Asymptotic behavior for the singularly perturbed damped Boussinesq equation

Abstract

Communicated by T. HishidaThis work is focused on the long-time behavior of solutions to the singularly perturbed damped Boussinesq equation in a 3D case u tt C 2 u u t f.u/ D g.x/, where > 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. (1.5) and its generalization form (see [3])which is used to describe the small nonlinear oscillations of elastic membranes in the presence of viscosity.

Uniform attractor for non‐autonomous Boussinesq‐type equation with critical nonlinearity

“…Bona and Sachs considered the local well‐posedness of solutions for with α = 1, β =− 1. Yang and Guo studied the existence and non‐existence of global weak solution of multi‐dimension Boussinesq equation The longtime dynamics of problem has been investigated in for the autonomous case. The asymptotic behaviors of solutions were obtained by Yang in a weaker space with non‐supercritical nonlinearity.…”

“…The global attractor was shown by Li and Fu in a stronger space with critical nonlinearity. Recently, Li and Yang studied the equation with supercritical nonlinearity, taking advantage of the smallness of ε , we established in an appropriate bounded phase space a finite‐dimensional global attractor as well as an exponential attractor of optimal regularity. We also mention the paper , where the dissipative term −Δ u t is replaced by a weaker one u t in .…”

“…The longtime dynamics of problem (1.1) has been investigated in [12][13][14] for the autonomous case. The asymptotic behaviors of solutions were obtained by Yang [14] in a weaker space with non-supercritical nonlinearity.…”

“…The global attractor was shown by Li and Fu [12] in a stronger space with critical nonlinearity. Recently, Li and Yang [13] studied the equation with supercritical nonlinearity,…”

“…The present paper is a continuation of the investigations in Refs. on studying the longtime behavior of solutions to problem . The feature of the model is that: (i) The nonlinearity f ( u ) has critical exponent.…”