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In this paper we consider the blow-up of solutions to a weakly coupled systemof semilinear damped wave equations in the scattering case with nonlinearitiesof mixed type, namely, in one equation a power nonlinearity and in the other asemilinear term of derivative type. The proof of the blow-up results is basedon an iteration argument. As expected, due to the assumptions on thecoefficients of the damping terms, we find as critical curve in the p-q planefor the pair of exponents (p,q) in the nonlinear terms the same one found byHidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupledsystem of semilinear wave equations with the same kind of nonlinearities. Inthe critical and not-damped case we provide a different approach from the testfunction method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of thesolution on the critical curve, improving in some cases the upper boundestimate for the lifespan. More precisely, we combine an iteration argumentwith the so-called slicing method to show the blow-up dynamic of a weightedversion of the functionals used in the subcritical case.

Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms