A New Condition for the Concavity Method of Blow-up Solutions to p-Laplacian Parabolic Equations

2017/06/20 by Soon-Yeong Chung, Chung, Soon-Yeong, Min-Jun Choi +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1706.06893

15 pages. arXiv admin note: text overlap with arXiv:1706.03494

arxiv created 2017/06/20 · arxiv updated 2017/06/22

Abstract

In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equations \begincases u_t(x,t)=div(|∇ u(x,t)|^p-2∇ u(x,t))+f(u(x,t)), (x,t)∈ Ω×(0,+∞), \newline u(x,t)=0, (x,t)∈∂ Ω×[0,+∞), \newline u(x,0)=u₀≥0, x∈Ω, \endcases where p≥2 and Ω is a bounded domain of ℝ^N (N≥1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition \mbox(C_p)\hspace1cm α∫₀^uf(s)ds ≤ uf(u)+βu^p+γ, u>0 for some α, β, γ>0 with 0<β≤\frac(α-p)λ_{1, p}p, where λ_{1, p} is the first eigenvalue of p-Laplacian Δ_p, and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C_p) improves the conditions ever known so far.

Citations

• A New Condition for Blow-up Solutions to Discrete Semilinear Heat Equations on Networks

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