Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

Abstract

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $u_{tt} + 2u_t - a_{ij} (u_t ,\nabla u)\partial _i \partial _j u = f$ corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation $ - a_{ij} (0,\nabla v)\partial _i \partial _j v = h$ . We then give conditions for the convergence, as t → ∞, of the solution of the evolution equation to its stationary state.