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September  2017, 16(5): 1673-1695. doi: 10.3934/cpaa.2017080

## Semilinear damped wave equation in locally uniform spaces

 1 Institute of Mathematics of the Czech Academy of Sciences, Prague, Žitná 25,115 67 Praha 1, Czech Republic 2 Department of Mathematical Analysis, Charles University, Prague, Sokolovská 83,186 75 Praha 8, Czech Republic

* Corresponding author

Received  September 2016 Revised  March 2017 Published  May 2017

We study a damped wave equation with a nonlinear damping in the locally uniform spaces and prove well-posedness and existence of a locally compact attractor. An upper bound on the Kolmogorov's $\varepsilon$-entropy is also established using the method of trajectories.

Citation: Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080
##### References:
 [1] P. Anthony and S. Zelik, Infinite-energy solutions for the Navier-Stokes equations in a strip revisited, Commun. Pure Appl. Anal., 13 (2014), 1361-1393. doi: 10.3934/cpaa.2014.13.1361. Google Scholar [2] J. M. Arrieta, A. Rodriguez-Bernal, J. W. Cholewa and T. Dlotko, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. Google Scholar [3] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009. Google Scholar [4] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. Google Scholar [5] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183. doi: 10.1090/memo/0912. Google Scholar [6] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd. , Chichester, 1994. Google Scholar [7] E. Feireisl, Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1051-1062. doi: 10.1017/S0308210500022630. Google Scholar [8] E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447. doi: 10.1006/jdeq.1995.1042. Google Scholar [9] M. Grasselli, D. Pražák and G. Schimperna, Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories, J. Differential Equations, 249 (2010), 2287-2315. doi: 10.1016/j.jde.2010.06.001. Google Scholar [10] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri PoincarÃ©, 17 (2016), 2555-2584. doi: 10.1007/s00023-016-0480-y. Google Scholar [11] L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323. doi: 10.1080/03605309508821133. Google Scholar [12] A. K. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719. doi: 10.1016/j.jde.2006.06.001. Google Scholar [13] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), vol. 50 of Progr. Nonlinear Differential Equations Appl. , Birkhäuser, Basel, 2002,197-216. Google Scholar [14] J. -L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. Google Scholar [15] A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768. Google Scholar [16] J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, Journal of Differential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. Google Scholar [17] D. Pražák, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dynam. Differential Equations, 14 (2002), 763-776. doi: 10.1023/A:1020756426088. Google Scholar [18] A. Savostianov, Infinite energy solutions for critical wave equation with fractional damping in unbounded domains, Nonlinear Anal., 136 (2016), 136-167. doi: 10.1016/j.na.2016.02.016. Google Scholar [19] C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, Journal of Differential Equations, 227 (2006), 427-443. doi: 10.1016/j.jde.2005.09.010. Google Scholar [20] M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101. doi: 10.1090/S0002-9947-08-04680-1. Google Scholar [21] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete Contin. Dyn. Syst., 11 (2004), 351-0392. doi: 10.3934/dcds.2004.11.351. Google Scholar [22] S. V. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dynam. Systems, 7 (2001), 593-641. doi: 10.3934/dcds.2001.7.593. Google Scholar [23] S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $Îµ$-entropy, Math. Nachr., 232 (2001), 129-179. doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.3.CO;2-K. Google Scholar

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##### References:
 [1] P. Anthony and S. Zelik, Infinite-energy solutions for the Navier-Stokes equations in a strip revisited, Commun. Pure Appl. Anal., 13 (2014), 1361-1393. doi: 10.3934/cpaa.2014.13.1361. Google Scholar [2] J. M. Arrieta, A. Rodriguez-Bernal, J. W. Cholewa and T. Dlotko, Linear parabolic equations in locally uniform spaces, Math. Models Methods Appl. Sci., 14 (2004), 253-293. doi: 10.1142/S0218202504003234. Google Scholar [3] L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009. Google Scholar [4] I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. Dynam. Differential Equations, 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x. Google Scholar [5] I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), ⅷ+183. doi: 10.1090/memo/0912. Google Scholar [6] A. Eden, C. Foias, B. Nicolaenko and R. Temam, Exponential Attractors for Dissipative Evolution Equations, vol. 37 of RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd. , Chichester, 1994. Google Scholar [7] E. Feireisl, Asymptotic behaviour and attractors for a semilinear damped wave equation with supercritical exponent, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 1051-1062. doi: 10.1017/S0308210500022630. Google Scholar [8] E. Feireisl, Global attractors for semilinear damped wave equations with supercritical exponent, J. Differential Equations, 116 (1995), 431-447. doi: 10.1006/jdeq.1995.1042. Google Scholar [9] M. Grasselli, D. Pražák and G. Schimperna, Attractors for nonlinear reaction-diffusion systems in unbounded domains via the method of short trajectories, J. Differential Equations, 249 (2010), 2287-2315. doi: 10.1016/j.jde.2010.06.001. Google Scholar [10] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri PoincarÃ©, 17 (2016), 2555-2584. doi: 10.1007/s00023-016-0480-y. Google Scholar [11] L. Kapitanski, Minimal compact global attractor for a damped semilinear wave equation, Comm. Partial Differential Equations, 20 (1995), 1303-1323. doi: 10.1080/03605309508821133. Google Scholar [12] A. K. Khanmamedov, Global attractors for wave equations with nonlinear interior damping and critical exponents, J. Differential Equations, 230 (2006), 702-719. doi: 10.1016/j.jde.2006.06.001. Google Scholar [13] H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, in Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), vol. 50 of Progr. Nonlinear Differential Equations Appl. , Birkhäuser, Basel, 2002,197-216. Google Scholar [14] J. -L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. Google Scholar [15] A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768. Google Scholar [16] J. Málek and D. Pražák, Large time behavior via the method of $\ell$-trajectories, Journal of Differential Equations, 181 (2002), 243-279. doi: 10.1006/jdeq.2001.4087. Google Scholar [17] D. Pražák, On finite fractal dimension of the global attractor for the wave equation with nonlinear damping, J. Dynam. Differential Equations, 14 (2002), 763-776. doi: 10.1023/A:1020756426088. Google Scholar [18] A. Savostianov, Infinite energy solutions for critical wave equation with fractional damping in unbounded domains, Nonlinear Anal., 136 (2016), 136-167. doi: 10.1016/j.na.2016.02.016. Google Scholar [19] C. Sun, M. Yang and C. Zhong, Global attractors for the wave equation with nonlinear damping, Journal of Differential Equations, 227 (2006), 427-443. doi: 10.1016/j.jde.2005.09.010. Google Scholar [20] M. Yang and C. Sun, Dynamics of strongly damped wave equations in locally uniform spaces: attractors and asymptotic regularity, Trans. Amer. Math. Soc., 361 (2009), 1069-1101. doi: 10.1090/S0002-9947-08-04680-1. Google Scholar [21] S. Zelik, Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities, Discrete Contin. Dyn. Syst., 11 (2004), 351-0392. doi: 10.3934/dcds.2004.11.351. Google Scholar [22] S. V. Zelik, The attractor for a nonlinear hyperbolic equation in the unbounded domain, Discrete Contin. Dynam. Systems, 7 (2001), 593-641. doi: 10.3934/dcds.2001.7.593. Google Scholar [23] S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $Îµ$-entropy, Math. Nachr., 232 (2001), 129-179. doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.3.CO;2-K. Google Scholar
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