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March  2011, 31(1): 119-138. doi: 10.3934/dcds.2011.31.119

Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

A. Kh. Khanmamedov 1,

1. 

Department of Mathematics, Faculty of Science, Hacettepe University, Beytepe 06800, Ankara, Turkey

Received  March 2010 Revised  December 2010 Published  June 2011

In this paper the long time behaviour of the solutions of the 3-D strongly damped wave equation is studied. It is shown that the semigroup generated by this equation possesses a global attractor in $H_{0}^{1}(\Omega )\times L_{2}(\Omega )$ and then it is proved that this is also a global attractor in $(H^{2}(\Omega )\cap H_{0}^{1}(\Omega ))\times H_{0}^{1}(\Omega )$.
Keywords: Attractors, strongly damped wave equations..
Mathematics Subject Classification: 35B41, 35L05, 35L7.
Citation: A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119
References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310.

[3]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping, arXiv:1010.4991.

[4]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, 195 (2008).

[5]

M. Conti and V. Pata, On the regulariaty of global attractors, Discrete Contin. Dynam. Systems, 25 (2009), 1209-1217.

[6]

B. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106.

[7]

V. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 50-54.

[8]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Diff. Equations, 247 (2009), 1120-1155.

[9]

A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187.

[10]

A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlinear Analysis, 72 (2010), 1993-1999.

[11]

A. Kh. Khanmamedov, A strong global attractor for 3-D wave equations with displacement dependent damping, Appl. Math. Letters, 23 (2010), 928-934.

[12]

J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," 1, Springer-Verlag, New York-Heidelberg, 1972.

[13]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188.

[14]

V. Pata and M. Squassina, On the strongly damped wave equation, Commun. Math. Phys., 253 (2005), 511-533.

[15]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.

[16]

V. Pata and S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping, Math. Methods Appl. Sci., 29 (2006), 1291-1306.

[17]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.

[18]

J. Simon, Compact sets in the space $L_p(0, T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96.

[19]

C. Sun, D. Cao and J. Duan, Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761.

[20]

M. Yang and C. Sun, Attractors for strongly damped wave equations, Nonlinear Analysis: Real World Applications, 10 (2009), 1097-1100.

[21]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.

[22]

S. Zhou, Global attractor for strongly damped nonlinear wave equations, Funct. Diff. Eqns., 6 (1999), 451-470.

show all references

References:
[1]

A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[2]

A. N. Carvalho and J. W. Cholewa, Attractors for strongly damped wave equations with critical nonlinearities, Pacific J. Math., 207 (2002), 287-310.

[3]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping, arXiv:1010.4991.

[4]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Memoirs of AMS, 195 (2008).

[5]

M. Conti and V. Pata, On the regulariaty of global attractors, Discrete Contin. Dynam. Systems, 25 (2009), 1209-1217.

[6]

B. Duffy, P. Freitas and M. Grinfeld, Memory driven instability in a diffusion process, SIAM J. Math. Anal., 33 (2002), 1090-1106.

[7]

V. Kalantarov, Attractors for some nonlinear problems of mathematical physics, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 152 (1986), 50-54.

[8]

V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation, J. Diff. Equations, 247 (2009), 1120-1155.

[9]

A. Kh. Khanmamedov, Global attractors for 2-D wave equations with displacement-dependent damping, Math. Methods Appl. Sci., 33 (2010), 177-187.

[10]

A. Kh. Khanmamedov, Remark on the regularity of the global attractor for the wave equation with nonlinear damping, Nonlinear Analysis, 72 (2010), 1993-1999.

[11]

A. Kh. Khanmamedov, A strong global attractor for 3-D wave equations with displacement dependent damping, Appl. Math. Letters, 23 (2010), 928-934.

[12]

J.-L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," 1, Springer-Verlag, New York-Heidelberg, 1972.

[13]

W. E. Olmstead, S. H. Davis, S. Rosenblat and W. L. Kath, Bifurcation with memory, SIAM J. Appl. Math., 46 (1986), 171-188.

[14]

V. Pata and M. Squassina, On the strongly damped wave equation, Commun. Math. Phys., 253 (2005), 511-533.

[15]

V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity, 19 (2006), 1495-1506.

[16]

V. Pata and S. Zelik, Global and exponential attractors for 3-D wave equations with displacement dependent damping, Math. Methods Appl. Sci., 29 (2006), 1291-1306.

[17]

R. Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988.

[18]

J. Simon, Compact sets in the space $L_p(0, T;B)$, Annali Mat. Pura Appl., 146 (1987), 65-96.

[19]

C. Sun, D. Cao and J. Duan, Non-autonomous wave dynamics with memory-asymptotic regularity and uniform attractor, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 743-761.

[20]

M. Yang and C. Sun, Attractors for strongly damped wave equations, Nonlinear Analysis: Real World Applications, 10 (2009), 1097-1100.

[21]

S. Zelik, Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent, Commun. Pure Appl. Anal., 3 (2004), 921-934.

[22]

S. Zhou, Global attractor for strongly damped nonlinear wave equations, Funct. Diff. Eqns., 6 (1999), 451-470.

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