Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential

Abstract

We prove existence and uniqueness of a global in time self-similar solution growing up as t → ∞ for the following reaction-diffusion equation with a singular potential ∂tu = ∆u^m + |x|^σ u^p posed in dimension N ≥ 2, with m > 1, σ ∈ (−2, 0) and 1 <p< 1 − σ (m − 1)/2. For the special case of dimension N = 1, the same holds true for σ ∈ (−1, 0) and similar ranges for m and p. The existence of this global solution prevents finite time blow-up even with m > 1 and p > 1, showing an interesting effect induced by the singular potential |x|^σ . This result is also applied to reaction-diffusion equations with general potentials V (x) to prevent finite time blow-up via comparison.