The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions.

Existence and continuity of global attractors for ternary mixtures of solids

<p style='text-indent:20px;'>In this paper, we study the long-time dynamics of a system modelinga mixture of three interacting continua with nonlinear damping, sources terms and subjected to small perturbations of autonomousexternal forces with a parameter <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula>, inspired by the modelstudied by Dell' Oro and Rivera [<xref ref-type="bibr" rid="b12">12</xref>]. We establish astabilizability estimate for the associated gradient dynamicalsystem, which as a consequence, implies the existence of a compactglobal attractor with finite fractal dimension andexponential attractors. This estimate is establishedindependent of the parameter <inline-formula><tex-math id="M2">\begin{document}$ \epsilon\in[0,1] $\end{document}</tex-math></inline-formula>. We also prove thesmoothness of global attractors independent of the parameter<inline-formula><tex-math id="M3">\begin{document}$ \epsilon\in[0,1] $\end{document}</tex-math></inline-formula>. Moreover, we show that the family of globalattractors is continuous with respect to the parameter <inline-formula><tex-math id="M4">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> ona residual dense set <inline-formula><tex-math id="M5">\begin{document}$ I_*\subset[0,1] $\end{document}</tex-math></inline-formula> in the same sense proposed inHoang et al. [<xref ref-type="bibr" rid="b15">15</xref>].</p>

Asymptotic dynamics of the solutions for a system of N-coupled fractional nonlinear Schrödinger equations

In the current issue, we consider a system of N-coupled weakly dissipative fractional Schrödinger equations with cubic nonlinearities. We will prove that the asymptotic dynamics of the solutions will be described by the existence of a regular compact global attractor with finite fractal dimension.

ATTRACTORS FOR A CAGINALP MODEL WITH A LOGARITHMIC POTENTIAL AND COUPLED DYNAMIC BOUNDARY CONDITIONS

We study the longtime behavior of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions for both the order parameter and the temperature. Due to the possible lack of distributional solutions, we deal with a suitable definition of solutions based on variational inequalities, for which we prove well-posedness and the existence of global and exponential attractors with finite fractal dimension.

Fractal Dimension of a Random Invariant Set and Applications

We prove an abstract result on random invariant sets of finite fractal dimension. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the random attractors of fractal dimension.