Geodesic
Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations
Applications of Mathematics, Tome 56 (2011) no. 5, pp. 425-457
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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.
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.
DOI : 10.1007/s10492-011-0025-0
Classification : 35A01, 35B35, 35B40, 35J15, 35J60, 35L15, 35L70
Keywords: quasilinear evolution equation; quasilinear elliptic equation; a priori estimates; global existence; asymptotic behavior; stationary solutions 
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     title = {Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations},
     journal = {Applications of Mathematics},
     pages = {425--457},
     year = {2011},
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     zbl = {1249.35072},
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Milani, Albert; Volkmer, Hans. Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations. Applications of Mathematics, Tome 56 (2011) no. 5, pp. 425-457. doi: 10.1007/s10492-011-0025-0

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