Blow-up conditions of nonlinear parabolic equations and systems under mixed nonlinear boundary conditions

Abstract

In this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations ut=∇⋅[ρ(u)∇u]+f(x,t,u),in Ω×(0,t∗),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ u_{t}=\ abla \\cdot \\bigl[\\rho (u)\ abla u \\bigr]+f(x,t,u),\\quad \ ext{in }\\Omega \ imes \\bigl(0,t^{*}\\bigr), $$\\end{document} under mixed nonlinear boundary conditions frac{partial u}{partial n}+theta (z)u=h(z,t,u) on Gamma _{1}times (0,t^{*}) and u=0 on Gamma _{2}times (0,t^{*}), where Ω is a bounded domain and Gamma _{1} and Gamma _{2} are disjoint subsets of a boundary ∂Ω. Here, f and h are real-valued C^{1}-functions and ρ is a positive C^{1}-function. To obtain the blow-up solutions, we introduce the following blow-up conditions: (Cρ):(2+ϵ)∫0uρ(w)f(x,t,w)dw≤uρ(u)f(x,t,u)+β1u2+γ1,(2+ϵ)∫0uρ2(w)h(z,t,w)dw≤uρ2(u)h(z,t,u)+β2u2+γ2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ (C_{\\rho})\\,:\\, \\begin{aligned} &(2+\\epsilon ) \\int _{0}^{u}\\rho (w)f(x,t,w)\\,dw\\leq u\\rho (u)f(x,t,u)+ \\beta _{1}u^{2}+\\gamma _{1}, \\\\ &(2+\\epsilon ) \\int _{0}^{u}\\rho ^{2}(w)h(z,t,w)\\,dw \\leq u\\rho ^{2}(u)h(z,t,u)+ \\beta _{2}u^{2}+ \\gamma _{2}, \\end{aligned} $$\\end{document} for xin Omega , zin partial Omega , t>0, and uin mathbb{R} for some constants ϵ, beta _{1}, beta _{2}, gamma _{1}, and gamma _{2} satisfying ϵ>0,β1+λR+1λSβ2≤ρm2λR2ϵand0≤β2≤ρm2λS2ϵ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\epsilon >0,\\quad \\beta _{1}+\\frac{\\lambda _{R}+1}{\\lambda _{S}}\\beta _{2} \\leq \\frac{\\rho _{m}^{2}\\lambda _{R}}{2}\\epsilon \\quad \ ext{and}\\quad 0 \\leq \\beta _{2}\\leq \\frac{\\rho _{m}^{2}\\lambda _{S}}{2}\\epsilon , $$\\end{document} where rho _{m}:=inf_{s>0}rho (s), lambda _{R} is the first Robin eigenvalue and lambda _{S} is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.