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

Abstract

Abstract In this paper, we firstly discuss blow-up phenomena for nonlinear parabolic equations u t = ∇ ⋅ [ ρ ( u ) ∇ u ] + f ( x , t , u ) , in Ω × ( 0 , t ∗ ) , $$ u_{t}=\nabla \cdot \bigl[\rho (u)\nabla u \bigr]+f(x,t,u),\quad \text{in }\Omega \times \bigl(0,t^{*}\bigr), $$ under mixed nonlinear boundary conditions ∂ u ∂ n + θ ( z ) u = h ( z , t , u ) $\frac{\partial u}{\partial n}+\theta (z)u=h(z,t,u)$ on Γ 1 × ( 0 , t ∗ ) $\Gamma _{1}\times (0,t^{*})$ and u = 0 $u=0$ on Γ 2 × ( 0 , t ∗ ) $\Gamma _{2}\times (0,t^{*})$ , where Ω is a bounded domain and Γ 1 $\Gamma _{1}$ and Γ 2 $\Gamma _{2}$ are disjoint subsets of a boundary ∂Ω. Here, f and h are real-valued C 1 $C^{1}$ -functions and ρ is a positive C 1 $C^{1}$ -function. To obtain the blow-up solutions, we introduce the following blow-up conditions: ( C ρ ) : ( 2 + ϵ ) ∫ 0 u ρ ( w ) f ( x , t , w ) d w ≤ u ρ ( u ) f ( x , t , u ) + β 1 u 2 + γ 1 , ( 2 + ϵ ) ∫ 0 u ρ 2 ( w ) h ( z , t , w ) d w ≤ u ρ 2 ( u ) h ( z , t , u ) + β 2 u 2 + γ 2 , $$ (C_{\rho})\,:\, \begin{aligned} &(2+\epsilon ) \int _{0}^{u}\rho (w)f(x,t,w)\,dw\leq u\rho (u)f(x,t,u)+ \beta _{1}u^{2}+\gamma _{1}, \\ &(2+\epsilon ) \int _{0}^{u}\rho ^{2}(w)h(z,t,w)\,dw \leq u\rho ^{2}(u)h(z,t,u)+ \beta _{2}u^{2}+ \gamma _{2}, \end{aligned} $$ for x ∈ Ω $x\in \Omega $ , z ∈ ∂ Ω $z\in \partial \Omega $ , t > 0 $t>0$ , and u ∈ R $u\in \mathbb{R}$ for some constants ϵ, β 1 $\beta _{1}$ , β 2 $\beta _{2}$ , γ 1 $\gamma _{1}$ , and γ 2 $\gamma _{2}$ satisfying ϵ > 0 , β 1 + λ R + 1 λ S β 2 ≤ ρ m 2 λ R 2 ϵ and 0 ≤ β 2 ≤ ρ m 2 λ S 2 ϵ , $$ \epsilon >0,\quad \beta _{1}+\frac{\lambda _{R}+1}{\lambda _{S}}\beta _{2} \leq \frac{\rho _{m}^{2}\lambda _{R}}{2}\epsilon \quad \text{and}\quad 0 \leq \beta _{2}\leq \frac{\rho _{m}^{2}\lambda _{S}}{2}\epsilon , $$ where ρ m : = inf s > 0 ρ ( s ) $\rho _{m}:=\inf_{s>0}\rho (s)$ , λ R $\lambda _{R}$ is the first Robin eigenvalue and λ S $\lambda _{S}$ is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.

Keywords

• Blow-up
• Mixed nonlinear boundary
• Nonlinear parabolic equation