Existence and nonexistence of time-global solutions to damped wave equation on half-line (Q1780773)

The authors compare the existence and the blowup properties of solutions to the semilinear heat equation \(\phi_t-\phi_{xx}=f(\phi)\) and the semilinear damped wave equation \(u_{tt}+u_t-u_{xx}=f(u)\), both with \(f(u)\sim | u| ^\rho\), posed in the half-space \(x,t\in (0,\infty)\), supplemented with the Dirichlet boundary conditions, and with suitable initial data. In particular, in the case of the first equation and for \(1<\rho\leq \rho_c=2\), they derive estimates (from above and below) of the blowup time. Next, an analogous result is shown for the damped wave equation.