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Asymptotic behavior for the singularly perturbed damped Boussinesq equation
Author(s) -
Li Ke,
Yang Zhijian
Publication year - 2014
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3167
Subject(s) - mathematics , attractor , bounded function , mathematical analysis , exponential function , work (physics) , exponential growth , nonlinear system , boussinesq approximation (buoyancy) , physics , heat transfer , natural convection , quantum mechanics , rayleigh number , thermodynamics
This work is focused on the long‐time behavior of solutions to the singularly perturbed damped Boussinesq equation in a 3D case ε u tt+Δ 2 u − Δ u t− Δf ( u ) = g ( x ) ,where ε > 0 is small enough. Without any growth restrictions on the nonlinearity f ( u ), we establish in an appropriate bounded phase space a finite dimensional global attractor as well as an exponential attractor of optimal regularity. The key step is the estimate of the difference between the solutions of the damped Boussinesq equation and the corresponding pseudo‐parabolic equation.Copyright © 2014 John Wiley & Sons, Ltd.