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

On the half-line R+=(0,∞)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 ρc(N,k)ρc(N,k) of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space View the MathML sourceDN,k=R+k×RN-k is given by ρc(N,k)=1+2/(N+k)ρc(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 ρc(1,1)=2ρc(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+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).