Blow‐up solutions to nonlinear parabolic equations with non‐autonomous reactions under the mixed boundary conditions

Chung, Soon‐Yeong; Hwang, Jaeho

doi:10.1002/mma.7131

Cited by 2 publications

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References 16 publications

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“…for all t ∈ (0, t * ). Using ρ ′ 1 ≤ 0, ρ ′ 2 ≤ 0, and the assumption (23), we obtain from similar way to (14) that 2 + ǫ 2 A ′ (t)B(t) < A(t)B ′ (t) for all t ∈ (0, t * ). Therefore, we can obtain…”

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confidence: 53%

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

Chung

Hwang

2022

Preprint

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mentioning

confidence: 53%

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

Chung

Hwang

2022

Preprint

Self Cite

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

Chung

Hwang

2022

Bound Value Probl

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

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Blow‐up solutions to nonlinear parabolic equations with non‐autonomous reactions under the mixed boundary conditions

Cited by 2 publications

References 16 publications

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

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

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

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