2020/04/12 by Razvan Gabriel Iagar, Iagar, Razvan Gabriel, Ariel Sánchez +1
Computer Science · Medicine · Engineering · #Advanced Mathematical Modeling in Engineering #Mathematical and Theoretical Epidemiology and Ecology Models #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2004.05650
We study the self-similar blow-up profiles associated to the following second\norder reaction-diffusion equation with strong weighted reaction and unbounded\nweight:
partialtu=
partialxx(um) + |x|
sigmaup, posed for\nx\∈ real, t\≥0, where m>1, 0<p<1 and \σ>2(1-p)/(m-1). As a\nfirst outcome, we show that finite time blow-up solutions in self-similar form\nexist for m+p>2 and \σ in the considered range, a fact that is\ncompletely new: in the already studied reaction-diffusion equation without\nweights there is no finite time blow-up when p<1. We moreover prove that, if\nthe condition m+p>2 is fulfilled, all the self-similar blow-up profiles are\ncompactly supported and there exist \two different interface behaviors\nfor solutions of the equation, corresponding to two different interface\nequations. We classify the self-similar blow-up profiles having both types of\ninterfaces and show that in some cases \global blow-up occurs, and in\nsome other cases finite time blow-up occurs \only at space infinity. We\nalso show that there is no self-similar solution if m+p<2, while the critical\nrange m+p=2 with \σ>2 is postponed to a different work due to\nsignificant technical differences.\n