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On a mixed problem with a nonlinear acoustic boundary condition for a non-locally reacting boundaries. (English) Zbl 1310.35153
Summary: We discuss global solvability to the mixed problem for a nonlinear wave equation in domains with non-locally reacting boundaries and nonlinear acoustic boundary condition. We prove the existence and uniqueness of solution. Additionally, using the method of Nakao we demonstrate the decay of the associated energy (Stability).

##### MSC:
 35L20 Initial-boundary value problems for second-order hyperbolic equations 35L70 Second-order nonlinear hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
 [1] Aassila, M.; Cavalcanti, M. M.; Domingos Cavalcanti, V. N., Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term, Calc. Var. Partial Differential Equations, 15, 2, 155-180, (2002) · Zbl 1009.35055 [2] T. Aubin, Some Nonlinear Problems in Riemannian Geometry, Springer Monographs in Mathematics, Berlin, 1998. · Zbl 0896.53003 [3] Beale, J. T.; Rosencrans, S. I., Acoustic boundary conditions, Bull. Amer. Math. Soc., 80, 6, 1276-1278, (1974) · Zbl 0294.35045 [4] Biezuner, R. J.; Montenegro, M., Best constant in second-order Sobolev inequalities on Riemannian manifolds and applications, J. Math. Pures Appl., 82, 457-502, (2003) · Zbl 1043.58010 [5] Brito, E. H., The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability, Appl. Anal., 13, 219-233, (1982) · Zbl 0458.35065 [6] Carrier, G. F., On the non-linear vibration problem of the elastic string, Quart. Appl. Math., 3, 157-165, (1945) · Zbl 0063.00715 [7] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Soriano, J. A., Asymptotic stability of the wave equation on compact surfaces and locally distributed damping—a sharp result, Trans. Amer. Math. Soc., 361, 9, 4561-4580, (2009) · Zbl 1179.35052 [8] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Soriano, J. A., Asymptotic stability of the wave equation on compact manifolds and locally distributed damping: a sharp result, Arch. Ration. Mech. Anal., 197, 3, 925-964, (2010) · Zbl 1232.58019 [9] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Fukuoka, R.; Toundykov, D., Stabilization of the damped wave equation with Cauchy-Ventcel boundary conditions, J. Evol. Equ., 9, 1, 143-169, (2009) · Zbl 1239.35020 [10] Cavalcanti, M. M.; Domingos Cavalcanti, V. D.; Lasiecka, I., Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236, 407-459, (2007) · Zbl 1117.35048 [11] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Prates Filho, J. S.; Soriano, J. A., Existence and uniform decay of solutions of a degenerate equation with nonlinear boundary damping and boundary memory source term, Nonlinear Anal. TMA, 38, 281-294, (1999) · Zbl 0933.35157 [12] Cavalcanti, M. M.; Lasiecka, I.; Toundykov, D., Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable, Trans. Amer. Math. Soc., 364, 11, 5693-5713, (2012) · Zbl 1408.35101 [13] Cousin, A. T.; Frota, C. L.; Larkin, N. A., Global solvability and decay of the energy for the nonhomogeneous Kirchhoff equation, Differential Integral Equations, 15, 10, 1219-1236, (2002) · Zbl 1011.35097 [14] de Caldas, C. S.Q.; Limaco, J.; Barreto, R. K.; Gamboa, P., Global existence and exponential decay estimates for a hyperbolic equation with a nonlinear dissipation, Rev. Mat. Apl., 22, 25-40, (2001) · Zbl 1098.35551 [15] Doronin, G. G.; Larkin, N. A., Global solvability for the quasilinear damped wave equation with nonlinear second-order boundary conditions, Nonlinear Anal. TMA, 50, 1119-1134, (2002) · Zbl 1002.35090 [16] Frigeri, S., On the convergence to stationary solutions for a semilinear wave equation with an acoustic boundary condition, Z. Anal. Anwend., 30, 181-191, (2011) · Zbl 1219.35143 [17] Frota, C. L.; Cousin, A. T.; Larkin, N. A., Existence of global solutions and energy decay for the carrier equation with dissipative term, Differential Integral Equations, 12, 4, 453-469, (1999) · Zbl 1014.35060 [18] Frota, C. L.; Cousin, A. T.; Larkin, N. A., Global solvability and asymptotic behaviour of a hyperbolic problem with acoustic boundary conditions, Funkcial. Ekvac., 44, 3, 471-485, (2001) · Zbl 1145.35433 [19] Frota, C. L.; Goldstein, J. A., Some nonlinear wave equations with acoustic boundary conditions, J. Differential Equations, 164, 92-109, (2000) · Zbl 0979.35105 [20] Frota, C. L.; Medeiros, L. A.; Vicente, A., Wave equation in domains with non-locally reacting boundary, Differential Integral Equations, 24, 11-12, 1001-1020, (2011) · Zbl 1249.35221 [21] Graber, P. J., Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system, Nonlinear Anal. TMA, 73, 3058-3068, (2010) · Zbl 1200.35193 [22] Graber, P. J., Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions, Nonlinear Anal. TMA, 74, 3137-3148, (2011) · Zbl 1217.35027 [23] Hebey, E., Nonlinear analysis on manifolds: Sobolev spaces and inequalities, (1999), American Mathematical Society [24] Koch, H.; Zuazua, E., A hybrid system of PDE’s arising in multi-structure interaction: coupling of wave equations in $$n$$ and $$n - 1$$ space dimensions, (Recent Trends in Partial Differential Equations, Contemp. Math., vol. 409, (2006), Amer. Math. Soc.), 55-77 · Zbl 1108.35110 [25] Lasiecka, I.; Triggiani, R.; Zhang, X., Nonconservative wave equations with unobserved Neumann B.C.: global uniqueness and observability in one shot, (Differential Geometric Methods in the Control of Partial Differential Equations, Boulder, CO, 1999, Contemp. Math., vol. 268, (2000), Amer. Math. Soc.), 227-325 · Zbl 1096.93503 [26] Lions, J. L., Quelques Méthodes de Résolution des problémes aux limites non linéaires, (1969), Dunod Gauthier-Villars Paris · Zbl 0189.40603 [27] Lions, J. L.; Magenes, E., Non homegeneuous boundary value problems and applications I, (1972), Springer Verlag Berlin · Zbl 0223.35039 [28] Liu, K.; Rao, B.; Zhang, X., Stabilization of the wave equations with potential and indefinite damping, J. Math. Anal. Appl., 269, 747-769, (2002) · Zbl 1004.35023 [29] Liu, W.; Yu, J., On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms, Nonlinear Anal. TMA, 74, 2175-2190, (2011) · Zbl 1215.35034 [30] Littman, W.; Taylor, S. W., Boundary feedback stabilization of a vibrating string with an interior point mass, (Nonlinear Problems in Mathematical Physics and Related Topics I, Int. Math. Ser., vol 1, (2002)), 271-287 · Zbl 1051.93048 [31] Nakao, M., A difference inequality and its application to nonlinear evolution equations, J. Math. Soc. Japan, 30, 4, 747-762, (1978) · Zbl 0388.35007 [32] Park, J. Y.; Park, S. H., Decay rate estimates for wave equations of memory type with acoustic boundary conditions, Nonlinear Anal. TMA, 74, 993-998, (2011) · Zbl 1202.35032 [33] Taylor, M. E., Partial differential equations I, (1996), Springer-Verlag New York [34] Vicente, A., Wave equation with acoustic/memory boundary conditions, Bol. Soc. Parana. Mat. (3), 27, 29-39, (2009) · Zbl 1227.35194 [35] Vicente, A.; Frota, C. L., Nonlinear wave equation with weak dissipative term in domains with non-locally reacting boundary, Wave Motion, 50, 2, 162-169, (2013) · Zbl 1360.76311 [36] Vitillaro, E., Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 86, 259-298, (2002)
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