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Energy decay rates for solutions of the Kirchhoff type wave equation with boundary damping and source terms. (English) Zbl 1415.35177

Summary: In this work, we are concerned with uniform stabilization for an initial-boundary value problem associated with the Kirchhoff type wave equation with feedback terms and memory condition at the boundary. We prove that the energy decays exponentially when the boundary damping term has a linear growth near zero and polynomially when the boundary damping term has a polynomial growth near zero. Furthermore, we study the decay rate of the energy without imposing any restrictive growth assumption on the damping term near zero.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35R09 Integro-partial differential equations
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[1] M. Aassila, M.M. Cavalcanti and J.A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Contr. Optim. 38 (2000), 1581–1602. · Zbl 0985.35008
[2] F. Alabau-Boussouira, J. Prüss and R. Zacher, Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels, C.R. Acad. Sci. Paris 347 (2009), 277–282. · Zbl 1175.35012
[3] G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlin. Anal. 73 (2010), 1952–1965. · Zbl 1197.35173
[4] ——–, Kirchhoff systems with nonlinear source and boundary damping terms, Comm. Pure Appl. Anal. 9 (2010), 1161–1188. · Zbl 1223.35082
[5] J.J. Bae, Global existence and decay for Kirchhoff type wave equation with boundary and localized dissipations in exterior domains, Funk. Ekvac. 47 (2004), 453–477. · Zbl 1206.35156
[6] J.J. Bae and M. Nakao, Existence problem for the Kirchhoff type wave equation with a localized weakly nonlinear dissipation in exterior domains, Discr. Cont. Dynam. Syst. 11 (2004), 731–743. · Zbl 1058.35167
[7] M.M. Cavalcanti, V.N. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Diff. Eqs. 203 (2004), 119–158. · Zbl 1049.35047
[8] E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ricerch. Mat. 8 (1959), 24–51. · Zbl 0199.44701
[9] G.C. Gorain, Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in \(\mathbb{R}^n\), J. Math. Anal. Appl. 319 (2006), 635–650. · Zbl 1098.35021
[10] ——–, Exponential energy decay estimates for the solutions of \(n\)-dimensional Kirchhoff type wave equation, Appl. Math. Comp. 177 (2006), 235–242. · Zbl 1331.35338
[11] T.G. Ha, Asymptotic stability of the semilinear wave equation with boundary damping and source term, C.R. Math. Acad. Sci. Paris 352 (2014), 213–218. · Zbl 1288.35076
[12] ——–, General decay estimates for the wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary, Appl. Math. Lett. 60 (2016), 43–49. · Zbl 1339.35041
[13] ——–, General decay rate estimates for viscoelastic wave equation with Balakrishnan-Taylor damping, Z. Angew. Math. Phys. 67 (2016), Art. 32. · Zbl 1353.35064
[14] ——–, Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions, Discr. Cont. Dynam. Syst. 36 (2016), 6899–6919. · Zbl 1357.35226
[15] ——–, Global existence and uniform decay of coupled wave equation of Kirchhoff type in a noncylindrical domain, J. Korean Math. Soc. 54 (2017), 1081–1097. · Zbl 1379.35185
[16] ——–, On viscoelastic wave equation with nonlinear boundary damping and source term, Comm. Pur. Appl. Anal. 9 (2010), 1543–1576. · Zbl 1211.35047
[17] ——–, Stabilization for the wave equation with variable coefficients and Balakrishnan-Taylor damping, Taiwanese J. Math. 21 (2017), 807–817. · Zbl 1394.35044
[18] T.G. Ha, D. Kim and I.H. Jung, Global existence and uniform decay rates for the semi-linear wave equation with damping and source terms, Comp. Math. Appl. 67 (2014), 692–707. · Zbl 1381.35103
[19] T.G. Ha and J.Y. Park, Global existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain, Nonlin. Anal. 74 (2011), 577–584. · Zbl 1208.35085
[20] T.G. Ha and J.Y. Park, Existence of solutions for the Kirchhoff type wave equation with memory term and acoustic boundary conditions, Numer. Funct. Anal. Optim. 31 (2010), 921–935. · Zbl 1203.35175
[21] ——–, Stabilization of the viscoelastic Euler-Bernoulli type equation with a local nonlinear dissipation, Dynam. PDE 6 (2009), 335–366. · Zbl 1195.35228
[22] H. Harrison, Plane and circular motion of a string, J. Acoust. Soc. Amer. 20 (1948), 874–875.
[23] V. Komornik, Exact Controllability And Stabilization. The multiplier method, John Wiley, Paris, 1994. · Zbl 0937.93003
[24] ——–, On the nonlinear boundary stabilization of the wave equation, Chinese Ann. Math. 14 (1993), 153–164. · Zbl 0804.35065
[25] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pure Appl. 69 (1990), 33–54. · Zbl 0636.93064
[26] J. Lagnese, Decay of solutions of the wave equation in a bounded region with boundary dissipation, J. Diff. Eqs. 50 (1983), 163–182. · Zbl 0536.35043
[27] I. Lasiecka and J. Ong, Global solvability and uniform decays of solution to quasilinear equation with nonlinear boundary dissipation, Comm. PDE 24 (1999), 2069–2107. · Zbl 0936.35031
[28] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Diff. Int. Eqs. 6 (1993), 507–533. · Zbl 0819.35098
[29] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM: Contr. Optim. Calc. Var. 4 (1999), 419–444. · Zbl 0923.35027
[30] A.H. Nayfeh and D.T. Mook, Nonlinear oscillations, Wiley, New York, 1979. · Zbl 0418.70001
[31] S. Nicaise and C. Pignotti, Stabilization of the wave equation with variable coefficients and boundary condition of memory type, Asympt. Anal. 50 (2006), 31–67. · Zbl 1139.35373
[32] L. Nirenberg, On elliptic partial differential equations, Ann. Scuol. Norm. Pisa 13 (1959), 115–162. · Zbl 0088.07601
[33] J.Y. Park, J.J. Bae and I.H. Jung, Uniform decay of solution for wave equation of Kirchhoff type with nonlinear boundary damping and memory term, Nonlin. Anal. 50 (2002), 871–884. · Zbl 1004.35020
[34] J.Y. Park and T.G. Ha, Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term, J. Math. Phys. 49 (2008), 053511.
[35] ——–, Energy decay for nondissipative distributed systems with boundary damping and source term, Nonlin. Anal. 70 (2009), 2416–2434. · Zbl 1169.35037
[36] ——–, Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions, J. Math. Phys. 50 (2009), 013506.
[37] J.Y. Park, T.G. Ha and Y.H. Kang, Energy decay rates for solutions of the wave equation with boundary damping and source term, Z. Angew. Math. Phys. 61 (2010), 235–265. · Zbl 1198.35034
[38] J. Prüss, Evolutionary integral equations and applications, Birkhäuser-Verlag, Basel, 1993.
[39] A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl. 71 (1992), 455–467. · Zbl 0832.35084
[40] M.L. Santos, J. Ferreira, D.C. Pereira and C.A. Raposo, Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary, Nonlinear Anal. 54 (2003), 959–976. · Zbl 1032.35140
[41] R. Triggiani, Wave equation on a bounded domain with boundary dissipation: An operator approach, J. Math. Anal. Appl. 137 (1989), 438–461. · Zbl 0686.35067
[42] E. Vitillaro, A potential well method for the wave equation with nonlinear source and boundary damping terms, Glasgow Math. J. 44 (2002), 375–395. · Zbl 1016.35048
[43] R. Zacher, Quasilinear parabolic integro-differential equations with nonlinear boundary conditions, Diff. Int. Eqs. 19 (2006), 1129–1156. · Zbl 1212.45015
[44] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. PDE 15 (1990), 205–235. · Zbl 0716.35010
[45] ——–, Uniform stabilization of the wave equation by nonlinear boundary feedback, SIAM J. Contr. Optim. 28 (1990), 466–477. · Zbl 0695.93090
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