Abstract
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 \({\mathbb {R}}^2,\)
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.
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References
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)
Alabau-Boussouira, F.: Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems. Appl. Math. Optim. 51, 51–105 (2005)
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)
Alabau-Boussouira, F., Cannarsa, P., Sforza, D.: Decay estimates for second order evolution equations with memory. J. Funct. Anal. 254, 1342–1372 (2008)
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)
Alves, C.O., Cavalcanti, M.M.: 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)
Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Appleby, J.A.D., Fabrizio, M., Lazzari, B., Reynolds, D.W.: On exponencial asymptotic stability in linear viscoelasticity. Math. Models Methods Appl. Sci. 16(10), 1677–1694 (2006)
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)
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)
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). (in French)
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., Oquendo, H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM J. Control Optim. 42(4), 1310–1324 (2003)
Cavalcanti, M.M., Cavalcanti, V.N.D., Fukuoka, R., Soriano, J.A.: Uniform stabilization of the wave equation on compact surfaces and locally distributed damping. Methods Appl. Anal. 15(4), 405–426 (2008)
Cavalcanti, M.M., Cavalcanti, V.N.D., Fukuoka, R., Soriano, J.A.: Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result. Trans. AMS 361(9), 4561–4580 (2009)
Cavalcanti, M.M., Cavalcanti, V.N.D., 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)
Cavalcanti, M.M., Cavalcanti, V.N.D., Nascimento, F.A.F.: Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation. Proc. Am. Math. Soc. 141(9), 3183–3193 (2013)
Cavalcanti, M.M., Cavalcanti, V.N.D., Lasiecka, I., Nascimento, F.A.F.: 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)
Cavalcanti, M.M., Fatori, L.H., Ma, T.F.: Attractors for wave equations with degenerate memory. J. Differ. Equ. 260(1), 56–83 (2016)
Cavalcanti, M.M., Cavalcanti, V.N.D., Silva, M.A.J., 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., Cavalcanti, V.N.D., 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(9), 4031–4064 (2018)
Cavalcanti, M.M., Cavalcanti, V.N.D., Martinez, V.H.G., Peralta, V.A., Vicente, A.: Stability for semilinear hyperbolic coupled system with frictional and viscoelastic localized damping. J. Differ. Equ. 269(10), 8212–8268 (2020)
Cavalcanti, M.M., Cavalcanti, V.N.D., Antunes, J.G.S., 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)
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)
Conti, M., Marchini, E.M., Pata, V.: A well posedness result for nonlinear viscoelastic equations with memory. Nonlinear Anal. 94, 206–216 (2004)
Conti, M., Marchini, E.M., Pata, V.: Non classical diffusion with memory. Math. Methods Appl. Sci. 38, 948–958 (2015)
Conti, M., Marchini, E.M., Pata, V.: Global attractors for nonlinear viscoelastic equations with memory. Commun. Pure Appl. Anal. 15(5), 1893–1913 (2016)
Dafermos, C.M.: Asymptotic stability in viscoelasticity. Arch. Ration. Mech. Anal. 37, 297–308 (1970)
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
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)
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)
Dehman, B., Lebeau, G., Zuazua, E.: Stabilization and control for the subcritical semilinear wave equation. Anna. Sci. Ec. Norm. Super. 36, 525–551 (2003)
Duistermaat, J.J., Hörmander, L.: Fourier integral operators. II. Acta Math. 128(3–4), 183–269 (1972)
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)
Fabrizio, M., Giorgi, C., Pata, V.: A new approach to equations with memory. Arch. Ration. Mech. Anal. 198(1), 189–232 (2010)
Gérard, P.: Microlocal defect measures. Commun. Partial Differ. Equ. 16, 1761–1794 (1991)
Gérard, P.: Oscillations and concentration effects in semilinear dispersive wave equations. J. Funct. Anal. 141(1), 60–98 (1996)
Giorgi, C., Marzocchi, A., Pata, V.: Asymptotic behavior of a semilinear problem in heat conduction with memory. NoDEA 5, 333–354 (1998)
Giorgi, C., Rivera, J.E.M., Pata, V.: Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl. 260, 83–99 (2001)
Grasselli, M., Pata, V.: Uniform attractors of non autonomous systems with memory. In: Lorenzi, A., Ruf, B. (eds.) Evolution Equations, Semigroups and Functional Analysis, pp. 155–178. Birkhauser, Boston (2002)
Guesmia, A., Messaoudi, S.A.: A general decay result for a viscoelastic equation in the presence of past and finite history memories. Nonlinear Anal. 13, 476–485 (2012)
Guo, Y., Rammaha, M.A., 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)
Hitrik, M.: Expansions and eigenfrequencies for damped wave equations. Journées équations aux Dérivées Partielles (Plestin-les-Grèves, 2001) 6, 10 (2001)
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)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer, Berlin (1985)
Joly, R., Laurent, C.: Stabilization for the semilinear wave equation with geometric control. J. Anal. PDE 6(5), 1089–1119 (2013)
Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Cham (2008)
Koch, H., Tataru, D.: Dispersive estimates for principally normal pseudodifferential operators. Commun. Pure Appl. Math. 58, 217–284 (2005)
Lasiecka, I., Tataru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 6, 507–533 (1993)
Lasiecka, I., Messaoudi, S.A., Mustafa, M.I.: Note on intrinsic decay rates for abstract wave equations with memory. J. Math. Phys. 54(3), 031504 (2013)
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)
Lebeau, G.: Equations des ondes amorties. Algebraic Geometric Methods in Maths. Physics, pp. 73–109 (1996)
Lions, J.L.: Quelques Methódes de Resolution des Probléms aux limites Non Lineéires. (1969)
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
Liu, K., Liu, Z.: Exponential decay of energy of vibrating strings with local viscoelasticity. ZAMP 53, 265–280 (2002)
Martinez, P.: A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complutense 12(1), 251–283 (1999)
Miller, L.: Escape function conditions for the observation, control, and stabilization of the wave equation. SIAM J. Control Optim. 41(5), 1554–1566 (2002)
Moser, J.: A sharp form of an inequality by N. Trudinger. Ind. Univ. Math. J. 20, 1077–1092 (1979)
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)
Nakao, M.: Energy decay for the wave equation with boundary and localized dissipations in exterior domains. Math. Nachr. 278(7–8), 771–783 (2005)
Ning, Z.-H.: Asymptotic behavior of the nonlinear Schrödinger equation on exterior domain. arXiv:1905.09540 (2019)
Pata, V.: Stability and exponential stability in linear viscoelasticity. Milan J. Math. 77, 333–360 (2009)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44. Springer, New York (1983)
Qin, T.: Asymptotic behavior of a class of abstract semilinear integrodifferential equations and applications. J. Math. Anal. Appl. 233(1), 130–147 (1999)
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)
Rivera, J.E.M., Peres Salvatierra, A.: Asymptotic behaviour of the energy in partially viscoelastic materials. Quart. Appl. Math. 59, 557–578 (2001)
Ruiz, A.: Unique continuation for weak solutions of the wave equation plus a potential. J. Math. Pures. Appl. 71, 455–467 (1992)
Simon, J.: Compact Sets in the space \(L^p(0, T; B)\). Ann. Math. Pura Appl. 146, 65–96 (1987)
Strauss, W.A.: On weak solutions of semilinear hyperbolic equations. Anais da Academis Brasileira de Ciências 71, 645–651 (1972)
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)
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)
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
Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17(5), 473–483 (1967)
Willem, M.: Minimax Theorems. Birkhäuser, Basel (1996)
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.: Observability inequalities for shallow shells. SIAM J. Control Optim. 38(6), 1729–1756 (2000)
Yao, P.-F.: Global smooth solutions for the quasilinear wave equation with boundary dissipation. J. Differ. Equ. 241(1), 62–93 (2007)
Yao, P.-F.: Boundary controllability for the quasilinear wave equation. Appl. Math. Optim. 61(2), 191–233 (2010)
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)
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)
Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Differ. Equ. 15(2), 205–235 (1990)
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J. G. S. Antunes: Research of José Guilherme Simion Antunes is partially supported by the CAPES Grant 88882.449189/2019-01. M. M. Cavalcanti: Research of Marcelo M. Cavalcanti is partially supported by the CNPq Grant 300631/2003-0. V. N. D. Cavalcanti: Research of Valéria N. Domingos Cavalcanti is partially supported by the CNPq Grant 304895/2003-2.
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Antunes, J.G.S., Cavalcanti, M.M., Cavalcanti, V.N.D. et al. Exponential Stability for the 2D Wave Model with Localized Memory in a Past History Framework and Nonlinearity of Arbitrary Growth. J Geom Anal 33, 39 (2023). https://doi.org/10.1007/s12220-022-01085-w
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DOI: https://doi.org/10.1007/s12220-022-01085-w