Nonexistence of Global Solutions for a Weakly Coupled System of Semilinear Damped Wave Equations of Derivative Type in the Scattering Case

In this paper, we consider the blow-up for solutions to a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case. The assumption on the time-dependent coefficients for the damping terms means that these coefficients are summable and nonnegative. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single semilinear equation, we employ Kato’s lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case, an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p - q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of semilinear not-damped wave equations with the same kind of nonlinearities.