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On the Structure of the Global Attractor for Infinite-Dimensional Non-Autonomous Dynamical Systems with Weak Convergence
The aim of this paper is to describe the structure of global attractors for infinite-dimensional non-autonomous dynamical systems with recurrent coefficients. We consider a special class of this type of systems (the so–called weak convergent systems). We study this problem in the framework of general
non-autonomous dynamical systems (cocycles). In particular, we apply the general results obtained in our previous paper to study the almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent,
asymptotically pseudo recurrent) solutions of different classes of differential equations (functional-differential equations, evolution equation with monotone operator, semi-linear parabolic equations)
Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem
We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB)
equations, i.e. scalar conservation laws with diffusive-dispersive
regularization. We review the existence of traveling wave solutions for these
two classes of evolution equations. For classical equations the traveling wave
problem (TWP) for a local KdVB equation can be identified with the TWP for a
reaction-diffusion equation. In this article we study this relationship for
these two classes of evolution equations with nonlocal diffusion/dispersion.
This connection is especially useful, if the TW equation is not studied
directly, but the existence of a TWS is proven using one of the evolution
equations instead. Finally, we present three models from fluid dynamics and
discuss the TWP via its link to associated reaction-diffusion equations
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