On a reaction-diffusion model for sterile insect release method with release on the boundary
Xin Li Xingfu Zou
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2509-2522 doi: 10.3934/dcdsb.2012.17.2509
We consider a partial differential equation model that describes the sterile insect release method (SIRM) in a bounded 1-dimensional domain (interval). Unlike everywhere-releasing in the domain as considered in previous works [17] and [14] , we propose the mechanism of releasing on the boundary only. We show existence of the fertile-free steady state and prove its stability under some conditions. By using the upper-lower solution method, we also show that under some other conditions there may exist a coexistence steady state. Biological implications of our mathematical results are that the SIRM with releasing only on the boundary can successfully eradicate the fertile insects as long as the strength of the sterile releasing is reasonably large, while the method may also fail if the releasing is not sufficient.
keywords: diffusion steady state coexistence. upper- lower solution Sterile insect release method stability
Invariant measures for complex-valued dissipative dynamical systems and applications
Xin Li Wenxian Shen Chunyou Sun
Discrete & Continuous Dynamical Systems - B 2017, 22(6): 2427-2446 doi: 10.3934/dcdsb.2017124

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.

keywords: Complex-valued dynamical systems complex invariant measure global attractor Ginzburg-Landau equation nonlinear Schrödinger equation
Dynamics for the damped wave equations on time-dependent domains
Feng Zhou Chunyou Sun Xin Li
Discrete & Continuous Dynamical Systems - B 2018, 23(4): 1645-1674 doi: 10.3934/dcdsb.2018068

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.

keywords: Non-autonomous dynamical systems wave equation time-dependent domain critical exponent pullback attractor

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