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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 and 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.

[2]

J. M. ArrietaA. Rodriguez-BernalJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

M. GrasselliD. 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.

[10]

V. KalantarovA. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y.

[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.

[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.

[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.

[14]

J. -L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[15]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768. 

[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.

[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.

[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.

[19]

C. SunM. 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.

[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.

[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.

[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.

[23]

S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $ε$-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.3.CO;2-K.

show all references

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.

[2]

J. M. ArrietaA. Rodriguez-BernalJ. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

M. GrasselliD. 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.

[10]

V. KalantarovA. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, Ann. Henri Poincaré, 17 (2016), 2555-2584.  doi: 10.1007/s00023-016-0480-y.

[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.

[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.

[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.

[14]

J. -L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969.

[15]

A. Mielke and G. Schneider, Attractors for modulation equations on unbounded domains-existence and comparison, Nonlinearity, 8 (1995), 743-768. 

[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.

[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.

[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.

[19]

C. SunM. 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.

[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.

[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.

[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.

[23]

S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $ε$-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.3.CO;2-K.

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