Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction

Abstract

We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and unbounded weight: ∂tu = ∂x x (um) + |x| σ u p, posed for x ∈ R, t ≥ 0, where m > 1, 0 < p < 1 such that m+ p = 2 and σ > 2 completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for σ > 2. Moreover all the blow-up profiles have compact support and their supports are localized: there exists an explicit η > 0 such that any blow-up profile satisfies supp f ⊆ [0, η]. This property is unexpected and contrasting with the range m+ p > 2. We also classify the possible behaviors of the profiles near the origin.