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Self-Similar Blow-Up Profiles for a Reaction-Diffusion Equation with Strong Weighted Reaction

Authors
  • Iagar, Razvan Gabriel1
  • Sánchez, Ariel1
  • 1 Universidad Rey Juan Carlos, Móstoles, 28933 , (Spain)
Type
Published Article
Journal
Advanced Nonlinear Studies
Publisher
De Gruyter
Publication Date
Aug 06, 2020
Volume
20
Issue
4
Pages
867–894
Identifiers
DOI: 10.1515/ans-2020-2104
Source
De Gruyter
Keywords
License
Yellow

Abstract

We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight: ∂t⁡u=∂x⁢x⁡(um)+|x|σ⁢up,\partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p}, posed for x∈ℝ{x\in\mathbb{R}}, t≥0{t\geq 0}, where m>1{m>1}, 0<p<1{0<p<1} and σ>2⁢(1-p)m-1{\sigma>\frac{2(1-p)}{m-1}}. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for m+p>2{m+p>2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when p<1{p<1}. We moreover prove that, if the condition m+p>2{m+p>2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if m+p<2{m+p<2}, while the critical range m+p=2{m+p=2} with σ>2{\sigma>2} is postponed to a different work due to significant technical differences.

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