Herein I define the global attractor for the semidynamical system (H, {S(t)}t≥0), where H is a Hilbert space and S(t) is a semigroup. In particular, I will consider the semigroup S(t) which acts on a relevant function space by S(t)u0 = u(t; u0), where u(t; u0) is the solution of a given partial differential equation at time t with initial condition u(0) = u0. By imposing regularity properties o...

In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn–Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unst...

In this work the existence of a global attractor for the solution semiflow of the coupled two-cell Brusselator model equations is proved. A grouping estimation method and a new decomposition approach are introduced to deal with the challenge in proving the absorbing property and the asymptotic compactness of this type of fourvariable reaction-diffusion systems with cubic autocatalytic nonlinear...

This paper addresses the long-time behaviour of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global attractor. The abstract results are applied to various classes of non convex evolution problems. In particular, we discuss the longtime behaviour of solutions ...

The initial boundary value problem for a nonlinear system of equations modeling the chevron patterns is studied in one and two spatial dimensions. existence an exponential attractor stabilization zero steady state solution through application finite-dimensional feedback control proved arbitrary fixed shown dimension along with relevant numerical results.

A phase-field system of coupled Allen–Cahn type PDEs describing grain growth is analyzed and simulated. In the periodic setting, we prove the existence and uniqueness of global weak solutions to the problem. Then we investigate the long-time behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. Namely, the problem possesses a global attractor as well...

In this paper, relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered. By introducing almost surely D–α-contracting property for random dynamical systems, we obtain a global random attractor of the stochastic wave equation νutt + ut − uν + f (uν) = √ νẆ endowed with Dirichlet boundary condition for any 0 < ν 1. The upper semicontinuity of thi...