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

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation u _{t t}+2u _t-a _ij(u _tu) α _iα _ju=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,v)α _iα _jv = h.We then give conditions for the convergence, as t-∞, of the solution of the evolution equation to its stationary state.