Existence of $ \mathcal{D}- $pullback attractor for an infinite dimensional dynamical system

Abstract

<p style='text-indent:20px;'>At the very beginning of the theory of finite dynamical systems, it was discovered that some relatively simple systems, even of ordinary differential equations, can generate very complicated (chaotic) behaviors. Furthermore these systems are extremely sensitive to perturbations, in the sense that trajectories with close but different initial data may diverge exponentially. Very often, the trajectories of such chaotic systems are localized, up to some transient process, in some subset of the phase space, the so-called strange attractors. Such subset have a very complicated geometric structure. They accumulate the nontrivial dynamics of the system.</p><p style='text-indent:20px;'>For a distributed system, whose time evolution is usually governed by partial differential equations (PDEs), the phase space X is (a subset of) an infinite dimensional function space. We will thus speak of infinite dimensional dynamical systems. Since the global existence and uniqueness of solutions has been proven for a large class of PDEs arising from different domains as Mechanics and Physics, it is therefore natural to investigate whether the features, in particular the notion of attractor, obtained for dynamical systems generated by systems of ODEs generalizes to systems of PDEs.</p><p style='text-indent:20px;'>In this paper we give a positive aftermath by proving the existence of pullback <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-attractor. The key point is to find a bounded family of pullback <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-absorbing sets then we apply the decomposition techniques and a method used in previous works to verify the pullback <inline-formula><tex-math id="M4">\begin{document}$ w $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M5">\begin{document}$ \mathcal{D} $\end{document}</tex-math></inline-formula>-limit compactness. It is based on the concept of the Kuratowski measure of noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions.</p>