Acta Mathematicae Applicatae Sinica, English Series最新文献

In this paper, we prove an existence and uniqueness theorem for backward doubly stochastic differential equations under a new kind of stochastic non-Lipschitz condition which involves stochastic and time-dependent condition. As an application, we use the result to obtain the existence of stochastic viscosity solution for some nonlinear stochastic partial differential equations under stochastic non-Lipschitz conditions.

This paper is devoted to solving a reflected backward stochastic differential equation (BSDE in short) with one continuous barrier and a quasi-linear growth generator g, which has a linear growth in (y, z), except the upper direction in case of y < 0, and is more general than the usual linear growth generator. By showing the convergence of a penalization scheme we prove existence and comparison theorem of the minimal L^p (p > 1) solutions for the reflected BSDEs. We also prove that the minimal L^p solution can be approximated by a sequence of L^p solutions of certain reflected BSDEs with Lipschitz generators.

The paper deals with a Cauchy problem for the chemotaxis system with the effect of fluid

where d ≥ 2. It is known that for each ϵ > 0 and all sufficiently small initial data (u₀, n₀, c₀) belongs to certain Fourier space, the problem possesses a unique global solution (u^ϵ, n^ϵ, c^ϵ) in Fourier space. The present work asserts that these solutions stabilize to (u^∞, n^∞, c^∞) as ϵ⁻¹ → 0. Moreover, we show that c^ϵ(t) has the initial layer as ϵ⁻¹ → 0. As one expects its limit behavior maybe give a new viewlook to understand the system.

In this paper, a stochastic Lotka-Volterra system with infinite delay is considered. A new concept of extinction, namely, the almost sure β-extinction is proposed and sufficient conditions for the solution to be almost sure β-extinction are obtained. When the positive equilibrium exists and the intensities of the noises are small enough, any solution of the system is attracted by the positive equilibrium. Finally, numerical simulations are carried out to support the results.

The purpose of this work is to investigate the existence and uniqueness of weak solutions to the initial-boundary value problem for a coupled system of an incompressible non-Newtonian fluid and the Vlasov equation. The coupling arises from the acceleration in the Vlasov equation and the drag force in the incompressible viscous non-Newtonian fluid with the stress tensor of a power-law structure for (pgeqslant {11over 5}). The main idea of the existence analysis is to reformulate the coupled system by means of a so-called truncation function. The advantage of the new formulation is to control the external force term (G=-int_mathbb{{R}^{d}}(mathbf{u}-mathbf{v})fdmathbf{v} (d=2,3)). The global existence of weak solutions to the reformulated system is shown by using the Faedo-Galerkin method and weak compactness techniques. We further prove the uniqueness of weak solutions to the considered system.

In this paper, we study a class of time-periodic population model with dispersal. It is well known that the existence of the periodic traveling fronts has been established. However, the uniqueness and stability of such fronts remain unsolved. In this paper, we first prove the uniqueness of non-critical periodic traveling fronts. Then, we show that all non-critical periodic traveling fronts are exponentially asymptotically stable.

In this work, a novel time-stepping (overline{L1}) formula is developed for a hidden-memory variable-order Caputo’s fractional derivative with an initial singularity. This formula can obtain second-order accuracy and an error estimate is analyzed strictly. As an application, a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model. Numerical experiments are provided to support our theoretical results.

Chaotic dynamics of the van der Pol-Duffing oscillator subjected to periodic external and parametric excitations with delayed feedbacks are investigated both analytically and numerically in this manuscript. With the Melnikov method, the critical value of chaos arising from homoclinic or heteroclinic intersections is derived analytically. The feature of the critical curves separating chaotic and non-chaotic regions on the excitation frequency and the time delay is investigated analytically in detail. The monotonicity of the critical value to the excitation frequency and time delay is obtained rigorously. It is presented that there may exist a special frequency for this system. With this frequency, chaos in the sense of Melnikov may not occur for any excitation amplitudes. There also exists a uncontrollable time delay with which chaos always occurs for this system. Numerical simulations are carried out to verify the chaos threshold obtained by the analytical method.

Based on the martingale difference divergence, a recently proposed metric for quantifying conditional mean dependence, we introduce a consistent test of U-type for the goodness-of-fit of linear models under conditional mean restriction. Methodologically, our test allows heteroscedastic regression models without imposing any condition on the distribution of the error, utilizes effectively important information contained in the distance of the vector of covariates, has a simple form, is easy to implement, and is free of the subjective choice of parameters. Theoretically, our mathematical analysis is of own interest since it does not take advantage of the empirical process theory and provides some insights on the asymptotic behavior of U-statistic in the framework of model diagnostics. The asymptotic null distribution of the proposed test statistic is derived and its asymptotic power behavior against fixed alternatives and local alternatives converging to the null at the parametric rate is also presented. In particular, we show that its asymptotic null distribution is very different from that obtained for the true error and their differences are interestingly related to the form expression for the estimated parameter vector embodied in regression function and a martingale difference divergence matrix. Since the asymptotic null distribution of the test statistic depends on data generating process, we propose a wild bootstrap scheme to approximate its null distribution. The consistency of the bootstrap scheme is justified. Numerical studies are undertaken to show the good performance of the new test.

Kawasaki disease (KD) is an acute, febrile, systemic vasculitis that mainly affects children under five years of age. In this paper, we propose and study a class of 5-dimensional ordinary differential equation model describing the vascular endothelial cell injury in the lesion area of KD. This model exhibits forward/backward bifurcation. It is shown that the vascular injury-free equilibrium is locally asymptotically stable if the basic reproduction number R₀ < 1. Further, we obtain two types of suffcient conditions for the global asymptotic stability of the vascular injury-free equilibrium, which can be applied to both the forward and backward bifurcation cases. In addition, the local and global asymptotic stability of the vascular injury equilibria and the presence of Hopf bifurcation are studied. It is also shown that the model is permanent if the basic reproduction number R₀ > 1, and some explicit analytic expressions of ultimate lower bounds of the solutions of the model are given. Our results suggest that the control of vascular injury in the lesion area of KD is not only correlated with the basic reproduction number R₀, but also with the growth rate of normal vascular endothelial cells promoted by the vascular endothelial growth factor.