Existence and nonexistence of time-global solutions to damped wave equation on half-line

Abstract

On the half-line R + = ( 0 , ∞ ) the initial-boundary value problems with null-Dirichlet boundary for both the semilinear heat equation and damped wave equation are considered. The critical exponent ρ c ( N , k ) of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space D N , k = R + k × R N - k is given by ρ c ( N , k ) = 1 + 2 / ( N + k ) (J. Appl. Math. Phys. 39 (1988) 135–149; Arch. Rational Mech. Anal. 109 (1990) 63–71). Since the damped wave equation is expected to be close to the heat equation (J. Differential Equations 191 (2003) 445–469; Math. Z. 244 (2003) 631–649), the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation. However, there is no blow-up result on the halved space for the damped wave equation. In this paper, the exponent ρ c ( 1 , 1 ) = 2 is shown to be critical for the existence and nonexistence of time-global solution to both the semilinear heat equation and damped wave equation on the half-line R + , together with the derivation of the blow-up time. For the proof the explicit formulas of solutions are used in a similar fashion to those in Li and Zhou (Discrete Continuous Dynamic Systems 1 (1995) 503–520).