Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method

By Masahiro Ikeda, M. Sobajima. Nonlinear Analysis, 2019

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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 ) = e f ( x ) , x ∈ C Σ , τ ∂ t u ( x , 0 ) = τ e 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 e > 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.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66a492af2cb489b195692dd89c2673ae1ae1fbee