Analytic Smoothing Effect of the Time Variable for the Spatially Homogeneous Landau Equation

In this work,we study the Cauchy problem of the spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework.We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L^(2)initial datum for positive time.So that the smoothing effect of the Cauchy problem for the spatially homogeneous Landau equation with hard potentials is exactly same as heat equation.

The analytic smoothing effect of solutions for the nonlinear spatially homogeneous Landau equation with hard potentials

In this work,we study the Cauchy problem of the nonlinear spatially homogeneous Landau equation with hard potentials in a close-to-equilibrium framework.We prove that the solution to the Cauchy problem with the initial datum in L^(2)enjoys an analytic regularizing effect,and the evolution of the analytic radius is the same as that of heat equations.

Hitting Probabilities and the Hausdorff Dimension of the Inverse Images of a Class of Anisotropic Random Fields

Let X = {X（t）：t ∈ R^N} be an anisotropic random field with values in R^d.Under certain conditions on X,we establish upper and lower bounds on the hitting probabilities of X in terms of respectively Hausdorff measure and Bessel-Riesz capacity.We also obtain the Hausdorff dimension of its inverse image,and the Hausdorff and packing dimensions of its level sets.These results are applicable to non-linear solutions of stochastic heat equations driven by a white in time and spatially homogeneous Gaussian noise and anisotropic Guassian random fields.

Dynamical analysis of a reaction-diffusion SEI epidemic model with nonlinear incidence rate

In this paper,a reaction-diffusion SEI epidemic model with nonlinear incidence rate is proposed.The well-posedness of solutions is studied,including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions.The basic reproduction numbers are given in both heterogeneous and homogeneous environments.For spatially heterogeneous environment,by the comparison principle of the diffusion system,the infection-free steady state is proved to be globally asymptotically stable if R_(0)<1,if R_(0)>1,the system will be persistent and admit at least one positive steady state.For spatially homogenous environment,by constructing a Lyapunov function,the infect ion-free steady state is proved to be globally asymptotically stable if,R_(0)<1,and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if R_(0)>1.Finally,two examples are given via numerical simulations,and then some control strategies are also presented by the sensitive analysis.

Some models with repulsion effect on superinfecting viruses by infected cells

In this paper,we study some models with repulsion effect on superinfecting viruses by infected cells{■T/■t=DT△T(x,t)-■(TФ(T,I)■I)+h(x)-dTT(x,t)-β(x)T(x,t)V(x,t),■I/■T=DT△I+β(x)T(x,t)V(x,t)-dII(xt,),■I/■T=■(Dv(I)■V)+γ(x)I(x,t)-dvV(x,t),where T(x,t),I(x,t)and V(x,t)are the density of uninfected cells,infected cells and viruses at time t at location x,respectively.The functions h(x),β(x)andγ(x)are assumed to be positive,continuous and bounded.h(x)denotes the production rate of uninfected cells.The infection rate isβ(x)and the functionγ(x)is the production rate of free viruses.Andβ(x)T(x,t)V(x,t)is the rate of transfer from uninfected cells to infected cells.The positive constants dT,dI and dV denote the death rate of uninfected cells,infected cells and viruses,respectively.The stability of the infection-free equilibrium solution and infection equilibrium solution is discussed.It is shown that if the basic reproduction number R0≤1 then the chemotaxis has no effect,that is,the infection-free constant solution is stable.For the system with chemotactic sensitivityФ(T,I)=1-T,if R0>1,then the infection constant solution will be unstable under some conditions.

A Remark about Time-Analyticity of the Linear Landau Equation with Soft Potential

In this note,we study the Cauchy problem of the linear spatially homogeneous Landau equation with soft potentials.We prove that the solution to the Cauchy problem enjoys the analytic regularizing effect of the time variable with an L2 initial datum for positive time.So that the smoothing effect of Cauchy problem for the linear spatially homogeneous Landau equation with soft potentials is similar to the heat equation.