2021/08/20 by Iagar, Razvan Gabriel, Muñoz, Ana I., Sánchez, Ariel
#35A24 #35B44 #35C06 #35K10 #35K65 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2108.09088
We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: ∂tu=Δum+|x|σup, posed in any space dimension x\inRN, t≥0 and with exponents m>1, p∈(0,1) and σ>2(1-p)/(m-1). We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are compactly supported and might present two different types of interface behavior and three different possible good behaviors near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of σ. This paper generalizes in dimension N>1 previous results by the authors in dimension N=1 and also includes some finer classification of the profiles for σ large that is new even in dimension N=1.