A New Condition for the Concavity Method of Blow-up Solutions to Semilinear Heat Equations

2017/05/16 by Soon‐Yeong Chung, Soon-Yeong Chung, Chung, Soon-Yeong +2
Computer Science · Mathematics · #35K57 #39A14 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations #math.AP #msc:35K57 #msc:39A14

paper · pdf · doi:10.48550/arxiv.1705.05629

7 pages

arxiv created 2017/05/16 · openalex publication_date 2017/05/16 · arxiv updated 2017/05/17 · openalex created_date 2017/05/26 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the semilinear heat equations under Dirichlet boundary condition u_t(x,t)=Δu(x,t)+f(u(x,t)), (x,t)∈ Ω×(0,+∞), u(x,t)=0, (x,t)∈∂ Ω×[0,+∞), u(x,0)=u₀≥0, x∈Ω, where Ω is a bounded domain of ℝ^N (N≥1) with smooth boundary ∂Ω. The main contribution of our work is to introduce a new condition (C) α∫₀^uf(s)ds ≤ uf(u)+βu²+γ, u>0 for some α, β, γ>0 with 0<β≤\frac(α-2)λ₀2, where λ₀ is the first eigenvalue of Laplacian Δ, and we use the concavity method to obtain the blow-up solutions to the semilinear heat equations. In fact, it will be seen that the condition (C) improves the conditions known so far.

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