Self-similar blow-up profiles for a reaction-diffusion equation with\n critically strong weighted reaction

2020/06/01 by Razvan Gabriel Iagar, Iagar, Razvan Gabriel, Ariel Sánchez +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2006.01076

openalex publication_date 2020/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify the self-similar blow-up profiles for the following\nreaction-diffusion equation with critical strong weighted reaction and\nunbounded weight:
partial_tu=
partial_xx(u^m) + |x|
^sigmau^p, posed\nfor x\∈ real, t\≥0, where m>1, 0<p<1 such that m+p=2 and\n\σ>2 completing the analysis performed in a recent work where this very\ninteresting critical case was left aside. We show that finite time blow-up\nsolutions in self-similar form exist for \σ>2. Moreover all the blow-up\nprofiles have compact support and their supports are \localized: there\nexists an explicit \η>0 such that any blow-up profile satisfies rm\nsupp ,f\⊆[0,\η]. This property is unexpected and contrasting with\nthe range m+p>2. We also classify the possible behaviors of the profiles near\nthe origin.\n

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