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Exponential stability for the 2D wave model with localized memory in a past history framework and nonlinearity of arbitrary growth. (English) Zbl 1504.35060

Summary: In this paper, we discuss the asymptotic stability as well as the wellposedness of the viscoelastic damped wave equation posed on a bounded domain \(\Omega\) of \({\mathbb{R}}^2\), \[ \partial_t^2u - \Delta u+ \displaystyle \int_0^\infty g(s)\mathrm{div}[a(x)\nabla u(\cdot ,t-s)]\,\text{{d}}s + b(x) \partial_tu + f(u)=0, \text{ in }\Omega \times{\mathbb{R}}_+, \] subject to a locally distributed viscoelastic effect driven by a nonnegative function \(a(x)\) which is positive around the entire neighborhood of \(\partial \Omega\) and supplemented with a frictional damping \(b(x)\ge 0\) acting effectively on \(\partial A\) where \(A=\{x\in \Omega \big / a(x)=0\}\). Assuming that well-known geometric control condition \((\omega^\prime , T_0)\) holds, supposing that the relaxation function \(g\) is bounded by a function that decays exponentially to zero and the function \(f\) possesses an arbitrary growth, we show that the solutions to the corresponding partial viscoelastic model decay exponentially to zero. We can also treat the focusing case for those solutions with energy less than \(d\) of the ground state, where \(d\) is the level of the Mountain Pass Theorem.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
35R09 Integro-partial differential equations
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[1] Alabau-Boussouira, F., Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51, 1, 61-105 (2005) · Zbl 1107.35077 · doi:10.1007/s00245
[2] Alabau-Boussouira, F., Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51, 51-105 (2005) · Zbl 1107.35077 · doi:10.1007/s00245
[3] Alabau-Boussouira, F.; Cannarsa, P., A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I, 347, 867-872 (2009) · Zbl 1179.35058 · doi:10.1016/j.crma.2009.05.011
[4] Alabau-Boussouira, F.; Cannarsa, P.; Sforza, D., Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254, 1342-1372 (2008) · Zbl 1145.35025 · doi:10.1016/j.jfa.2007.09.012
[5] Aloui, L.; Ibrahim, S.; Nakanishi, K., Exponential energy decay for damped Klein-Gordon equation with nonlinearities of arbitrary growth, Commun. Partial Differ. Equ., 36, 5, 797-818 (2011) · Zbl 1243.35122 · doi:10.1080/03605302.2010.534684
[6] Alves, CO; Cavalcanti, MM, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differ. Equ., 34, 3, 377-411 (2009) · Zbl 1172.35043 · doi:10.1007/s00526-008-0188-z
[7] Ambrosetti, A.; Rabinowitz, PH, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[8] Appleby, JAD; Fabrizio, M.; Lazzari, B.; Reynolds, DW, On exponencial asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16, 10, 1677-1694 (2006) · Zbl 1114.45003 · doi:10.1142/S0218202506001674
[9] Bardos, C.; Lebeau, G.; Rauch, J., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30, 5, 1024-1065 (1992) · Zbl 0786.93009 · doi:10.1137/0330055
[10] Bellassoued, M., Decay of solutions of the elastic wave equation with a localized dissipation, Annales de la Faculté des Sciences de Toulouse, XII, 3, 267-301 (2003) · Zbl 1073.35036
[11] Burq, N.; Gérard, P., Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. [A necessary and sufficient condition for the exact controllability of the wave equation], C. R. Acad. Sci. Paris Sér. I Math., 325, 7, 749-752 (1997) · Zbl 0906.93008 · doi:10.1016/S0764-4442(97)80053-5
[12] Burq, N., Gérard, P.: Contrôle Optimal des équations aux dérivées partielles. (2001) http://www.math.u-psud.fr/ burq/articles/coursX.pdf
[13] Cavalcanti, MM; Oquendo, HP, Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42, 4, 1310-1324 (2003) · Zbl 1053.35101 · doi:10.1137/S0363012902408010
[14] Cavalcanti, MM; Cavalcanti, VND; Fukuoka, R.; Soriano, JA, Uniform stabilization of the wave equation on compact surfaces and locally distributed damping, Methods Appl. Anal., 15, 4, 405-426 (2008) · Zbl 1183.35200 · doi:10.4310/MAA.2008.v15.n4.a1
[15] Cavalcanti, MM; Cavalcanti, VND; Fukuoka, R.; Soriano, JA, Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result, Trans. AMS, 361, 9, 4561-4580 (2009) · Zbl 1179.35052 · doi:10.1090/S0002-9947-09-04763-1
[16] Cavalcanti, MM; Cavalcanti, VND; Fukuoka, R.; Soriano, JA, 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 · doi:10.1007/s00205-009-0284-z
[17] Cavalcanti, MM; Cavalcanti, VND; Nascimento, FAF, Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation, Proc. Am. Math. Soc., 141, 9, 3183-3193 (2013) · Zbl 1286.35031 · doi:10.1090/S0002-9939-2013-11869-1
[18] Cavalcanti, MM; Cavalcanti, VND; Lasiecka, I.; Nascimento, FAF, Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. Ser. B, 19, 7, 1987-2012 (2014) · Zbl 1326.35041
[19] Cavalcanti, MM; Fatori, LH; Ma, TF, Attractors for wave equations with degenerate memory, J. Differ. Equ., 260, 1, 56-83 (2016) · Zbl 1323.35106 · doi:10.1016/j.jde.2015.08.050
[20] Cavalcanti, MM; Cavalcanti, VND; Silva, MAJ; de Souza Franco, AY, Exponential stability for the wave model with localized memory in a past history framework, J. Differ. Equ., 264, 6535-6584 (2018) · Zbl 1404.35034 · doi:10.1016/j.jde.2018.01.044
[21] Cavalcanti, MM; Cavalcanti, VND; Fukuoka, R.; Pampu, AB; Astudillo, M., Uniform decay rate estimates for the semilinear wave equation in inhomogeneous medium with locally distributed nonlinear damping, Nonlinearity, 31, 9, 4031-4064 (2018) · Zbl 1397.35025 · doi:10.1088/1361-6544/aac75d
[22] Cavalcanti, MM; Cavalcanti, VND; Martinez, VHG; Peralta, VA; Vicente, A., Stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping, J. Differ. Equ., 269, 10, 8212-8268 (2020) · Zbl 1453.35022 · doi:10.1016/j.jde.2020.06.013
[23] Cavalcanti, MM; Cavalcanti, VND; Antunes, JGS; Vicente, A., Stability for the wave equation in an unbounded domain with finite measure and with nonlinearities of arbitrary growth, J. Differ. Equ., 318, 230-269 (2022) · Zbl 1484.35048 · doi:10.1016/j.jde.2022.02.029
[24] Chueshov, I.; Eller, M.; Lasiecka, I., On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Commun. Partial Differ. Equ., 27, 1901-1951 (2002) · Zbl 1021.35020 · doi:10.1081/PDE-120016132
[25] Conti, M.; Marchini, EM; Pata, V., A well posedness result for nonlinear viscoelastic equations with memory, Nonlinear Anal., 94, 206-216 (2004) · Zbl 1282.35249 · doi:10.1016/j.na.2013.08.015
[26] Conti, M.; Marchini, EM; Pata, V., Non classical diffusion with memory, Math. Methods Appl. Sci., 38, 948-958 (2015) · Zbl 1315.35037 · doi:10.1002/mma.3120
[27] Conti, M.; Marchini, EM; Pata, V., Global attractors for nonlinear viscoelastic equations with memory, Commun. Pure Appl. Anal., 15, 5, 1893-1913 (2016) · Zbl 1347.35045 · doi:10.3934/cpaa.2016021
[28] Dafermos, CM, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37, 297-308 (1970) · Zbl 0214.24503 · doi:10.1007/BF00251609
[29] Dafermos, C.M.: Asymptotic behavior of solutions of evolution equations. Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., (1977), pp. 103-123, Publ. Math. Res. Center Univ. Wisconsin, 40, Academic Press, New York-London, 1978 · Zbl 0499.35015
[30] Danese, V.; Geredeli, P.; Pata, V., Exponential attractors for abstract equations with memory and applications to viscoelasticity, Discrete Contin. Dyn. Syst., 35, 7, 2881-2904 (2015) · Zbl 1336.35077 · doi:10.3934/dcds.2015.35.2881
[31] Daoulatli, M.; Lasiecka, I.; Toundykov, D., Uniform energy decay for a wave equation with partialy supported nonlinear boundary dissipation without growth restrictions, DCDS-S, 2, 1 (2009) · Zbl 1172.35443 · doi:10.3934/dcdss.2009.2.67
[32] Dehman, B.; Lebeau, G.; Zuazua, E., Stabilization and control for the subcritical semilinear wave equation, Anna. Sci. Ec. Norm. Super., 36, 525-551 (2003) · Zbl 1036.35033 · doi:10.1016/S0012-9593(03)00021-1
[33] Duistermaat, JJ; Hörmander, L., Fourier integral operators. II, Acta Math., 128, 3-4, 183-269 (1972) · Zbl 0232.47055 · doi:10.1007/BF02392165
[34] Duyckaerts, T.; Zhang, X.; Zuazua, E., On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 1-41 (2008) · Zbl 1248.93031 · doi:10.1016/j.anihpc.2006.07.005
[35] Fabrizio, M.; Giorgi, C.; Pata, V., A new approach to equations with memory, Arch. Ration. Mech. Anal., 198, 1, 189-232 (2010) · Zbl 1245.45010 · doi:10.1007/s00205-010-0300-3
[36] Gérard, P., Microlocal defect measures, Commun. Partial Differ. Equ., 16, 1761-1794 (1991) · Zbl 0770.35001 · doi:10.1080/03605309108820822
[37] Gérard, P., Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal., 141, 1, 60-98 (1996) · Zbl 0868.35075 · doi:10.1006/jfan.1996.0122
[38] Giorgi, C.; Marzocchi, A.; Pata, V., Asymptotic behavior of a semilinear problem in heat conduction with memory, NoDEA, 5, 333-354 (1998) · Zbl 0912.45009 · doi:10.1007/s000300050049
[39] Giorgi, C.; Rivera, JEM; Pata, V., Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260, 83-99 (2001) · Zbl 0982.35021 · doi:10.1006/jmaa.2001.7437
[40] Grasselli, M.; Pata, V.; Lorenzi, A.; Ruf, B., Uniform attractors of non autonomous systems with memory, Evolution Equations, Semigroups and Functional Analysis, 155-178 (2002), Boston: Birkhauser, Boston · Zbl 1039.34074 · doi:10.1007/978-3-0348-8221-7_9
[41] Guesmia, A.; Messaoudi, SA, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal., 13, 476-485 (2012) · Zbl 1239.35156 · doi:10.1016/j.nonrwa.2011.08.004
[42] Guo, Y.; Rammaha, MA; Sakuntasathien, S.; Titi, E.; Toundykov, D., Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differ. Equ., 257, 10, 3778-3812 (2014) · Zbl 1302.35259 · doi:10.1016/j.jde.2014.07.009
[43] Hitrik, M., Expansions and eigenfrequencies for damped wave equations, Journées équations aux Dérivées Partielles (Plestin-les-Grèves, 2001), 6, 10 (2001) · Zbl 1213.35329
[44] Hörmander, L.: The propagation of singularities for solutions of the Dirichlet problem. In Pseudodifferential operators and applications (Notre Dame, Ind., 1984), volume 43 of Proc. Sympos. Pure Math., pp. 157-165. Amer. Math. Soc., Providence, RI (1985) · Zbl 0618.35015
[45] Hörmander, L., The Analysis of Linear Partial Differential Operators (1985), Berlin: Springer, Berlin · Zbl 0601.35001
[46] Joly, R.; Laurent, C., Stabilization for the semilinear wave equation with geometric control, J. Anal. PDE, 6, 5, 1089-1119 (2013) · Zbl 1329.35062 · doi:10.2140/apde.2013.6.1089
[47] Jost, J., Riemannian Geometry and Geometric Analysis (2008), Cham: Springer, Cham · Zbl 1143.53001
[48] Koch, H.; Tataru, D., Dispersive estimates for principally normal pseudodifferential operators, Commun. Pure Appl. Math., 58, 217-284 (2005) · Zbl 1078.35143 · doi:10.1002/cpa.20067
[49] Lasiecka, I.; Tataru, D., Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping, Differ. Integral Equ., 6, 507-533 (1993) · Zbl 0803.35088
[50] Lasiecka, I.; Messaoudi, SA; Mustafa, MI, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54, 3, 031504 (2013) · Zbl 1282.74018 · doi:10.1063/1.4793988
[51] Laurent, C., On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold, J. Funct. Anal., 260, 5, 1304-1368 (2011) · Zbl 1244.35012 · doi:10.1016/j.jfa.2010.10.019
[52] Lebeau, G.: Equations des ondes amorties. Algebraic Geometric Methods in Maths. Physics, pp. 73-109 (1996) · Zbl 0863.58068
[53] Lions, J.L.: Quelques Methódes de Resolution des Probléms aux limites Non Lineéires. (1969) · Zbl 0189.40603
[54] Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Vol. 1. (French) Travaux et Recherches Mathématiques, No. 17 Dunod, Paris 1968 xx+372 pp · Zbl 0165.10801
[55] Liu, K.; Liu, Z., Exponential decay of energy of vibrating strings with local viscoelasticity, ZAMP, 53, 265-280 (2002) · Zbl 0999.35012
[56] Martinez, P., A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complutense, 12, 1, 251-283 (1999) · Zbl 0940.35034
[57] Miller, L., Escape function conditions for the observation, control, and stabilization of the wave equation, SIAM J. Control Optim., 41, 5, 1554-1566 (2002) · Zbl 1032.35117 · doi:10.1137/S036301290139107X
[58] Moser, J., A sharp form of an inequality by N. Trudinger, Ind. Univ. Math. J., 20, 1077-1092 (1979) · Zbl 0203.43701 · doi:10.1512/iumj.1971.20.20101
[59] Nakao, M.: Decay and global existence for nonlinear wave equations with localized dissipations in general exterior domains. New trends in the theory of hyperbolic equations, 213-299, Oper. Theory Adv. Appl., 159, Birkhäuser, Basel (2005) · Zbl 1130.35094
[60] Nakao, M., Energy decay for the wave equation with boundary and localized dissipations in exterior domains, Math. Nachr., 278, 7-8, 771-783 (2005) · Zbl 1071.35072
[61] Ning, Z.-H.: Asymptotic behavior of the nonlinear Schrödinger equation on exterior domain. arXiv:1905.09540 (2019)
[62] Pata, V., Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77, 333-360 (2009) · Zbl 1205.45010 · doi:10.1007/s00032-009-0098-3
[63] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44 (1983), New York: Springer, New York · Zbl 0516.47023
[64] Qin, T., Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications, J. Math. Anal. Appl., 233, 1, 130-147 (1999) · Zbl 0981.45007 · doi:10.1006/jmaa.1999.6271
[65] Rauch, J.; Taylor, M., Decay of solutions to n on dissipative hyperbolic systems on compact manifolds, Commun. Pure Appl. Math., 28, 4, 501-523 (1975) · Zbl 0295.35048 · doi:10.1002/cpa.3160280405
[66] Rivera, JEM; Peres Salvatierra, A., Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59, 557-578 (2001) · Zbl 1028.35025 · doi:10.1090/qam/1848535
[67] Ruiz, A., Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures. Appl., 71, 455-467 (1992) · Zbl 0832.35084
[68] Simon, J., Compact Sets in the space \(L^p(0, T; B)\), Ann. Math. Pura Appl., 146, 65-96 (1987) · Zbl 0629.46031 · doi:10.1007/BF01762360
[69] Strauss, WA, On weak solutions of semilinear hyperbolic equations, Anais da Academis Brasileira de Ciências, 71, 645-651 (1972) · Zbl 0217.13104
[70] Tataru, D., The \(X_\theta^s\) spaces and unique continuation for solutions to the semilinear wave equation, Commun. Partial Differ. Equ., 2, 841-887 (1996) · Zbl 0853.35017 · doi:10.1080/03605309608821210
[71] Toundykov, D., Optimal decay rates for solutions of nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponents source terms under mixed boundary, Nonlinear Anal. T. M. A., 67, 2, 512-544 (2007) · Zbl 1117.35050 · doi:10.1016/j.na.2006.06.007
[72] Triggiani, R., Yao, P.F.: Carleman estimates with no lower-Order terms for general Riemannian wave equations. Global uniqueness and observability in one shot, phAppl. Math. and Optim 46 (Sept./ Dec. 2002) 331-375. Special issue dedicated to J. L, Lions · Zbl 1030.35018
[73] Trudinger, NS, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17, 5, 473-483 (1967) · Zbl 0163.36402
[74] Willem, M., Minimax Theorems (1996), Basel: Birkhäuser, Basel · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1
[75] 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) · Zbl 0951.35069 · doi:10.1137/S0363012997331482
[76] Yao, P-F, Observability inequalities for shallow shells, SIAM J. Control Optim., 38, 6, 1729-1756 (2000) · Zbl 0974.35013 · doi:10.1137/S0363012999338692
[77] Yao, P-F, Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differ. Equ., 241, 1, 62-93 (2007) · Zbl 1214.35037 · doi:10.1016/j.jde.2007.06.014
[78] Yao, P-F, Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., 61, 2, 191-233 (2010) · Zbl 1185.93018 · doi:10.1007/s00245-009-9088-7
[79] Yao, P.-F.: Modeling and Control in Vibrational and Structural Dynamics. A Differential Geometric Approach, Chap-man & Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, Boca Raton, FL, (2011) · Zbl 1229.74002
[80] Zhang, X., Explicit observability estimates for the wave equation with lower order terms by means of Carleman inequalities, SIAM J. Cont. Optim., 3, 812-834 (2000) · Zbl 0982.35059 · doi:10.1137/S0363012999350298
[81] Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping, Commun. Partial Differ. Equ., 15, 2, 205-235 (1990) · Zbl 0716.35010 · doi:10.1080/03605309908820684
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