Critical exponent for the Cauchy problem to the weakly coupled damped wave system

In this paper, we consider a system of weakly coupled semilinear damped wave equations. We determine the critical exponent for any space dimensions. Our proof of the global existence of solutions for supercritical nonlinearities is based on a weighted energy method, whose multiplier is appropriately modified in the case where one of the exponent of the nonlinear term is less than the so called Fujita's critical exponent. We also give estimates of the lifespan of solutions from above for subcritical nonlinearities.