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

Abstract

AbstractIn this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations$$ u_{t}=\nabla \cdot \bigl[\rho (u)\nabla u \bigr]+f(x,t,u),\quad \text{in }\Omega \times \bigl(0,t^{*}\bigr), $$ut=∇⋅[ρ(u)∇u]+f(x,t,u),in Ω×(0,t∗),under mixed nonlinear boundary conditions$\frac{\partial u}{\partial n}+\theta (z)u=h(z,t,u)$∂u∂n+θ(z)u=h(z,t,u)on$\Gamma _{1}\times (0,t^{*})$Γ1×(0,t∗)and$u=0$u=0on$\Gamma _{2}\times (0,t^{*})$Γ2×(0,t∗), where Ω is a bounded domain and$\Gamma _{1}$Γ1and$\Gamma _{2}$Γ2are disjoint subsets of a boundary∂Ω. Here,fandhare real-valued$C^{1}$C1-functions andρis a positive$C^{1}$C1-function. To obtain the blow-up solutions, we introduce the following blow-up conditions:$$ (C_{\rho})\,:\, \begin{aligned} &(2+\epsilon ) \int _{0}^{u}\rho (w)f(x,t,w)\,dw\leq u\rho (u)f(x,t,u)+ \beta _{1}u^{2}+\gamma _{1}, \\ &(2+\epsilon ) \int _{0}^{u}\rho ^{2}(w)h(z,t,w)\,dw \leq u\rho ^{2}(u)h(z,t,u)+ \beta _{2}u^{2}+ \gamma _{2}, \end{aligned} $$(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,for$x\in \Omega $x∈Ω,$z\in \partial \Omega $z∈∂Ω,$t>0$t>0, and$u\in \mathbb{R}$u∈Rfor some constantsϵ,$\beta _{1}$β1,$\beta _{2}$β2,$\gamma _{1}$γ1, and$\gamma _{2}$γ2satisfying$$ \epsilon >0,\quad \beta _{1}+\frac{\lambda _{R}+1}{\lambda _{S}}\beta _{2} \leq \frac{\rho _{m}^{2}\lambda _{R}}{2}\epsilon \quad \text{and}\quad 0 \leq \beta _{2}\leq \frac{\rho _{m}^{2}\lambda _{S}}{2}\epsilon , $$ϵ>0,β1+λR+1λSβ2≤ρm2λR2ϵand0≤β2≤ρm2λS2ϵ,where$\rho _{m}:=\inf_{s>0}\rho (s)$ρm:=infs>0ρ(s),$\lambda _{R}$λRis the first Robin eigenvalue and$\lambda _{S}$λSis the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.