## Long-time behavior of small solutions to quasilinear dissipative hyperbolic equation.(English)Zbl 1249.35072

Summary: 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 $$t\rightarrow \infty ,$$ of the solution of the evolution equation to its stationary state.

### MSC:

 35J15 Second-order elliptic equations 35J60 Nonlinear elliptic equations 35L15 Initial value problems for second-order hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations
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### References:

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