Remark on upper bound for lifespan of solutions to semilinear evolution equations in a two-dimensional exterior domain

Abstract

In this paper we consider the following initial-boundary value problem with the power type nonlinearity | u | p with 1 < p ≤ 2 in a two-dimensional exterior domain (0.1) { τ ∂ t 2 u ( x , t ) − Δ u ( x , t ) + e i ζ ∂ t u ( x , t ) = λ | u ( x , t ) | p , ( x , t ) ∈ Ω × ( 0 , T ) , u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , u ( x , 0 ) = ε f ( x ) , x ∈ Ω , ∂ t u ( x , 0 ) = ε g ( x ) , x ∈ Ω , where Ω is given by Ω = { x ∈ R 2 ; | x | > 1 } , ζ ∈ [ − π 2 , π 2 ] , λ ∈ C and τ ∈ { 0 , 1 } switches the parabolicity, dispersivity and hyperbolicity. Remark that 2 = 1 + 2 / N is well-known as the Fujita exponent. If p > 2 , then there exists a small global-in-time solution of (0.1) for some initial data small enough (see Ikehata [11] ), and if p < 2 , then global-in-time solutions cannot exist for any positive initial data (see Ogawa–Takeda [22] and Lai–Yin [14] ). The result is that for given initial data ( f , τ g ) ∈ H 0 1 ( Ω ) × L 2 ( Ω ) satisfying ( f + τ g ) log ⁡ | x | ∈ L 1 ( Ω ) with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to (0.1) is given as the following double exponential type when p = 2 : LifeSpan ( u ) ≤ exp ⁡ [ exp ⁡ ( C ε − 1 ) ] . The crucial idea is to use test functions which approximates the harmonic function log ⁡ | x | satisfying Dirichlet boundary condition and the technique modified from [9] .