Global solvability and asymptotics of semilinear parabolic Cauchy problems in are considered. Following the approach of A. Mielke [15] these problems are investigated in weighted Sobolev spaces. The paper provides also a theory of second order elliptic operators in such spaces considered over , . In particular, the generation of analytic semigroups and the embeddings for the domains of fractional powers of elliptic operators are discussed.
We investigate the long-time behaviour of solutions to the Korteweg-de Vries equation with a zero order dissipation and an additional forcing term, when the space variable varies over , and prove that it is described by a maximal compact attractor in .
Nous étudions le comportement pour les grands temps de l’équation de Schrödinger-Poisson (NLSP) avec un terme de force extérieure supplémentaire et un terme de dissipation d’ordre zéro, la variable d’espace étant dans un domaine borné de . Nous démontrons que ce comportement est décrit par un attracteur global de dimension de Hausdorff finie pour la topologie forte de .
Consider the family uₜ = Δu + G(u), t > 0, , , t > 0, , of semilinear Neumann boundary value problems, where, for ε > 0 small, the set is a thin domain in , possibly with holes, which collapses, as ε → 0⁺, onto a (curved) k-dimensional submanifold of . If G is dissipative, then equation has a global attractor . We identify a “limit” equation for the family , prove convergence of trajectories and establish an upper semicontinuity result for the family as ε → 0⁺.