Asymptotic behaviour of solutions of semilinear hyperbolic systems in arbitrary domains

Abstract

In this paper the long time asymptotic behaviour of solutions to semilnear first order hyperbolic systems including Maxwell's equations and the scalar wave-equation in an arbitrary spatial domain is investigated. Weak conergence to stationary states is proved. The possibly nonlinear damping-term may vanish on some subdomain and obeys on the other part of the domain a coerciveness condition, but it is not necessarily monotone. In the case that it is monotone also strong $L^q$-convergence is shown.