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# Article

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Keywords:
quasilinear evolution equation; quasilinear elliptic equation; a priori estimates; global existence; asymptotic behavior; stationary solutions
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.
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