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

• Published in: Boundary value problems 2022-06, Vol.2022 (1), p.1-19, Article 46
• Main Authors: Chung, Soon-Yeong, Hwang, Jaeho
• Format: Article
• Language: English
• Subjects: Analysis; Approximations and Expansions; Blow-up; Boundary conditions; Boundary value problems; Difference and Functional Equations; Eigenvalues; Mathematical analysis; Mathematics; Mathematics and Statistics; Mixed nonlinear boundary; Nonlinear parabolic equation; Nonlinear systems; Ordinary Differential Equations; Partial Differential Equations
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-022-01627-9

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 ∗ ) , under mixed nonlinear boundary conditions ∂ u ∂ n + θ ( z ) u = h ( z , t , u ) on Γ 1 × ( 0 , t ∗ ) and u = 0 on Γ 2 × ( 0 , t ∗ ) , where Ω is a bounded domain and Γ 1 and Γ 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: ( 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 ∈ Ω , z ∈ ∂ Ω , t > 0 , and u ∈ R for some constants ϵ , β 1 , β 2 , γ 1 , and γ 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 , where ρ m : = inf s > 0 ρ ( s ) , λ R is the first Robin eigenvalue and λ S is the first Steklov eigenvalue. Lastly, we discuss blow-up solutions for nonlinear parabolic systems.