LONG-TIME DYNAMICS OF THE STRONGLY DAMPED SEMILINEAR PLATE EQUATION IN RN

doi:10.1016/S0252-9602(18)30799-9

Abstract

Abstract:

We investigate the initial-value problem for the semilinear plate equation containing localized strong damping, localized weak damping and nonlocal nonlinearity. We prove that if nonnegative damping coefficients are strictly positive almost everywhere in the exterior of some ball and the sum of these coefficients is positive a.e. in Rⁿ, then the semigroup generated by the considered problem possesses a global attractor in H² (Rⁿ)×L² (Rⁿ). We also establish the boundedness of this attractor in H³ (Rⁿ)×H² (Rⁿ).

Key words: Wave equation, plate equation, global attractor

Azer KHANMAMEDOV, Sema YAYLA. LONG-TIME DYNAMICS OF THE STRONGLY DAMPED SEMILINEAR PLATE EQUATION IN R^N[J].Acta mathematica scientia,Series B, 2018, 38(3): 1025-1042.

Trendmd

References

[1] Pazy A. Semigroups of linear operators and applications to partial differential equations. New York:Springer, 1983
[2] Cazenave T, Haraux A. An introduction to semilinear evolution equations. New York:Oxford University Press, 1998
[3] Feireisl E. Attractors for semilinear damped wave equations on R³. Nonlinear Anal, 1994, 23:187-195
[4] Feireisl E. Asymptotic behavior and attractors for a semilinear damped wave equation with supercritical exponent. Proc Roy Soc Edinburgh, 1995, 125:1051-1062
[5] Belleri V, Pata V. Attractors for semilinear strongly damped wave equations on R³. Discrete Contin Dyn Syst, 2001, 7:719-735
[6] Conti M, Pata V, Squassina M. Strongly damped wave equations on R³ with critical nonlinearities. Commun Appl Anal, 2005, 9:161-176
[7] Yang Z, Ding P. Longtime dynamics of the Kirchhoff equation with strong damping and critical nonlinearity on RN. J Math Anal Appl, 2016, 434:1826-1851
[8] Khanmamedov A.Kh. Existence of a global attractor for the plate equation with a critical exponent in an unbounded domain. Applied Mathematics Letters, 2005, 18:827-832
[9] Khanmamedov A.Kh. Global attractors for the plate equation with a localized damping and a critical exponent in an unbounded domain. J Differential Equations, 2006, 225:528-548
[10] Wang B. Attractors for reaction diffusion equations in unbounded domains. Phys D, 1999, 128:41-52
[11] Ball J M. Global attractors for damped semilinear wave equations. Discrete Contin Dyn Syst, 2004, 10:31-52
[12] Arat Z, Khanmamedov A, Simsek S. Global attractors for the plate equation with nonlocal nonlinearity in unbounded domains. Dyn Partial Differ Equ, 2014, 11:361-379
[13] Khanmamedov A, Simsek S. Existence of the global attractor for the plate equation with nonlocal nonlinearity in Rn. Discrete Contin Dyn Syst Ser B, 2016, 21:151-172
[14] Chueshov I, Lasiecka I. Von Karman Evolution Equations:Well-posedness and long-time dynamics. New York:Springer, 2010
[15] Simon J. Compact sets in the space L_p(0, T; B). Annali Mat Pura Appl, 1987, 146:65-96
[16] Khanmamedov A.Kh. Global attractors for von Karman equations with nonlinear interior dissipation. J Math Anal Appl, 2006, 318:92-101
[17] Khanmamedov A.Kh. Global attractors for 2-D wave equations with displacement dependent damping. Math Methods Appl Sci, 2010, 33:177-187
[18] Babin A V, Vishik M I. Attractors of evolution equations. Amsterdam:North-Holland, 1992

LONG-TIME DYNAMICS OF THE STRONGLY DAMPED SEMILINEAR PLATE EQUATION IN R^N

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Lei ZHANG, Bin LIU. THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM [J]. Acta mathematica scientia,Series B, 2018, 38(2): 651-672.
[2]	Zaiyun ZHANG, Jianhua HUANG, Mingbao SUN. ALMOST CONSERVATION LAWS AND GLOBAL ROUGH SOLUTIONS OF THE DEFOCUSING NONLINEAR WAVE EQUATION ON R² [J]. Acta mathematica scientia,Series B, 2017, 37(2): 385-394.
[3]	LUO Shao-Ying, LIU Qi. SPHERICAL SYMMETRIC SOLUTIONS FOR THE MOTION OF RELATIVISTIC MEMBRANES AND NULL MEMBRANES IN THE REISSNER-NORDSTRÖM SPACE-TIME [J]. Acta mathematica scientia,Series B, 2014, 34(3): 917-931.
[4]	WANG Yong-Hai, WANG Ling-Zhi. TRAJECTORY ATTRACTORS FOR NONCLASSICAL DIFFUSION EQUATIONS WITH FADING MEMORY [J]. Acta mathematica scientia,Series B, 2013, 33(3): 721-737.
[5]	GUO Hong-Xia, CHEN Guo-Wang. A NOTE ON “THE CAUCHY PROBLEM FOR COUPLED IMBQ EQUATIONS” [J]. Acta mathematica scientia,Series B, 2013, 33(2): 375-392.
[6]	LUO Cao. LOCAL STABILITY OF TRAVELLING FRONTS FOR A DAMPED WAVE EQUATION [J]. Acta mathematica scientia,Series B, 2013, 33(1): 75-83.
[7]	Shun-Tang Wu. EXPONENTIAL DECAY FOR A NONLINEAR VISCOELASTIC EQUATION WITH SINGULAR KERNELS [J]. Acta mathematica scientia,Series B, 2012, 32(6): 2237-2246.
[8]	ZHANG Fa-Yong, GUO Bai-Ling. ATTRACTORS FOR FULLY DISCRETE FINITE DIFFERENCE SCHEME OF DISSIPATIVE ZAKHAROV EQUATIONS [J]. Acta mathematica scientia,Series B, 2012, 32(6): 2431-2452.
[9]	CHEN Xue-Yong, LIU Wei-An. GLOBAL ATTRACTOR FOR A DENSITY-DEPENDENT SENSITIVITY CHEMOTAXIS MODEL [J]. Acta mathematica scientia,Series B, 2012, 32(4): 1365-1375.
[10]	Jeong Ja Bae. ON TRANSMISSION PROBLEM FOR KIRCHHOFF TYPE WAVE EQUATION WITH A LOCALIZED NONLINEAR DISSIPATION IN BOUNDED DOMAIN [J]. Acta mathematica scientia,Series B, 2012, 32(3): 893-906.
[11]	CHEN Jiao. GLOBAL EXISTENCE AND L^p ESTIMATES FOR SOLUTIONS OF DAMPED WAVE EQUATION WITH NONLINEAR CONVECTION IN#br# MULTI-DIMENSIONAL SPACE [J]. Acta mathematica scientia,Series B, 2012, 32(3): 1167-1180.
[12]	Donatella Donatelli, Pierangelo Marcati. LOW MACH NUMBER LIMIT ON EXTERIOR DOMAINS [J]. Acta mathematica scientia,Series B, 2012, 32(1): 164-176.
[13]	Liu Xiaomin, Wang Zhen. THE RIEMANN PROBLEM FOR THE NONLINEAR DEGENERATE WAVE EQUATIONS [J]. Acta mathematica scientia,Series B, 2011, 31(6): 2313-2322.
[14]	FAN Li-Li, LIU Hong-Xia, YIN Hui. DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN#br# MULTI-DIMENSIONAL HALF SPACE [J]. Acta mathematica scientia,Series B, 2011, 31(4): 1389-1410.
[15]	Jeong Ja Bae. GLOBAL EXISTENCE FOR QUASILINEAR \|WAVE EQUATION WITH A LOCALIZED WEAKLY NONLINEAR DISSIPATION IN EXTERIOR DOMAINS [J]. Acta mathematica scientia,Series B, 2009, 29(5): 1203-1215.