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

summary:We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $$ u_{tt} + 2 u_t - a_{ij}(u_t,\nabla u)\partial _i\partial _j u = f $$ 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\to \infty $, of the solution of the evolution equation to its stationary state