Global dynamics of damped Boussinesq equations.

摘要

This research deals with global dynamics of the following nonlinear Boussinesq equation with weak damping, utt+dut+auxxxx +buxx-kux 2uxx+gu2 xx=f,t0,xe0,1 , u0,t=u1, t=ux1,t =ux1,t=0, t≥0, ux,0=u0 x,ut x,0=u1x ,xe0,1 , where $d,a,$ and k are positive constants, $b$ and $g$ are real constants, and f = f( x) is a time-invariant function. The damped Boussinesq equation is a fourth-order hyperbolic PDE which includes one anti-dissipative lower-order linear term and two strongly nonlinear terms.; The first part of the research (Chapter 2) proves the existence of a global attractor for the solution semiflow of the Boussinesq evolutionary equation in the energy space under the condition . Due to the aforementioned inherent difficulties, in each step of showing the absorbing property and the κ-contracting property of the solution semiflow toward this goal, various obstacles have been surmounted. A new technique in terms of the dual exchange of proving a precompact pseudometric with the gradients is contributed to the crucial component of this proof. At the end of this part, it is shown that the Hausdorff dimension of the global attractor admits a finite upper bound by the Lyapunov exponent approach and a priori estimation.; The second part of this research (Chapter 3) is devoted to the proof of the existence of inertial sets for the semiflow generated by the Boussinesq equation. The key in this proof is to generalize the technical conditions and to show that the pivotal squeezing property is satisfied by this solution semiflow.; The third, and the last, part of this research (Chapter 4) makes a new contribution to the investigation of lattice dynamcal systems booming rapidly in the recent decade. The existence of a global attractor and its upper semicontinuity proved here is the first result, as far as we are aware, concerning the lattice differential equations of second-order in time variable and fourth-order in spatial variable with nonlinearity involving the gradients, which is closely related to the studied Boussinesq equation. An original proof of the uniform asymptotical estimates on the “tail portion” of the lattice solutions is the key to confirming the asymptotical compactness for such a dynamical system defined on an unbounded domain.