Self-Similar Blow-Up Profiles for a Reaction-Diffusion Equation with Strong Weighted Reaction

Abstract

Abstract We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight: ∂ t ⁡ u = ∂ x ⁢ x ⁡ ( u m ) + | x | σ ⁢ u p , \partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p}, posed for x ∈ ℝ {x\in\mathbb{R}} , t ≥ 0 {t\geq 0} , where m > 1 {m>1} , 0 < p < 1 {0<p<1} and σ > 2 ⁢ ( 1 - p ) m - 1 {\sigma>\frac{2(1-p)}{m-1}} . As a first outcome, we show that finite time blow-up solutions in self-similar form exist for m + p > 2 {m+p>2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when p < 1 {p<1} . We moreover prove that, if the condition m + p > 2 {m+p>2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if m + p < 2 {m+p<2} , while the critical range m + p = 2 {m+p=2} with σ > 2 {\sigma>2} is postponed to a different work due to significant technical differences.