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Finite-dimensional attractors for the quasi-linear strongly-damped wave equation. (English) Zbl 1183.35053

The paper is concerned with the quasilinear strongly damped wave equation
\[ \partial^2_t u-\gamma\partial _t\Delta _x u-\Delta_xu+f(u) =\nabla_x\cdot\varphi'(\nabla_x u)+g \]
in a smooth bounded domain \(\Omega\subset\mathbb{R}^3\) with Dirichlet boundary condition \(u|_{\partial\Omega}=0\) and prescribed initial condition \((u(0),\partial_t u(0))\). Here \(\gamma>0\) is a positive constant and \(g\in L^2(\Omega)\) an external force. The nonlinearities \(f\in C^2(\mathbb{R})\), \(\varphi\in C^2(\mathbb{R}^3)\) satisfy \(f'(s)\sim|s|^q\) for some \(q>0\) and \(\varphi''(\eta)\sim|\eta|^{p-1}\) for some \(p\in[1,5)\).
The authors prove the existence of a unique weak solution such that \((u(t),\partial_t u(t))\) lies in the energy space \((W^{1,p+1}(\Omega)\cap L^{q+2}(\Omega))\times L^2(\Omega)\). If the initial conditions satisfy appropriate regularity conditions then the weak solution is in fact a strong solution. In addition to solving the Cauchy problem they show the existence of a global attractor for the semigroup associated with weak energy solutions.

MSC:

35B41 Attractors
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35L72 Second-order quasilinear hyperbolic equations
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[1] Acerbi, E.; Mingione, G., Gradient estimates for a class of parabolic systems, Duke Math. J., 136, 2, 285-320 (2007) · Zbl 1113.35105
[2] Andrews, G.; Ball, J. M., Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations, 44, 306-341 (1982) · Zbl 0501.35011
[3] Arrieta, J.; Carvalho, A. N.; Hale, J. K., A damped hyperbolic equation with critical exponent, Comm. Partial Differential Equations, 17, 841-866 (1992) · Zbl 0815.35067
[4] Aviles, P.; Sandefur, J., Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58, 404-427 (1985) · Zbl 0572.34004
[5] Babin, A. V.; Vishik, M. I., Attractors of Evolutionary Equations (1992), North-Holland: North-Holland Amsterdam · Zbl 0778.58002
[6] Ball, J. M.; Holmes, P. J.; James, R. D.; Pego, R. L.; Swart, P. J., On the dynamics of fine structure, J. Nonlinear Sci., 1, 17-70 (1991) · Zbl 0791.35030
[7] Banks, H. T.; Gilliam, D. S.; Shubov, V. I., Global solvability for damped abstract nonlinear hyperbolic systems, Differential Integral Equations, 10, 309-332 (1997) · Zbl 0892.47063
[8] Berkaliev, Z., Attractor of nonlinear evolutionary equation of viscoelasticity, Moscow Univ. Math. Bull., 40, 5, 61-63 (1985) · Zbl 0609.47076
[9] Bruschi, S.; Carvalho, A.; Cholewa, J.; Dlotko, T., Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equations, J. Dynam. Differential Equations, 18, 767-814 (2006) · Zbl 1103.35020
[10] Carvalho, A.; Cholewa, J., Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207, 287-310 (2002) · Zbl 1060.35082
[11] Carvalho, A.; Cholewa, J.; Dlotko, T., Strongly damped wave problems: Bootstrapping and regularity of solutions, J. Differential Equations, 244, 9, 2310-2333 (2008) · Zbl 1151.35056
[12] Carvalho, A.; Cholewa, J., Regularity of solutions on the global attractor for a semilinear damped wave equation, J. Math. Anal. Appl., 337, 2, 932-948 (2008) · Zbl 1139.35026
[13] Carvalho, A.; Cholewa, J., Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Math. Anal. Appl., 310, 2, 557-578 (2005) · Zbl 1077.35031
[14] Chen, F.; Guo, B.; Wang, P., Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations, 147, 231-241 (1998) · Zbl 0912.35111
[15] Cholewa, J. W.; Dlotko, T., Strongly damped wave equation in uniform spaces, Nonlinear Anal., 64, 174-187 (2006) · Zbl 1083.35066
[16] Clements, J., Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26, 745-752 (1974) · Zbl 0252.35044
[17] Clements, J., On the existence and uniqueness of solutions of the equation \(u_{t t} - \partial \sigma_i(u_{x_i}) / \partial x_i - \Delta u_t = f\), Canad. Math. Bull., 18, 181-187 (1975)
[18] Dai, Zh.; Guo, B., Exponential attractors of the strongly damped nonlinear wave equations, (Recent Advances in Differential Equations. Recent Advances in Differential Equations, Kunming, 1997. Recent Advances in Differential Equations. Recent Advances in Differential Equations, Kunming, 1997, Pitman Res. Notes Math. Ser., vol. 386 (1998), Longman: Longman Harlow), 149-159, (English summary) · Zbl 0929.35014
[19] Damascelli, L., Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, 493-516 (1998) · Zbl 0911.35009
[20] Ebihara, Y., On some nonlinear evolution equations with strong dissipation, J. Differential Equations, 30, 149-164 (1978) · Zbl 0362.35016
[21] Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R., Exponential Attractors for Dissipative Evolution Equations (1994), Masson: Masson Paris · Zbl 0842.58056
[22] Efendiev, M.; Miranville, A.; Zelik, S., Exponential attractors for a nonlinear reaction-diffusion system in \(R^3\), C. R. Math. Acad. Sci. Paris, 330, 713-718 (2000) · Zbl 1151.35315
[23] Engler, H., Existence of radially symmetric solutions of strongly damped wave equations, (Nonlinear Semigroups, Partial Differential Equations and Attractors. Nonlinear Semigroups, Partial Differential Equations and Attractors, Washington, DC, 1985. Nonlinear Semigroups, Partial Differential Equations and Attractors. Nonlinear Semigroups, Partial Differential Equations and Attractors, Washington, DC, 1985, Lecture Notes in Math., vol. 1248 (1987), Springer: Springer Berlin), 40-51
[24] Engler, H., Strong solutions for strongly damped quasilinear wave equations, (The Legacy of Sonya Kovalevskaya. The Legacy of Sonya Kovalevskaya, Cambridge, MA, Amherst, MA, 1985. The Legacy of Sonya Kovalevskaya. The Legacy of Sonya Kovalevskaya, Cambridge, MA, Amherst, MA, 1985, Contemp. Math., vol. 64 (1987), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 219-237
[25] Engler, H., Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Math. Z., 202, 251-259 (1989) · Zbl 0697.73033
[26] Fabrie, P.; Galushinski, C.; Miranville, A.; Zelik, S., Uniform exponential attractors for a singular perturbed damped wave equation, Discrete Contin. Dyn. Syst., 10, 211-238 (2004) · Zbl 1060.35011
[27] Feireisl, E.; Petzeltova, H., Global existence for a quasi-linear evolution equation with a non-convex energy, Trans. Amer. Math. Soc., 354, 4, 1421-1434 (2002) · Zbl 0985.35093
[28] Findleky, W. N.; Lai, J. S.; Onaran, K. O., Creep and Relaxation of Nonlinear Viscoelastic Materials (1976), North-Holland: North-Holland Amsterdam, New York
[29] Fitzgibbon, W. E., Strongly damped quasilinear evolution equations, J. Math. Anal. Appl., 79, 536-550 (1981) · Zbl 0476.35040
[30] Friedman, A.; Necas, J., Systems of nonlinear wave equations with nonlinear viscosity, Pacific J. Math., 135, 29-55 (1988) · Zbl 0685.35070
[31] Gajewski, H.; Gröger, K.; Zacharias, K., Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38 (1974), Akademie-Verlag: Akademie-Verlag Berlin · Zbl 0289.47029
[32] Ghidaglia, J.-M.; Marzocchi, A., Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22, 879-895 (1991) · Zbl 0735.35015
[33] Greenberg, J. M.; MacCamy, R. C.; Mizel, V. J., On the existence, uniqueness, and stability of solutions of the equation \(\sigma^\prime(u_x) u_{x x} + \lambda u_{x t x} = \rho_0 u_{t t}\), J. Math. Mech., 17, 707-728 (1967/1968) · Zbl 0157.41003
[34] Hale, J. K., Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25 (1988), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0642.58013
[35] Ikehata, R.; Matsuyama, T.; Nakao, M., Global solutions to the initial-boundary value problem for the quasilinear viscoelastic wave equation with a perturbation, Funkcial. Ekvac., 40, 293-312 (1997) · Zbl 0891.35105
[36] Kalantarov, V. K., Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152, 50-54 (1986) · Zbl 0621.35022
[37] V.K. Kalantarov, Global behavior of solutions to nonlinear problems of mathematical physics of classical and nonclassical type, Postdoc thesis, Leningrad, 1988; V.K. Kalantarov, Global behavior of solutions to nonlinear problems of mathematical physics of classical and nonclassical type, Postdoc thesis, Leningrad, 1988
[38] Kawashima, S.; Shibata, Y., Global existence and exponential stability of small solutions to nonlinear viscoelasticity, Comm. Math. Phys., 148, 189-208 (1992) · Zbl 0779.35066
[39] Kobayashi, T.; Pecher, H.; Shibata, Y., On a global in time existence theorem of smooth solutions to a nonlinear wave equation with viscosity, Math. Ann., 296, 215-234 (1993) · Zbl 0788.35001
[40] Kozhanov, A. I.; Larkin, N. A.; Yanenko, N. N., A mixed problem for a class of third-order equations, Sibirsk. Mat. Zh., 22, 81-86 (1981) · Zbl 0499.35034
[41] Knowles, J. K., One finite antiplane shear for incompressible elastic material, J. Aust. Math. Soc. Ser. B, 19, 400-415 (1975/1976) · Zbl 0363.73045
[42] Levine, H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5, 138-146 (1974) · Zbl 0243.35069
[43] Málek, J.; Pražák, D., Large time behaviour via the method of \(l\)-trajectories, J. Differential Equations, 181, 243-279 (2002) · Zbl 1187.37113
[44] Maslov, V. P.; Mosolov, P. P., Nonlinear Wave Equations Perturbed by Viscous Terms (2000), Walter de Gruyter: Walter de Gruyter Berlin, New York · Zbl 0951.35002
[45] Massatt, P., Limiting behavior for strongly damped nonlinear wave equations, J. Differential Equations, 48, 334-349 (1983) · Zbl 0561.35049
[46] Nakao, M.; Nanbu, T., Existence of global (bounded) solutions for some nonlinear evolution equations of second order, Math. Rep. College General Ed. Kyushu Univ., 10, 67-75 (1975)
[47] Ono, K., On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20, 151-177 (1997) · Zbl 0878.35081
[48] Park, J. Y.; Bae, J. J., On the existence of solutions of strongly damped nonlinear wave equations, Int. J. Math. Math. Sci., 23, 369-382 (2000) · Zbl 0960.35069
[49] Pata, V.; Squassina, M., On the strongly damped wave equation, Comm. Math. Phys., 253, 3, 511-533 (2005) · Zbl 1068.35077
[50] Pata, V.; Zelik, S., Smooth attractors for strongly damped wave equations, Nonlinearity, 19, 1495-1506 (2006) · Zbl 1113.35023
[51] Pecher, H., On global regular solutions of third order partial differential equations, J. Math. Anal. Appl., 73, 278-299 (1980) · Zbl 0429.35057
[52] Pinter, G. A., Global attractor for damped abstract nonlinear hyperbolic systems, Nonlinear Anal., 53, 653-668 (2003) · Zbl 1024.35020
[53] Rybka, P., Viscous damping prevents propagation of singularities in the system of viscoelasticity, Proc. Roy. Soc. Edinburgh Sect. A, 127, 1067-1074 (1997) · Zbl 0890.35156
[54] Webb, G. F., Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32, 631-643 (1980) · Zbl 0414.35046
[55] Yamada, Y., Quasilinear wave equations and related nonlinear evolution equations, Nagoya Math. J., 84, 31-83 (1981) · Zbl 0472.35052
[56] Yang, M.; Sun, C., Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361, 2, 1069-1101 (2009) · Zbl 1159.37022
[57] Yang, Zh., Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms, J. Math. Anal. Appl., 300, 218-243 (2004) · Zbl 1060.35086
[58] Yang, Zh., Cauchy problem for quasi-linear wave equations with viscous damping, J. Math. Anal. Appl., 320, 859-881 (2006) · Zbl 1112.35128
[59] Tvedt, B., Quasilinear equations for viscoelasticity of strain-rate type, Arch. Ration. Mech. Anal., 189, 237-281 (2008) · Zbl 1147.74008
[60] Zelik, S., Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete Contin. Dyn. Syst., 11, 351-392 (2004) · Zbl 1059.35018
[61] Zelik, S., The attractor for a nonlinear reaction-diffusion system with a supercritical nonlinearity and it’s dimension, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 24, 1-25 (2000)
[62] Zelik, S., Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3, 921-934 (2004) · Zbl 1197.35168
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