## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Foundations of Data Science
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
- Electronic Research Announcements
- Mathematics in Engineering

### Open Access Journals

In this paper, we study the dynamics of a non-autonomous reaction diffusion model with the fractional diffusion on the whole space. We firstly prove the existence of a $(L^2,L^2)$ pullback $\mathscr{D}_μ$ -attractor of this model. Then we show that the pullback $\mathscr{D}_μ$ -attractor attract the $\mathscr{D}_μ$ class (especially all $L^2$ -bounded set) in $L^{2+δ}$-norm for any $δ∈[0,∞)$. Moreover, the solution of the model is shown to be continuous in $H^s$ with respect to initial data under a slightly stronger condition on external forcing term. As an application, we prove that the $(L^2,L^2)$ pullback $\mathscr{D}_{μ}$-attractor indeed attract the class of $\mathscr{D}_{μ}$ in $H^s$ -norm, and thus the existence of a $(L^2, H^s)$ pullback $\mathscr{D}_μ$ -attractor is obtained.

In this paper, the dynamical behavior of weakly damped wave equations with a sup-cubic nonlinearity is considered in locally uniform spaces. We first prove the global well-posedness of the Shatah-Struwe solutions, then we obtain the existence of the $ \big(H_{lu}^{1}(\mathbb{R}^{3})\times L_{lu}^{2}(\mathbb{R}^{3}),H_{\rho}^{1}(\mathbb{R}^{3})\times L_{\rho}^{2}(\mathbb{R}^{3})\big) $-global attractor for the Shatah-Struwe solutions semigroup of this equation. The results are crucially based on the recent extension of Strichartz estimates to the case of bounded domains.

In this work, we extend the classical real-valued framework to deal with complex-valued dissipative dynamical systems. With our new complex-valued framework and using generalized complex Banach limits, we construct invariant measures for continuous complex semigroups possessing global attractors. In particular, for any given complex Banach limit and initial data $u_{0}$, we construct a unique complex invariant measure $\mu$ on a metric space which is acted by a continuous semigroup $\{S(t)\}_{t\geq 0}$ possessing a global attractor $\mathcal{A}$. Moreover, it is shown that the support of $\mu$ is not only contained in global attractor $\mathcal{A}$ but also in $\omega(u_{0})$. Next, the structure of the measure $\mu$ is studied. It is shown that both the real and imaginary parts of a complex invariant measure are invariant signed measures and that both the positive and negative variations of a signed measure are invariant measures. Finally, we illustrate the main results of this article on the model examples of a complex Ginzburg-Landau equation and a nonlinear Schrödinger equation and construct complex invariant measures for these two complex-valued equations.

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

## Year of publication

## Related Authors

## Related Keywords

[Back to Top]