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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 & 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.  Google Scholar

[2]

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

[3]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \arXiv{1010.4991}., ().   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

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.  Google Scholar

[2]

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

[3]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \arXiv{1010.4991}., ().   Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

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

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