Exponential Attractor for the Boussinesq Equation with Strong Damping and Clamped Boundary Condition

Abstract

The paper studies the existence of exponential attractor for the Boussinesq equation with strong damping and clamped boundary conditionutt-Δu+Δ2u-Δut-Δg(u)=f(x). The main result is concerned with nonlinearitiesg(u)with supercritical growth. In that case, we construct a bounded absorbing set with further regularity and obtain quasi-stability estimates. Then the exponential attractor is established in natural energy spaceV2×H.

Asymptotic behavior of the Boussinesq equation with nonlocal weak damping and arbitrary growth nonlinear function

“…Besides, such equations can model not only the oscillation of the nonlinear strings but also the two-dimensional irrigational flows of an inviscid liquid in a uniform rectangular channel as µ > 0 ( [8,9]); meanwhile, they can also be exploited to describe the propagation of ion-sound waves in a uniform isotropic plasma and nonlinear lattice waves as µ < 0 ( [3,4]). To the best of our knowledge, there are quite a lot of profound researches to the Boussinesq equations from various view of dynamical system, see [10][11][12][13][14][15][16][17][18] and references therein. For instance, in [10] the finite time blow-up of solution, existence and uniqueness of local mild solution were achieved for the cauchy problem of dissipative Boussinesq equations.…”

“…As far as we know, global attractor is a key concept to study the long-time behavior of solutions for dissipative nonlinear evolution equations coming from physics and mechanics as well as atmospheric sciences and so on, please refer to [22][23][24][25][26] and references therein. In the matter of Boussinesq equations, study of global attractor has attracted lots of mathematicians, see [12][13][14][15][16][17][18]. In these literatures, Li and…”

Global attractors for the semilinear beam equation with localized viscosity

“…is studied by some authors (see [6,10,[23][24][25]). Equation (1.1) can be seen as a generalized version of the Boussinesq equation with localized internal damping.…”