Separate variable blow-up patterns for a reaction–diffusion equation with critical weighted reaction. (April 2022) - Iagar, Razvan Gabriel; Sánchez, Ariel
Abstract: We study the separate variable blow-up patterns associated to the following second order reaction–diffusion equation: ∂ t u = Δ u m + | x | σ u m, posed for x ∈ R N, t ≥ 0, where m > 1, dimension N ≥ 2 and σ > 0 . A new and explicit critical exponent σ c = 2 ( m − 1 ) ( N − 1 ) 3 m + 1 is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent σ c and also depends strongly on whether the dimension N ≥ 4 or N ∈ 2, 3 . These results extend previous works of the authors in dimension N = 1 .