Δum + K |x|2 um, 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.