1. Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method.
- Author
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Ikeda, Masahiro and Sobajima, Motohiro
- Subjects
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NUMERICAL solutions to parabolic differential equations , *NUMERICAL solutions to hyperbolic differential equations , *EVOLUTION equations , *BOUNDARY value problems , *BLOWING up (Algebraic geometry) , *DIFFERENTIAL inequalities - Abstract
Abstract This paper is concerned with the blowup phenomena for initial–boundary value problem (0.1) τ ∂ t 2 u (x , t) − Δ u (x , t) + a (x) ∂ t u (x , t) = λ | u (x , t) | p , (x , t) ∈ C Σ × (0 , T) , u (x , t) = 0 , (x , t) ∈ ∂ C Σ × (0 , T) , u (x , 0) = ε f (x) , x ∈ C Σ , τ ∂ t u (x , 0) = τ ε g (x) , x ∈ C Σ , where C Σ is a cone-like domain in R N (N ≥ 2) defined as C Σ = int r ω ∈ R N ; r ≥ 0 , ω ∈ Σ with a connected open set Σ in S N − 1 with smooth boundary ∂ Σ. If N = 1 , then we only consider two cases C Σ = (0 , ∞) and C Σ = R. Here a (x) is a non-zero coefficient of ∂ t u which could be complex-valued and space-dependent, λ ∈ ℂ is a fixed constant, and ε > 0 is a small parameter. The constants τ = 0 , 1 switch the parabolicity and hyperbolicity of the problem (0.1). The result proposes an argument for sharp upper lifespan estimates of solutions to (0.1) by a test function method based on Mitidieri and Pokhozhaev (2001). The crucial idea is to introduce an ordinary differential inequality with a parameter as a variable. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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