American Institute of Mathematical Sciences

The asymptotic behaviour of an autonomous neural field lattice system with delays is investigated. It is based on the Amari model, but with the Heaviside function in the interaction term replaced by a sigmoidal function. First, the lattice system is reformulated as an infinite dimensional ordinary delay differential equation on weighted sequence state space \begin{document}$ \ell_\rho^2 $\end{document} under some appropriate assumptions. Then the global existence and uniqueness of its solution and its formulation as a semi-dynamical system on a suitable function space are established. Finally, the asymptotic behaviour of solution of the system is investigated, in particular, the existence of a global attractor is obtained.

In this paper, two properties of the pullback attractor for a 2D non-autonomous micropolar fluid flows with delay on unbounded domains are investigated. First, we establish the \begin{document}$ H^1 $\end{document}-boundedness of the pullback attractor. Further, with an additional regularity limit on the force and moment with respect to time t, we remark the \begin{document}$ H^2 $\end{document}-boundedness of the pullback attractor. Then, we verify the upper semicontinuity of the pullback attractor with respect to the domains.

In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary \begin{document}$ p>2 $\end{document} order nonlinearity and in any space dimension \begin{document}$ N \geqslant 1 $\end{document}. It is proved that the weak solutions can be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document}-continuous in initial data for arbitrarily large \begin{document}$ \gamma \geqslant 2 $\end{document} (independent of the physical parameters of the system), i.e., can converge in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document} as the corresponding initial values converge in \begin{document}$ L^2 $\end{document}. In fact, the system is shown to be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document}-smoothing in a H\begin{document}$ \ddot{\rm o} $\end{document}lder way. Applying this to the global attractor we find that, with external forcing only in \begin{document}$ L^2 $\end{document}, the attractor \begin{document}$ \mathscr{A} $\end{document} attracts bounded subsets of \begin{document}$ L^2 $\end{document} in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document}, and that every translation set \begin{document}$ \mathscr{A}-z_0 $\end{document} of \begin{document}$ \mathscr{A} $\end{document} for any \begin{document}$ z_0\in \mathscr{A} $\end{document} is a finite dimensional compact subset of \begin{document}$ L^\gamma\cap H_0^1 $\end{document}. The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order \begin{document}$ p>2 $\end{document} of the nonlinearity and the space dimension \begin{document}$ N \geqslant 1 $\end{document}.

We study the random dynamics for the stochastic non-autonomous Kuramoto-Sivashinsky equation in the possibly non-dissipative case. We first prove the existence of a pullback attractor in the Lebesgue space of odd functions, then show that the fiber of the odd pullback attractor semi-converges to a nonempty compact set as the time-parameter goes to minus infinity and finally prove the measurability of the attractor. In a word, we obtain a longtime stable random attractor formed from odd functions. A key tool is the existence of a bridge function between Lebesgue and Sobolev spaces of odd functions.