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### Open Access Journals

DCDS-B

In this paper, we prove the existence of a $(L_{lu}^2(\mathbb{R}^N)\times
L_{lu}^2(\mathbb{R}^N),L_{\rho}^2(\mathbb{R}^N)\times L_{\rho}^2(\mathbb{R}^N))$-global
attractor for the solution semigroup generated by the Gray-Scott
equations on unbounded domains of space dimension $N\leq3.$

DCDS

In this paper, we consider the upper semicontinuity of pullback attractors for a nonautonomous Kirchhoff wave model with strong damping. For this purpose, some necessary abstract results are established.

DCDS

The present paper is devoted to the well-posedness issue of solutions of a full system of
the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations.
By means of Littlewood-Paley analysis
we prove the global well-posedness of solutions in the Besov spaces
$\dot{B}_{2,1}^\frac1{2}\times B_{2,1}^\frac3{2}\times B_{2,1}^\frac3{2}$ provided the norm of initial data is small
enough in the sense that
\begin{align*}
\big(\|u_0^h\|_{\dot{B}_{2,1}^\frac1{2}}
+\|E_0\|_{B_{2,1}^\frac{3}{2}}+\|B_0\|_{B_{2,1}^\frac{3}{2}}\big)\exp
\Big\{\frac{C_0}{\nu^2}\|u_0^3\|_{\dot{B}_{2,1}^\frac1{2}}^2\Big\}\leq c_0,
\end{align*}
for some sufficiently small constant $c_0.$

DCDS-B

In this paper we study the long time behavior of the three
dimensional Navier-Stokes-Voight model of viscoelastic
incompressible fluid for the autonomous and nonautonomous cases. A
useful decomposition method is introduced to overcome the
difficulties in proving the asymptotical regularity of the 3D
Navier-Stokes-Voight equations. For the autonomous case, we prove
the existence of global attractor when the external forcing belongs
to $V'.$ For the nonautonomous case, we only assume that
$f(x,t)$ is translation bounded instead of translation compact,
where $f=Pg$ and $P$ is the Helmholz-Leray orthogonal
projection. By means of this useful decomposition methods, we prove
the asymptotic regularity of solutions of 3D Navier-Stokes-Voight
equations and also obtain the existence of the uniform attractor.
Finally, we describe the structure of the uniform attractor and its
regularity.

DCDS

For weakly damped non-autonomous hyperbolic equations,
we introduce a new concept Condition (C*), denote the set of all
functions satisfying Condition (C*) by L

^{2}_{C* }$(R;X)$ which are translation bounded but not translation compact in $L^2$_{loc}$(R;X)$, and show that there are many functions satisfying Condition (C*); then we study the uniform attractors for weakly damped non-autonomous hyperbolic equations with this new class of time dependent external forces $g(x,t)\in $ L^{2}_{C* }$(R;X)$ and prove the existence of the uniform attractors for the family of processes corresponding to the equation in $H^1_0\times L^2$ and $D(A)\times H^1_0$.
DCDS-B

In this paper, we are concerned with
some properties of the global attractor of weakly damped wave
equations. We get the existence of multiple stationary solutions for
wave equations with weakly damping. Furthermore, we provide some
approaches to verify the small neighborhood of the origin is an
attracting domain which is important to obtain
the multiple equilibrium points in global attractor.

keywords:
wave equations.
,
global
attractor
,
Lyapunov functional
,
$Z_2$ index
,
equilibrium points

DCDS

The present paper is devoted to the well-posedness issue of solutions to
the $3$-$D$ incompressible magnetohydrodynamic(MHD) equations
with horizontal dissipation and horizontal magnetic diffusion.
By means of anisotropic Littlewood-Paley analysis
we prove the global well-posedness of solutions in the anisotropic Sobolev spaces
of type $H^{0,s_0}(\mathbb{R}^3)$ with $s_0>\frac1{2}$ provided the norm of initial data is small
enough in the sense that
\begin{align*}
(\|u_n^h(0)\|_{H^{0,s_0}}^2+\|B_n^h(0)\|_{H^{0,s_0}}^2)\exp
\Big\{C_1(\|u_0^3\|_{H^{0,s_0}}^4+\|B_0^3\|_{H^{0,s_0}}^4)\Big\}\leq\varepsilon_0,
\end{align*}
for some sufficiently small constant $\varepsilon_0.$

DCDS

Globally exponential $κ-$dissipativity, a new concept of dissipativity for semigroups, is introduced. It provides a more general criterion for the exponential attraction of some evolutionary systems. Assuming that a semigroup $\{S(t)\}_{t≥q 0}$ has a bounded absorbing set, then $\{S(t)\}_{t≥q 0}$ is globally exponentially $κ-$dissipative if and only if there exists a compact set $\mathcal{A}^*$ that is positive invariant and attracts any bounded subset exponentially. The set $\mathcal{A}^*$ need not be finite dimensional. This result is illustrated with an application to a damped semilinear wave equation on a bounded domain.

DCDS

The
existence and structure of uniform attractors in $V$ is proved for
nonautonomous 2D Navier-stokes equations on bounded domain with a
new class of external forces, termed

*normal*in $L_{l o c}^2(\mathbb R; H)$ (see Definition 3.1), which are translation bounded but not translation compact in $L_{l o c}^2(\mathbb R; H)$. To this end, some abstract results are established. First, a characterization on the existence of uniform attractor for a family of processes is presented by the concept of measure of noncompactness as well as a method to verify it. Then, the structure of the uniform attractor is obtained by constructing skew product flow on the extended phase space with weak topology. Finally, the uniform attractor of a process is identified with that of a family of processes with symbols in the closure of the translation family of the original symbol in a Banach space with weak topology.
DCDS-B

This paper is devoted to the existence of pullback attractors for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation. We first prove the existence of pullback absorbing sets in $H$ and $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8), and then we prove the existence of a pullback attractor in $H$ by the Sobolev compactness embedding theorem. Finally, we obtain the existence of a pullback attractor in $V$ for the process $\{U(t,\tau)\}_{t\geq \tau}$ associated with (1)-(8) by verifying the pullback $\mathcal{D}$ condition $(PDC)$.

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