Longtime Dynamics of Hyperbolic Evolutionary Equations in Unbounded Domains and Lattice Systems

This dissertation is a contribution to the study of longtime dynamics of evolutionary equations in unbounded domains and of lattice systems. It is of particular interest to prove the existence of global attractors for solutions of such equations. To this end, one needs in general some type of asymptotical compactness. In the case that the evolutionary PDE is defined on a bounded domain Ω in space, asymptotical compactness follows from the regularity estimates and the compactness of the Sobolev embeddings and therefore the existence of attractors has been established for most of the dissipative equations of mathematical physics in a bounded domain. The problem is more challenging when Ω is unbounded since the Sobolev embeddings are no longer compact, so that the usual regularity estimates may not be sufficient. To overcome this obstacle of compactness, A.V. Babin and M.I. Vishik introduced some weighted Sobolev spaces. In their pioneering paper [2], they established the existence of a global attractor for the reaction-diffusion equation ut − ν∆u + f(u) + λu = g, x ∈ R N Lately, a new technique of ”tail estimation” has been introduced by B. Wang [49] to prove the existence of global attractors for the reaction-diffusion equation (1) in the usual Hilbert space L 2 (R N ). In this research we take on the same approach to prove the existence of attracting sets for some nonlinear wave equations and hyperbolic lattice systems. The dissertation is organized as follows. In the first part (Chapter 2), we prove the existence of a global attractor in H10 (R N ) × L 2 (R N ) for the wave equation utt + λut − ∆u + u + f(u) = g, t \u3e 0, x ∈ R N . Removing the coercive mass term u from (2), we achieve the same result for the more challenging equation utt + λut − ∆u + f(u) = g, t \u3e 0, x ∈ Ω where Ω is a domain of R N bounded only in one direction. The second part of the dissertation deals with some lattice systems. We establish in Chapter 3 the existence of global attractor for the equation u¨i + λu˙i − (ui−1 − 2ui + ui+1) + f(ui) = gi , i ∈ Z which is a spatial discretization of (3)