Global dynamics of dissipative generalized KdV equations and Boussinesq equations

This dissertation work is on the global dynamics of two different types of nonlinear partlial differential equations: the dissipative fifth-order generalized KdV equations and the damped Boussinesq equations.

The fifth-order KdV equations of concern take a significant position in the hierarchy of nonlinear dispersive equations. As hyperbolic evolutionary equations, the relationship between the Boussinesq equations and the nonlinear wave equations is analogous to the relationship of Kuramoto-Sivashinsky equations versus reaction-diffusion equations. For these two types of nonlinear evolutionary equations, the long-time behavior of their solutions is open along with many other issues.

In the first part of this dissertation, the following fifth-order KdV equation is studied, $u\sb{t} + u\sb{xxxxx} + (\lambda u\sb{xxxx} - \eta{u}\sb{xx} + \gamma{u}) + au\sp2u\sb{x} - 2bu\sb{x}u\sb{xx} - buu\sb{xxx} = f, (x,t) \in\Re\times\Re\sp{+},$ where $\lambda$, $\eta$, $\gamma$ $>$ 0, a, b $\in \Re$ are constants, and f is a time-invariant function, with the 2$\pi$-periodic boundary condition and the initial condition u(x,0) = $u\sb0$. By using the elliptic regularization to study the perturbed equations in a framework based on analytic linear semigroups and nonlinear mild solutions, the global existence and regularity of solutions of the perturbed equations are established. Then, a priori estimates in the Sobolev state spaces are conducted to obtain the absorbing property of the semiflow, that leads to the existence of a global attractor in H$\sbsp{3}{2\pi}$.

A main result in Part I is the existence of inertial manifolds for the dissipative fifth-order generalized KdV equations. The approach in this proof consists of two phases. In the first phase, it is shown that there exists an $\epsilon\sb0 >$ 0 such that for 0 $< \epsilon \leq \epsilon\sb0$ the corresponding perturbed equations possess inertial manifolds with uniformly bounded dimensions as $\epsilon \rightarrow$ 0$\sp+$, by means of the non-self-adjoint theory of inertial manifolds and a series of delicate analysis. Then in the second phase, letting $\epsilon \rightarrow$ 0$\sp+$, it is proved that the family of the graph mappings has compactness and that the limit graph turns out to be an inerlial manifold for the original fifth-order KdV equation.

The second part of this dissertation deals with the following one-dimensional damped Boussinesq equation, $u\sb{tt} + au\sb{t} + u\sb{xxxx} - \beta u\sb{xx} - ku\sbsp{2}{x}u\sb{xx} - \gamma(u\sp2)\sb{xx} = 0$ in a bounded domain $\Omega$ = (0,1) with Dirichlet boundary conditions, where $\beta$, $\gamma \in \Re, k >$ 0 are constants, and a $>$ 0 is a damping coefficient. This equation is formulated into a nonlinear evolutionary equation in the energy space. A main result is the existence of a global attractor in that space. The crucial difficulty, which is in common to almost all the hyperbolic evolutionary equations, is to prove the contracting property of the solution semigroup in terms of Kuratowski measure as a substitution of the strict compactness. This difficulty is overcome by means of a precompact pseudometric and sophisticated manipulation of differential inequalities for suitably defined Lyapunov-like functionals. It is stressed here that this approach is applicable lo attack the same issue for other hyperbolic-type equations.

In the second part, a linear stabilization and another sharper nonlinear stabilization of the Boussinesq equations are also achieved by the energy method.