Global solvability and behavior of solutions of the cauchy problem for a system of two semilinear hyperbolic equations with dissipation

We obtain a sufficient condition for the existence of global solutions of Cauchy problems for systems of two semilinear hyperbolic equations with dissipation and find the decay rate of the solutions. We derive a sufficient condition for the absence of global solutions of systems of two semilinear hyperbolic inequalities. The proof of the absence of global solutions is performed on the basis of the test function method developed by S.I. Pokhozhaev, L. Veron, E. Mitidieri, et al.