Abstract
In this paper, we prove a stability result for a nonlinear wave equation, defined in a bounded domain of \({\mathbb {R}}^N\), \(N\ge 2\), with time-dependent coefficients. The smooth boundary of \(\Omega \) is \(\Gamma =\Gamma _0\cup \Gamma _1\) such that \(\Sigma ={\overline{\Gamma }}_0\cap {\overline{\Gamma }}_1\ne \emptyset \). On \(\Gamma _0\) we consider the homogeneous Dirichlet boundary condition and on \(\Gamma _1\) we consider the Neumann boundary condition with damping term. The presence of time-dependent coefficients and, moreover, of the singularities generated by the condition \(\Sigma \ne \emptyset \) brings some technical difficulties. The tools are the combination of appropriate functional with the techniques due to Bey, Loheac, and Moussaoui [2] and new technical arguments.
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References
Astudillo, M., Cavalcanti, M.M., Fukuoka, R., Gonzalez Martinez, V.H.: Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin-Voigt damping. Math. Nachr. 291(14–15), 2145–2159 (2018)
Bey, R., Lohéac, J.-P., Moussaoui, M.: Singularities of the solutions of a mixed problem for a general second order elliptic equation and boundary stabilization of the wave equation. J. Math. Pures Appl. 78, 1043–1067 (1999)
Boiti, C., Manfrin, R.: On the asymptotic boundedness of the energy of solutions of the wave equation \(u_{tt}-a(t)\Delta u=0\). Ann. Univ. Ferrara 58, 251–289 (2012)
Burq, N., Gérard, P.: Contrôle Optimal des équations aux dérivées partielles (2001). http://www.math.u-psud.fr/~burq/articles/coursX.pdf
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Fukuoka, R., Pampu, A.B., Astudillo, M.: Uniform decay rate estimates for the semilinear wave equation in inhomogeneous medium with locally distributed nonlinear damping. Nonlinearity 31, 4031–4064 (2018)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Frota, C.L., Vicente, A.: Stability for semilinear wave equation in an inhomogeneous medium with frictional localized damping and acoustic boundary conditions. SIAM J. Control Optim. 58(4), 2411–2445 (2020)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Gonzalez Martinez, V.H., Peralta, V.A., Vicente, A.: Stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping. J. Differ. Equ. 269, 8212–8268 (2020)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Jorge Silva, M.A., de Souza Franco, A.Y.: Exponential stability for the wave model with localized memory in a past history framework. J. Differ. Equ. 264, 6535–6584 (2018)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Mansouri, S., Gonzalez Martinez, V.H., Hajjej, Z., Astudillo Rojas, M.R.: Asymptotic stability for a strongly coupled Klein-Gordon system in an inhomogeneous medium with locally distributed damping. J. Differ. Equ. 268(2), 447–489 (2020)
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.P.: Existence and boundary stabilization of a nonlinear hyperbolic equation with time-dependent coefficients. Electron. J. Differ. Equ. 1998(08), 1–21 (1998)
Coclite, G.M., Goldstein, G.R., Goldstein, J.A.: Stability estimates for parabolic problems with Wentzell boundary conditions. J. Differ. Equ. 245, 2595–2626 (2008)
Coclite, G.M., Goldstein, G.R., Goldstein, J.A.: Stability of Parabolic Problems with nonlinear Wentzell boundary conditions. J. Differ. Equ. 246(6), 2434–2447 (2009)
Coclite, G.M., Goldstein, G.R., Goldstein, J.A.: Wellposedness of Nonlinear Parabolic Problems with nonlinear Wentzell boundary conditions. Adv. Differ. Equ. 16(9–10), 895–916 (2011)
Coclite, G.M., Goldstein, G.R., Goldstein, J.A.: Stability estimates for nonlinear hyperbolic problems with nonlinear Wentzell boundary conditions. Z. Angew. Math. Phys. 64, 733–753 (2013)
Coclite, G.M., Favini, A., Goldstein, G.R., Goldstein, J.A., Romanelli, S.: Continuous dependence on the boundary conditions for the Wentzell Laplacian. Semigroup Forum 77(1), 101–108 (2008)
Coclite, G.M., Favini, A., Goldstein, G.R., Goldstein, J.A.: Romanelli, S: Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Comm. Pure Appl. Anal. 13(1), 419–433 (2014)
Coclite, G.M., Favini, A., Gal, C.G., Goldstein, G.R., Goldstein, J.A., Obrecht, E., Romanelli, S.: The Role of Wentzell Boundary Conditions in Linear and Nonlinear Analysis. In: S. Sivasundaran (ed) Advances in Nonlinear Analysis: Theory, Methods and Applications vol. 3, p. 279-292, Cambridge Scientific Publishers Ltd., Cambridge (2009)
Coclite, G.M., Florio, G., Ligabò, M., Maddalena, F.: Nonlinear waves in adhesive strings. SIAM J. Appl. Math. 77(2), 347–360 (2017)
Coclite, G.M., Florio, G., Ligabò, M., Maddalena, F.: Adhesion and debonding in a model of elastics string. Comput. Math. Appl. 78, 189–1909 (2019)
Cornilleau, P., Lohéac, J.-P., Osses, A.: Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers. J. Dyn. Control Syst. 16(2), 163–188 (2010)
Grisvard, P.: Contrôlabilité exacte des solutions de l’equation des ondes en présence de singularités. J. Math. pures et appl. 68, 215–259 (1989)
Grisvard, P.: Singularities in Boundary Value Problems. Springer, Berlin (1992)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, London (1982)
Liu, Y., Yao, P.F.: Exact boundary controllability for the wave equation with time-dependent and variable coefficients. In: Proceedings of the 33rd Chinese Control Conference, 28–30 (2014)
Liu, Y., Li, J., Yao, P.F.: Decay Rates of the Hyperbolic Equation in an Exterior Domain with Half-Linear and Nonlinear Boundary Dissipations. J. Syst. Sci. Complex 29, 657–680 (2016)
Reissig, M., Smith, J.: \(L^p-L^q\) estimate for wave equation with bounded time dependent coefficient. Hokkaido Math. J. 34, 541–586 (2005)
Yao, P.F.: On the observability inequalities for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37(5), 1568–1599 (1999)
Yao, P.F.: Modeling and Control in Vibrational and Structural dynamics. A differential geometric approach. In: Chapman Hall/CRC Applied Mathematics and Nonlinear Science Series. CRC Press, Boca Raton, FL (2011)
Yao, P.F.: Global smooth solutions for the quasilinear wave equation with boundary dissipation. J. Differ. Equ. 241, 62–93 (2007)
Yao, P.F.: Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation. Chin. Ann. Math. Ser. B 31B(1), 59–70 (2010)
Funding
Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0. Research of Valéria N. Domingos Cavalcanti is partially supported by the CNPq Grant 304895/2003-2.
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Cavalcanti, M.M., Domingos Cavalcanti, V.N. & Vicente, A. Stability for a nonlinear hyperbolic equation with time-dependent coefficients and boundary damping. Z. Angew. Math. Phys. 73, 221 (2022). https://doi.org/10.1007/s00033-022-01856-z
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DOI: https://doi.org/10.1007/s00033-022-01856-z