Pullback attractors for a semilinear heat equation on time-varying domains

Abstract

The existence of a pullback attractor is established for the nonautonomous dynamical system generated by the weak solutions of a semilinear heat equation on time-varying domains with homogeneous Dirichlet boundary conditions. It is assumed that the spatial domains O t in R N are obtained from a bounded base domain O by a C 2 -diffeomorphism, which is continuously differentiable in the time variable, and are contained, in the past, in a common bounded domain.

Pullback-Forward Dynamics for Damped Schrödinger Equations with Time-Dependent Forcing

“…This paper deals with dynamics for the nonautonomous Schrödinger equation; that is, the force is time-dependent. To the best of our knowledge, there is no literature treating nonautonomous dynamics (including random dynamics) for the Schrödinger equation, even in the simple case for the existence of a pullback attractor, although the theory and application of pullback attractors had been widely developed for many other PDEs (see [12][13][14][15][16]), and for pullback random attractors, see, for example, [17][18][19][20].…”

Higher-order asymptotic attraction of pullback attractors for a reaction–diffusion equation in non-cylindrical domains

“…On the other hand, the problems with domains changing in time aroused many authors' interest as well, for example, see [6][7][8][9][10][11][12] and so on. Recently, under the condition of bounded domains increasing with respect to time, that is, the condition (1.1), Kloeden, Marín-Rubio and Real considered in the pioneer work [9], applying the penalty method, the existence and uniqueness of a variational solution for (1.3) as well as the existence of a (L 2 , L 2 ) pullback D λ -attractor for the dynamical system generated by (1.3) has been established.…”

Low‐order optimal regulation of parabolic PDEs with time‐dependent domain

“…In addition, only few contributions have reported the study of parabolic PDEs with time‐varying domain, in which main results are focused on establishing existence and regularity properties of the solution. These include development of transformations to map the PDE onto a new time‐invariant spatial domain and evolution of continuously differentiable diffeomorphisms . Among contributions along this line, a design of nonlinear distributed state observers for systems with moving boundaries using stochastic methods is notable.…”