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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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