American Institute of Mathematical Sciences

In this paper, we prove the local well-posedness of strong solutions to the density-dependent incompressible magneto-micropolar system with vacuum. There is no assuming compatibility condition on the initial data.

Recently, a transformation of the vertices of a regular triangulation of \begin{document}$ {\mathbb {R}}^n $\end{document} with vertices in the lattice \begin{document}$ \mathbb{Z}^n $\end{document} was introduced, which distributes the vertices with approximate rotational symmetry properties around the origin. We prove that the simplices of the transformed triangulation are \begin{document}$ (h, d) $\end{document}-bounded, a type of non-degeneracy particularly useful in the numerical computation of Lyapunov functions for nonlinear systems using the CPA (continuous piecewise affine) method. Additionally, we discuss and give examples of how this transformed triangulation can be used together with a Lyapunov function for a linearization to compute a Lyapunov function for a nonlinear system with the CPA method using considerably fewer simplices than when using a regular triangulation.

In this work we study the synchronization of ring-structured cellular neural networks modeled by the lattice FitzHugh-Nagumo equations with boundary feedback. Through the uniform estimates of solutions and the analysis of dissipative dynamics, the synchronization of this type neural networks is proved under the condition that the boundary gap signal exceeds the adjustable threshold.

In this paper, we study the qualitative behavior of hyperbolic system arising from chemotaxis models. Firstly, by establishing a new product estimates in multi-dimensional Besov space \begin{document}$ \dot{B}_{2, r}^{\frac d2}(\mathbb{R}^d)(1\leq r\leq \infty) $\end{document}, we establish the global small solutions in multi-dimensional Besov space \begin{document}$ \dot{B}_{2, r}^{\frac d2-1}(\mathbb{R}^d) $\end{document} by the method of energy estimates. Then we study the asymptotic behavior and obtain the optimal decay rate of the global solutions if the initial data are small in \begin{document}$ B_{2, 1}^{\frac{d}{2}-1}(\mathbb{R}^d)\cap \dot{B}_{1, \infty}^0(\mathbb{R}^d) $\end{document}.

The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic boundary condition. As a control parameter crosses a critical value, the trivial stable equilibrium solution will lose its stability, and undergoes a dynamic transition to a new physical state, described by a local attractor. The main result of this paper is to fully characterize the type and detailed structure of the transition using dynamic transition theory [7]. In particular, employing techniques from center manifold theory, we reduce this infinite dimensional problem to a finite one since the space on which the exchange of stability occurs is finite dimensional. The problem then reduces to analysis of single or double Hopf bifurcations, and we completely classify the possible phase changes depending on the dispersion for every spacial period.

In this work, we study some reaction-diffusion equations set in two habitats which model the spatial dispersal of the triatomines, vectors of Chagas disease. We prove in particular that the dispersal operator generates an analytic semigroup in an adequate space and we prove the local existence of the solution for the corresponding Cauchy problem.

In this paper, we make a mathematical analysis of an age-structured model with diffusion including a generalized Beverton-Holt fertility function. The existence of periodic wave train solutions of the age structure model with diffusion are investigated by using the theory of integrated semigroup and a Hopf bifurcation theorem for second order semi-linear equations. We also carry out numerical simulations to illustrate these results.

In this paper, a stochastic SIRS epidemic model with nonlinear incidence and vaccination is formulated to investigate the transmission dynamics of infectious diseases. The model not only incorporates the white noise but also the external environmental noise which is described by semi-Markov process. We first derive the explicit expression for the basic reproduction number of the model. Then the global dynamics of the system is studied in terms of the basic reproduction number and the intensity of white noise, and sufficient conditions for the extinction and persistence of the disease are both provided. Furthermore, we explore the sensitivity analysis of \begin{document}$ R_0^s $\end{document} with each semi-Markov switching under different distribution functions. The results show that the dynamics of the entire system is not related to its switching law, but has a positive correlation to its mean sojourn time in each subsystem. The basic reproduction number we obtained in the paper can be applied to all piecewise-stochastic semi-Markov processes, and the results of the sensitivity analysis can be regarded as a prior work for optimal control.

In this paper, we shall study the initial-boundary value problem of a chemotaxis model with signal-dependent diffusion and sensitivity as follows

\begin{document}$ \begin{cases} u_t = \nabla\cdot(\gamma(v)\nabla u-\chi(v)u\nabla v)+\alpha u F(w) +\theta u-\beta u^2, &x\in \Omega, \; \; t>0,\\ v_t = D\Delta v+u-v,& x\in \Omega, \; \; t>0,\\ w_t = \Delta w-uF(w),& x\in \Omega, \; \; t>0,\\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0,&x\in \partial\Omega, \; \; t>0,\\ u(x,0) = u_0(x), v(x,0) = v_0(x),w(x,0) = w_0(x), & x\in\Omega, \end{cases} \;\;(*)$ \end{document}

in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^2 $\end{document} with smooth boundary, where \begin{document}$ \alpha,\beta, D $\end{document} are positive constants, \begin{document}$ \theta\in \mathbb{R} $\end{document} and \begin{document}$ \nu $\end{document} denotes the outward normal vector of \begin{document}$ \partial \Omega $\end{document}. The functions \begin{document}$ \chi(v),\gamma(v) $\end{document} and \begin{document}$ F(v) $\end{document} satisfy

● \begin{document}$ (\gamma(v),\chi(v))\in [C^2[0,\infty)]^2 $\end{document} with \begin{document}$ \gamma(v)>0,\gamma'(v)<0 $\end{document} and \begin{document}$ \frac{|\chi(v)|+|\gamma'(v)|}{\gamma(v)} $\end{document} is bounded;

● \begin{document}$ F(w)\in C^1([0,\infty)), F(0) = 0,F(w)>0 \ \mathrm{in}\; (0,\infty)\; \mathrm{and}\; F'(w)>0 \ \mathrm{on}\ \ [0,\infty). $\end{document}

We first prove that the existence of globally bounded solution of system (*) based on the method of weighted energy estimates. Moreover, by constructing Lyapunov functional, we show that the solution \begin{document}$ (u,v,w) $\end{document} will converge to \begin{document}$ (0,0,w_*) $\end{document} in \begin{document}$ L^\infty $\end{document} with some \begin{document}$ w_*\geq0 $\end{document} as time tends to infinity in the case of \begin{document}$ \theta\leq 0 $\end{document}, while if \begin{document}$ \theta>0 $\end{document}, the solution \begin{document}$ (u,v,w) $\end{document} will asymptotically converge to \begin{document}$ (\frac{\theta}{\beta},\frac{\theta}{\beta},0) $\end{document} in \begin{document}$ L^\infty $\end{document}-norm provided \begin{document}$ D>\max\limits_{0\leq v\leq \infty}\frac{\theta|\chi(v)|^2}{16\beta^2\gamma(v)} $\end{document}.

Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. We next study the equilibria of the deterministic model. Our main contribution is a stability result of the endemic equilibrium in the case \begin{document}$ \mathcal{R}_0>1 $\end{document}. Finally we compare the equilibria with those of the homogeneous model, and with those of isolated patches.

We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.

The paper is devoted to establishing the long-time behavior of solutions for the wave equation with nonlocal strong damping: \begin{document}$ u_{tt}-\Delta u-\|\nabla u_{t}\|^{p}\Delta u_{t}+f(u) = h(x). $\end{document} It proves the well-posedness by means of the monotone operator theory and the existence of a global attractor when the growth exponent of the nonlinearity \begin{document}$ f(u) $\end{document} is up to the subcritical and critical cases in natural energy space.

In this paper, the nonlinear dynamics of a SIRS epidemic model with vertical transmission rate of neonates, nonlinear incidence rate and nonlinear recovery rate are investigated. We focus on the influence of public available resources (especially the number of hospital beds) on disease control and transmission. The existence and stability of equilibria are analyzed with the basic reproduction number as the threshold value. The conditions for the existence of transcritical bifurcation, Hopf bifurcation, saddle-node bifurcation, backward bifurcation and the normal form of Bogdanov-Takens bifurcation are obtained. In particular, the coexistence of limit cycle and homoclinic cycle, and the coexistence of stable limit cycle and unstable limit cycle are also obtained. This study indicates that maintaining enough number of hospital beds is very crucial to the control of the infectious diseases no matter whether the immunity loss population are involved or not. Finally, numerical simulations are also given to illustrate the theoretical results.

We are interested in the existence and stability of traveling waves of arbitrary amplitudes to a chemotaxis model with porous medium diffusion. We first make a complete classification of traveling waves under specific relations among the biological parameters. Then we show all these traveling waves are asymptotically stable under appropriate perturbations. The proof is based on a Cole-Hopf transformation and the energy method.

The paper investigates the existence and the continuity of uniform attractors for the non-autonomous Kirchhoff wave equations with strong damping: \begin{document}$ u_{tt}-(1+\epsilon\|\nabla u\|^{2})\Delta u-\Delta u_{t}+f(u) = g(x,t) $\end{document}, where \begin{document}$ \epsilon\in [0,1] $\end{document} is an extensibility parameter. It shows that when the nonlinearity \begin{document}$ f(u) $\end{document} is of optimal supercritical growth \begin{document}$ p: \frac{N+2}{N-2} = p^*: (ⅰ) the related evolution process has in natural energy space \begin{document}$ \mathcal{H} = (H^1_0\cap L^{p+1})\times L^2 $\end{document} a compact uniform attractor \begin{document}$ \mathcal{A}^{\epsilon}_{\Sigma} $\end{document} for each \begin{document}$ \epsilon\in [0,1] $\end{document}; (ⅱ) the family of compact uniform attractor \begin{document}$ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $\end{document} is continuous on \begin{document}$ \epsilon $\end{document} in a residual set \begin{document}$ I^*\subset [0,1] $\end{document} in the sense of \begin{document}$ \mathcal{H}_{ps} ( = (H^1_0\cap L^{p+1,w})\times L^2) $\end{document}-topology; (ⅲ) \begin{document}$ \{\mathcal{A}^{\epsilon}_{\Sigma}\}_{\epsilon\in [0,1]} $\end{document} is upper semicontinuous on \begin{document}$ \epsilon\in [0,1] $\end{document} in \begin{document}$ \mathcal{H}_{ps} $\end{document}-topology.

This paper derives a weak convergence theorem of the time discretization of the slow component for a two-time-scale stochastic evolutionary equations on interval [0, 1]. Here the drift coefficient of the slow component is cubic with linear coupling between slow and fast components.

In this paper, we study a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces. We establish the smooth convergence of unstable manifolds of these equations. As an example, we study the stochastic reaction-diffusion equations on thin domains.

We consider the Cauchy problem of nonhomogeneous magneto-micropolar fluid equations with zero density at infinity in the entire space \begin{document}$ \mathbb{R}^2 $\end{document}. We show that for the initial density allowing vacuum, the strong solution exists globally if a weighted density is bounded from above. It should be noted that our blow-up criterion is independent of micro-rotational velocity and magnetic field.

In this paper, we consider the stabilization problem of 1-D Schrödinger equation with internal difference-type control. Different from the other existing approaches of controller design, we introduce a new approach of controller design so called the parameterization controller. At first, we rewrite the system with internal difference-type control as a cascaded system of a transport equation and Schödinger equation; Further, to stabilize the system under consideration, we construct a target system that has exponential stability. By selecting the solution of nonlocal and singular initial value problem as parameter function and defining a bounded linear transformation, we show that the transformation maps the closed-loop system to the target system; Finally, we prove that the transformation is bounded inverse. Hence the closed-loop system is equivalent to the target system.

We study the dynamics of a two-layer baroclinic quasi-geostrophic model. We prove that the semigroup \begin{document}$ \{S(t)\}_{t\geq 0} $\end{document} associated with the solutions of the model has a global attractor in both \begin{document}$ {{\dot H}_{p}}^1(\Omega) $\end{document} and \begin{document}$ {{\dot H}_{p}}^2(\Omega) $\end{document}. Also we show that for any viscosity \begin{document}$ \mu>0 $\end{document}, there is an open and dense set of forcing \begin{document}$ \mathcal G\subset{{\dot H}_{p}}^0(\Omega) $\end{document} such that for each \begin{document}$ G = (g_1, g_2)\in \mathcal G $\end{document}, the set \begin{document}$ S(G, \mu) \subset {{\dot H}_{p}}^4(\Omega) $\end{document} of the steady state problem is non–empty and finite.

The objective of this paper is to study the fractal dimension of global attractor for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions. Inspired by the idea of the \begin{document}$ \ell $\end{document}-trajectory method, we prove the existence of a finite dimensional global attractor in an auxiliary normed space for the Cahn-Hilliard-Brinkman system with dynamic boundary conditions and estimate the fractal dimension of the global attractor in the original phase space for this system by defining a Lipschitz mapping from the auxiliary normed space into the original phase space.

The paper considers a \begin{document}$ n $\end{document}-patch model with migration terms, where each patch follows a logistic law. First, we give some properties of the total equilibrium population. In some particular cases, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of the \begin{document}$ n $\end{document} carrying capacities. Second, in the case of perfect mixing, i.e when the migration rate tends to infinity, the total population follows a logistic law with a carrying capacity which in general is different from the sum of the \begin{document}$ n $\end{document} carrying capacities. Finally, for the three-patch model we show numerically that the increase in number of patches from two to three gives a new behavior for the dynamics of the total equilibrium population as a function of the migration rate.

In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions, which is interesting in its own right. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations.

In this paper, the stochastic vector-host model has been proposed and analysed using nice properties of piecewise deterministic Markov processes (PDMPs). A threshold for the stochastic model is derived whose sign determines whether the disease will eventually disappear or persist. We show mathematically the existence of scenarios where switching plays a significant role in surprisingly reversing the long-term properties of deterministic systems.

In this paper, we consider the time-fractional Volterra integro-differential equations with Caputo derivative. For globally Lispchitz source term, we investigate the global existence for a mild solution. The main tool is to apply the Banach fixed point theorem on some new weighted spaces combining some techniques on the Wright functions. For the locally Lipschitz case, we study the existence of local mild solutions to the problem and provide a blow-up alternative for mild solutions. We also establish the problem of continuous dependence with respect to initial data. Finally, we present some examples to illustrate the theoretical results.