Solutions of some nonlinear parabolic equations with initial blow-up

2008/09/10 by Waad Al Sayed, Sayed, Waad Al, Лаурент Верон +2
Computer Science · Mathematics · #35K60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #math.AP #msc:35K60

paper · pdf · doi:10.48550/arxiv.0809.1805

arxiv created 2008/09/10 · openalex publication_date 2008/09/10 · arxiv updated 2009/12/01 · openalex created_date 2021/06/22 · openalex updated_date 2026/07/28

Abstract

We study the existence and uniqueness of solutions of ∂_tu-Δu+u^q=0 (q>1) in Ω× (0,∞) where Ω⊂\mathbb R^N is a domain with a compact boundary, subject to the conditions u=f≥ 0 on ∂Ω× (0,∞) and the initial condition lim_{t→ 0}u(x,t)=∞. By means of Brezis' theory of maximal monotone operators in Hilbert spaces, we construct a minimal solution when f=0, whatever is the regularity of the boundary of the domain. When ∂Ω satisfies the parabolic Wiener criterion and f is continuous, we construct a maximal solution and prove that it is the unique solution which blows-up at t=0.

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