A MIXED PROBLEM FOR SYSTEMS OF SEMILINEAR HYPERBOLIC EQUATIONS WITH NONLINEAR DISSIPATION AND NONLINEAR SOURCE

We study a mixed problem for systems of semilinear hyperbolic equations with nonlinear dissipation and focusing, as well as with ‘defocusing’ sources. Using the combined method of regularization and the Galerkin method, the theorem of local solvability is proved. For systems with a focusing source, an a priori estimate is obtained, from which follows the corresponding theorem on global solvability. In the case of systems of hyperbolic equations with a ‘defocusing’ source, it is proven that if the order of growth of the variables does not exceed the order of growth of the dissipative term, then there is global solvability.