Large time dynamics for non-autonomous weakly damped KdV-Burgers equations on the real line

Abstract

The large time dynamics of the weakly damped non-autonomous Korteweg-de Vries-Burgers equation \begin{equation*} u_t + u_{xxx}+ uu_x + λu - μu_{xx} = g(x,t) \end{equation*} in the phase space $L^2(\mathbb{R})$ is investigated. Due to the low regularity of the initial data, the framework of Bourgain spaces is employed. We first obtain the existence of a unique pullback attractor $\{{\mathcal{A}}_{\mu }(t){\}}_{t\in \mathbb{R}}$ in $L^2(\mathbb{R})$ for each $\mu \ge 0$. Then we show an optimal asymptotic smoothing effect of the equation, namely the pullback attractor is compact in $H^3(\mathbb{R})$. These results are proved by using an energy equation method, splitting solutions into parts with low and high frequencies and exploiting a dispersive regularization. Furthermore, the attractors are shown to be upper semicontinuous as $\mu \to 0$ and have finite fractal dimensions in $H^1(\mathbb{R})$.