American Institute of Mathematical Sciences

In this paper, we consider a Lewy-Stampacchia-type inequality for the fractional Laplacian on a bounded domain in Euclidean space. Using this inequality, we can show the well-posedness of fractional-type anomalous unidirectional diffusion equations. This study is an extension of the work by Akagi-Kimura (2019) for the standard Laplacian. However, there exist several difficulties due to the nonlocal feature of the fractional Laplacian. We overcome those difficulties employing the Caffarelli-Silvestre extension of the fractional Laplacian.

This paper is concerned with system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. Under appropriate assumptions on the time-varying delay term and time-dependent weights, we obtain exponential stability estimates by using the multiplicative technique, and prove the equivalence between stabilization and observability.

Worldwide, human population is increasing continuously and this has magnified the level of pollutants in the environment. Pollutants affect the human population as well as the environmental ecology including rainfall. Here, we formulate a mathematical model comprising ordinary differential equations to see the effect of human population and pollution caused by human population on the dynamics of rainfall. In the modeling process, it is assumed that the augmentation in the density of human population increases the concentration of pollutants; however, decreases the rate of formation of cloud droplets. It is also assumed that pollutants have negative impact on human population and affect the precipitation. The feasibility of all equilibrium and their stability properties are discussed. Further, to capture the effect of environmental randomness, the proposed model is also analyzed by incorporating white noise terms. For the proposed stochastic model, we have established the existence and uniqueness of global positive solution. It is also shown that system possesses a unique stationary distribution with some restrictions. The model analysis reveals that rainfall may decrease or increase due to the anthropogenic emission of pollutants in the atmospheric environment. Finally, for the validation of analytical findings, numerical simulations are presented.

In this paper, we study the influence of spatial-dependent variables on the basic reproduction ratio (\begin{document}$ \mathcal{R}_0 $\end{document}) for a scalar reaction-diffusion equation model. We first investigate the principal eigenvalue of a weighted eigenvalue problem and show the influence of spatial variables. We then apply these results to study the effect of spatial heterogeneity and dimension on the basic reproduction ratio for a spatial model of rabies. Numerical simulations also reveal the complicated effects of the spatial variables on \begin{document}$ \mathcal{R}_0 $\end{document} in two dimensions.

We consider the \begin{document}$ (\omega,Q) $\end{document}-periodic problem for a system of delay differential equations, where \begin{document}$ Q $\end{document} is an invertible matrix. Existence and multiplicity of solutions is proven under different conditions that extend well-known results for the periodic case \begin{document}$ Q = I $\end{document} and anti-periodic case \begin{document}$ Q = -I $\end{document}. In particular, the results apply to biological models with mixed terms of Nicholson, Lasota or Mackey type, and also vectorial versions of Nicholson or Mackey-Glass models.

A two-dimensional stage-structured model for the interactive wild and sterile mosquitoes is derived where the wild mosquito population is composed of larvae and adult classes and only sexually active sterile mosquitoes are included as a function given in advance. The strategy of constant releases of sterile mosquitoes is considered but periodic and impulsive releases are more focused on. Local stability of the origin and the existence of a positive periodic solution are investigated. While mathematical analysis is more challenging, numerical examples demonstrate that the model dynamics, determined by thresholds of the release amount and the release waiting period, essentially match the dynamics of the alike one-dimensional models. It is also shown that richer dynamics are exhibited from the two-dimensional stage-structured model.

This paper is devoted to an SEIR epidemic model with variable recruitment and both exposed and infected populations having infectious in a spatially heterogeneous environment. The basic reproduction number is defined and the existence of endemic equilibrium is obtained, and the relationship between the basic reproduction number and diffusion coefficients is established. Then the global stability of the endemic equilibrium in a homogeneous environment is investigated. Finally, the asymptotic profiles of endemic equilibrium are discussed, when the diffusion rates of susceptible, exposed and infected individuals tend to zero or infinity. The theoretical results show that limiting the movement of exposed, infected and recovered individuals can eliminate the disease in low-risk sites, while the disease is still persistent in high-risk sites. Therefore, the presence of exposed individuals with infectious greatly increases the difficulty of disease prevention and control.

In this paper, we investigate the effect of spontaneous infection and advection for a susceptible-infected-susceptible epidemic reaction-diffusion-advection model in a heterogeneous environment. The existence of the endemic equilibrium is proved, and the asymptotic behaviors of the endemic equilibrium in three cases (large advection; small diffusion of the susceptible population; small diffusion of the infected population) are established. Our results suggest that the advection can cause the concentration of the susceptible and infected populations at the downstream, and the spontaneous infection can enhance the persistence of infectious disease in the entire habitat.

This paper investigates the existence of strong global and exponential attractors and their robustness on the perturbed parameter for an extensible beam equation with nonlocal energy damping in \begin{document}$ \Omega\subset{\mathbb R}^N $\end{document}: \begin{document}$ u_{tt}+\Delta^2 u-\kappa\phi(\|\nabla u\|^2)\Delta u-M(\|\Delta u\|^2+\|u_t\|^2)\Delta u_t+f(u) = h $\end{document}, where \begin{document}$ \kappa \in \Lambda $\end{document} (index set) is an extensibility parameter, and where the "strong" means that the compactness, the attractiveness and the finiteness of the fractal dimension of the attractors are all in the topology of the stronger space \begin{document}$ {\mathcal H}_2 $\end{document} where the attractors lie in. Under the assumptions that either the nonlinearity \begin{document}$ f(u) $\end{document} is of optimal subcritical growth or even \begin{document}$ f(u) $\end{document} is a true source term, we show that (ⅰ) the semi-flow originating from any point in the natural energy space \begin{document}$ {\mathcal H} $\end{document} lies in the stronger strong solution space \begin{document}$ {\mathcal H}_2 $\end{document} when \begin{document}$ t>0 $\end{document}; (ⅱ) the related solution semigroup \begin{document}$ S^\kappa(t) $\end{document} has a strong \begin{document}$ ({\mathcal H},{\mathcal H}_2) $\end{document}-global attractor \begin{document}$ {\mathscr A}^\kappa $\end{document} for each \begin{document}$ \kappa $\end{document} and the family of \begin{document}$ {\mathscr A}^\kappa, \kappa\in \Lambda $\end{document} is upper semicontinuous on \begin{document}$ \kappa $\end{document} in the topology of stronger space \begin{document}$ {\mathcal H}_2 $\end{document}; (ⅲ) \begin{document}$ S^\kappa(t) $\end{document} has a strong \begin{document}$ ({\mathcal H},{\mathcal H}_2) $\end{document}-exponential attractor \begin{document}$ \mathfrak {A}^\kappa_{exp} $\end{document} for each \begin{document}$ \kappa $\end{document} and it is Hölder continuous on \begin{document}$ \kappa $\end{document} in the topology of \begin{document}$ {\mathcal H}_2 $\end{document}. These results break through long-standing existed restriction for the attractors of the extensible beam models in energy space and show the optimal topology properties of them in the stronger phase space.

In this paper, we present the procedure of generalization and implementation of the Cauchy-Born approximation to the calculation of stress at finite temperature for alloy system in which the effects of inner displacement should be incorporated. With the help of quasi-harmonic approximation, a closed form of the first Piola-Kirchhoff stress is derived as a summation of pure deformation contribution and linear term due to thermal effects. For alloy system with periodic boundary condition, a further simplified formulation of stress based on some invariance constraints is derived in reciprocal space by using Fourier transformation, in which the temperature effect can be efficiently taking account. Several numerical examples are performed for various crystalline systems to validate our generalization procedure of finite temperature Cauchy-Born (FTCB) method for alloy.

In this paper, we consider a stochastic predator-prey model with general functional response, which is perturbed by nonlinear Lévy jumps. Firstly, We show that this model has a unique global positive solution with uniform boundedness of \begin{document}$ \theta\in(0,1] $\end{document}-th moment. Secondly, we obtain the threshold for extinction and exponential ergodicity of the one-dimensional Logistic system with nonlinear perturbations. Then based on the results of Logistic system, we introduce a new technique to study the ergodic stationary distribution for the stochastic predator-prey model with general functional response and nonlinear jump-diffusion, and derive the sufficient and almost necessary condition for extinction and ergodicity.

This paper focuses on the spread dynamics of an HIV/AIDS model with multiple stages of infection and treatment, which is disturbed by both white noise and telegraph noise. Switching between different environmental states is governed by Markov chain. Firstly, we prove the existence and uniqueness of the global positive solution. Then we investigate the existence of a unique ergodic stationary distribution by constructing suitable Lyapunov functions with regime switching. Furthermore, sufficient conditions for extinction of the disease are derived. The conditions presented for the existence of stationary distribution improve and generalize the previous results. Finally, numerical examples are given to illustrate our theoretical results.

This paper concerns the null controllability for a class of stochastic singular parabolic equations with the convection term in one dimensional space. Due to the singularity, we first transfer to study an approximate nonsingular system. Next we establish a new Carleman estimate for the backward stochastic singular parabolic equation with convection term and then an observability inequality for the adjoint system of the approximate system. Based on this observability inequality and an approximate argument, we obtain the null controllability result.

This paper is concerned with the propagation dynamics in a diffusive susceptible-infective nonisolated-isolated-removed model that describes the recurrent outbreaks of childhood diseases. To model the spatial-temporal modes on disease spreading, we study the traveling wave solutions and the initial value problem with special decay condition. When the basic reproduction ratio of the corresponding kinetic system is larger than one, we define a threshold that is the minimal wave speed of traveling wave solutions as well as the spreading speed of some components. From the viewpoint of mathematical epidemiology, the threshold is monotone decreasing in the rate at which individuals leave the infective and enter the isolated classes.

We investigate a mathematical model of tumor–immune system interactions with oncolytic virus therapy (OVT). Susceptible tumor cells may become infected by viruses that are engineered specifically to kill cancer cells but not healthy cells. Once the infected cancer cells are destroyed by oncolysis, they release new infectious virus particles to help kill surrounding tumor cells. The immune system constructed includes innate and adaptive immunities while the adaptive immunity is further separated into anti-viral or anti-tumor immune cells. The model is first analyzed by studying boundary equilibria and their stability. Numerical bifurcation analysis is performed to investigate the outcomes of the oncolytic virus therapy. The model has a unique tumor remission equilibrium, which is unlikely to be stable based on the parameter values given in the literature. Multiple stable positive equilibria with tumor sizes close to the carrying capacity coexist in the system if the tumor is less antigenic. However, as the viral infection rate increases, the OVT becomes more effective in the sense that the tumor can be dormant for a longer period of time even when the tumor is weakly antigenic.

In this article we consider a singularly perturbed Allen-Cahn problem \begin{document}$ u_t = \epsilon^2(a^2u_x)_x+b^2(u-u^3) $\end{document}, for \begin{document}$ (x,t)\in (0,1)\times\mathbb{R}^+ $\end{document}, supplied with no-flux boundary condition. The novelty here lies in the fact that the nonnegative spatial inhomogeneities \begin{document}$ a(\cdot) $\end{document} and \begin{document}$ b(\cdot) $\end{document} are allowed to vanish at some points in \begin{document}$ (0,1) $\end{document}. Using the variational concept of \begin{document}$ \Gamma $\end{document}-convergence we prove that, for \begin{document}$ \epsilon $\end{document} small, such degeneracy of \begin{document}$ a(\cdot) $\end{document} and \begin{document}$ b(\cdot) $\end{document} induces the existence of stable stationary solutions which develop internal transition layer as \begin{document}$ \epsilon\to 0 $\end{document}.

In this paper we consider the reaction diffusion equation \begin{document}$ u_t = u_{xx} + f(u) $\end{document} with bistable-bistable type of nonlinearities, that is, \begin{document}$ f $\end{document} has five nonnegative zeros: \begin{document}$ 0<\alpha_1 <\alpha_2<\alpha_3 <\alpha_4 $\end{document}, and it is of bistable type on \begin{document}$ [0,\alpha_2] $\end{document} and \begin{document}$ [\alpha_2, \alpha_4] $\end{document}. We study the asymptotic behavior for the solutions under different conditions for \begin{document}$ k_4 : = \int_0^{\alpha_4} f(s) ds $\end{document} and \begin{document}$ k_2: = \int_0^{\alpha_2} f(s) ds $\end{document}. In case \begin{document}$ k_4 > k_2 > 0 $\end{document} (resp. \begin{document}$ k_4 > k_2 = 0 $\end{document}, \begin{document}$ k_2 < 0 < k_4 $\end{document}, \begin{document}$ k_2 < k_4 = 0 $\end{document}), we find 5 (resp. 3, 3, 1) possible choices for the \begin{document}$ \omega $\end{document}-limit of the solution.

We study an optimal control problem affine in two-dimensional bounded control, in which there is a singular point of the second order. In the neighborhood of the singular point we find optimal spiral-like solutions that attain the singular point in finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle \begin{document}$ S^{1} $\end{document}. The problem is related to the control of an inverted spherical pendulum in the neighborhood of the upper unstable equilibrium.

This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales. Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. Then, we consider nonlinear time-varying dynamical systems on time scales and establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning synchronization of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.

In this paper, we aim to investigate the dynamic transition of the Klausmeier-Gray-Scott (KGS) model in a rectangular domain or a square domain. Our research tool is the dynamic transition theory for the dissipative system. Firstly, we verify the principle of exchange of stability (PES) by analyzing the spectrum of the linear part of the model. Secondly, by utilizing the method of center manifold reduction, we show that the model undergoes a continuous transition or a jump transition. For the model in a rectangular domain, we discuss the transitions of the model from a real simple eigenvalue and a pair of simple complex eigenvalues. our results imply that the model bifurcates to exactly two new steady state solutions or a periodic solution, whose stability is determined by a non-dimensional coefficient. For the model in a square domain, we only focus on the transition from a real eigenvalue with algebraic multiplicity 2. The result shows that the model may bifurcate to an \begin{document}$ S^{1} $\end{document} attractor with 8 non-degenerate singular points. In addition, a saddle-node bifurcation is also possible. At the end of the article, some numerical results are performed to illustrate our conclusions.

In a recent article [16], the authors gave a starting point of the study on a series of problems concerning the initial boundary value problem and control theory of Biharmonic NLS in some non-standard domains. In this direction, this article deals to present answers for some questions left in [16] concerning the study of the cubic fourth order Schrödinger equation in a star graph structure \begin{document}$ \mathcal{G} $\end{document}. Precisely, consider \begin{document}$ \mathcal{G} $\end{document} composed by \begin{document}$ N $\end{document} edges parameterized by half-lines \begin{document}$ (0,+\infty) $\end{document} attached with a common vertex \begin{document}$ \nu $\end{document}. With this structure the manuscript proposes to study the well-posedness of a dispersive model on star graphs with three appropriated vertex conditions by using the boundary forcing operator approach. More precisely, we give positive answer for the Cauchy problem in low regularity Sobolev spaces. We have noted that this approach seems very efficient, since this allows to use the tools of Harmonic Analysis, for instance, the Fourier restriction method, introduced by Bourgain, while for the other known standard methods to solve partial differential partial equations on star graphs are more complicated to capture the dispersive smoothing effect in low regularity. The arguments presented in this work have prospects to be applied for other nonlinear dispersive equations in the context of star graphs with unbounded edges.

The paper deals with the transformation of a weakly nonlinear system of differential equations in a special form into a simplified form and its relation to the normal form and averaging. An original method of simplification is proposed, that is, a way to determine the coefficients of a given nonlinear system in order to simplify it. We call this established method the degree equalization method, it does not require integration and is simpler and more efficient than the classical Krylov-Bogolyubov method of normalization. The method is illustrated with several examples and provides an application to the analysis of cardiac activity modelled using van der Pol equation.

The societal impact of traffic is a long-standing and complex problem. We focus on the estimation of ground-level ozone production due to vehicular traffic. We propose a comprehensive computational approach combining four consecutive modules: a traffic simulation module, an emission module, a module for the main chemical reactions leading to ozone production, and a module for the diffusion of gases in the atmosphere. The traffic module is based on a second-order traffic flow model, obtained by choosing a special velocity function for the Collapsed Generalized Aw-Rascle-Zhang model. A general emission module is taken from literature, and tuned on NGSIM data together with the traffic module. Last two modules are based on reaction-diffusion partial differential equations. The system of partial differential equations describing the main chemical reactions of nitrogen oxides presents a source term given by the general emission module applied to the output of the traffic module. We use the proposed approach to analyze the ozone impact of various traffic scenarios and describe the effect of traffic light timing. The numerical tests show the negative effect of vehicles restarts on emissions, and the consequent increase in pollutants in the air, suggesting to increase the length of the green phase of traffic lights.

This paper deals with the dynamical properties of the quasilinear parabolic-parabolic chemotaxis system

\begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(\frac{u}{v} \nabla v)+\mu u- \mu u^{2}, \, \, \, &x\in\Omega, \, \, \, t>0, \\ v_{t} = \Delta v-v+u, &x\in\Omega, \, \, \, t>0, \end{array} \right. \end{eqnarray*} $\end{document}

under homogeneous Neumann boundary conditions in a convex bounded domain \begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document}, \begin{document}$ n\geq2 $\end{document}, with smooth boundary. \begin{document}$ \chi>0 $\end{document} and \begin{document}$ \mu>0 $\end{document}, \begin{document}$ D(u) $\end{document} is supposed to satisfy the behind properties

\begin{document}$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $\end{document}

It is shown that there is a positive constant \begin{document}$ m_{*} $\end{document} such that

\begin{document}$ \begin{equation*} \begin{split} \int_{\Omega}u\geq m_{*} \end{split} \end{equation*} $\end{document}

for all \begin{document}$ t\geq0 $\end{document}. Moreover, we prove that the solution is globally bounded. Finally, it is asserted that the solution exponentially converges to the constant stationary solution \begin{document}$ (1, 1) $\end{document}.

Competition stems from the fact that resources are limited. When multiple competitive species are involved with spatial diffusion, the dynamics becomes even complex and challenging. In this paper, we investigate the invasive speed to a diffusive three species competition system of Lotka-Volterra type. We first show that multiple species share a common spreading speed when initial data are compactly supported. By transforming the competitive system into a cooperative system, the determinacy of the invasive speed is studied by the upper-lower solution method. In our work, for linearly predicting the invasive speed, we concentrate on finding upper solutions only, and don't care about the existence of lower solutions. Similarly, for nonlinear selection of the spreading speed, we focus only on the construction of lower solutions with fast decay rate. This greatly develops and simplifies the ideas of past references in this topic.