SOLVABILITY OF A PARABOLIC-HYPERBOLIC TYPE CHEMOTAXIS SYSTEM IN 1-DIMENSIONAL DOMAIN

In this paper, we use contraction mapping principle, operator-theoretic approach and some uniform estimates to establish local solvability of the parabolic-hyperbolic type chemotaxis system with fixed boundary in 1-dimensional domain. In addition, local solvability of the free boundary problem is considered by straightening the free boundary.

LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS

Let Ω be a bounded open domain in Rn with smooth boundary Ω, X =（X1,X2,... ,Xm） be a system of real smooth vector fields defined on Ω and the bound-ary Ω is non-characteristic for X. If X satisfies the HSrmander＇s condition, then the vectorfield is finitely degenerate and the sum of square operator △X =m∑j=1 X2 j is a finitely de-generate elliptic operator. In this paper, we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △x on Ω.

GLOBAL EXISTENCE,EXPONENTIAL DECAY AND BLOW-UP IN FINITE TIME FOR A CLASS OF FINITELY DEGENERATE SEMILINEAR PARABOLIC EQUATIONS

In this paper,we study the initial-boundary value problem for the semilinear parabolic equations ut-△Xu=|u|p-1u,where X=（X1,X2,…,Xm） is a system of real smooth vector fields which satisfy the H?rmander’s condition,and △X=∑j=1m Xj2 is a finitely degenerate elliptic operator.Using potential well method,we first prove the invariance of some sets and vacuum isolating of solutions.Finally,by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy,and also we discuss the asymptotic behavior of the global solutions.

CONTROL STRATEGIES FOR A TUMOR-IMMUNE SYSTEM WITH IMPULSIVE DRUG DELIVERY UNDER A RANDOM ENVIRONMENT

This paper mainly studies the stochastic character of tumor growth in the presence of immune response and periodically pulsed chemotherapy.First,a stochastic impulsive model describing the interaction and competition among normal cells,tumor cells and immune cells under periodically pulsed chemotherapy is established.Then,sufficient conditions for the extinction,non-persistence in the mean,weak and strong persistence in the mean of tumor cells are obtained.Finally,numerical simulations are performed which not only verify the theoretical results derived but also reveal some specific features.The results show that the growth trend of tumor cells is significantly affected by the intensity of noise and the frequency and dose of drug deliveries.In clinical practice,doctors can reduce the randomness of the environment and increase the intensity of drug input to inhibit the proliferation and growth of tumor cells.

THE VLASOV-POISSON-BOLTZMANN SYSTEM NEAR MAXWELLIANS FOR LONG-RANGE INTERACTIONS

In this article, we are concerned with the construction of global smooth small-amplitude solutions to the Cauchy problem of the one species Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions. Compared with the former result obtained by Duan and Liu in [12] for the two species model, we do not ask the initial perturbation to satisfy the neutral condition and our result covers all physical collision kernels for the full range of intermolecular repulsive potentials.

GEVREY REGULARITY WITH WEIGHT FOR INCOMPRESSIBLE EULER EQUATION IN THE HALF PLANE

In this work we prove the weighted Gevrey regularity of solutions to the incompressible Euler equation with initial data decaying polynomially at infinity. This is motivated by the well-posedness problem of vertical boundary layer equation for fast rotating fluid. The method presented here is based on the basic weighted L;-estimate, and the main difficulty arises from the estimate on the pressure term due to the appearance of weight function.

HITTING PROBABILITIES OF WEIGHTED POISSON PROCESSES WITH DIFFERENT INTENSITIES AND THEIR SUBORDINATIONS

In this article,we study the hitting probabilities of weighted Poisson processes and their subordinated versions with different intensities.Furthermore,we simulate and analyze the asymptotic properties of the hitting probabilities in different weights and give an example in the case of subordination.

HYPOELLIPTIC ESTIMATE FOR SOME COMPLEX VECTOR FIELDS

1 Introduction and Main Results LetΩ■Rnbe a neighborhood of 0,and denote by%the square root of—1.We considerthe following system of complex vector fieldsPj=■xj-i(■xjφ(x))■t,j=1,…,n,(x,t)∈Ω×R,(1.1)whereφ(x)is a real-valued function defined in.

CONTINUOUS FINITE ELEMENT METHODS FOR REISSNER-MINDLIN PLATE PROBLEM

On triangle or quadrilateral meshes, two finite element methods are proposed for solving the Reissner-Mindlin plate problem either by augmenting the Galerkin formulation or modifying the plate-thickness. In these methods, the transverse displacement is approximated by conforming （bi）linear macroelements or （bi）quadratic elements, and the rotation by conforming （bi）linear elements. The shear stress can be locally computed from transverse displacement and rotation. Uniform in plate thickness, optimal error bounds are obtained for the transverse displacement, rotation, and shear stress in their natural norms. Numerical results are presented to illustrate the theoretical results.

The Discontinuous Galerkin Method by Patch Reconstruction for Helmholtz Problems

This paper develops and analyzes interior penalty discontinuous Galerkin(IPDG)method by patch reconstruction technique for Helmholtz problems.The technique achieves high order approximation by locally solving a discrete least-squares over a neighboring element patch.We prove a prior error estimates in the L 2 norm and energy norm.For each fixed wave number k,the accuracy and efficiency of the method up to order five with high-order polynomials.Numerical examples are carried out to validate the theoretical results.

Topology inference of uncertain complex dynamical networks and its applications in hidden nodes detection

The topological structure of a complex dynamical network plays a vital role in determining the network＇s evolutionary mecha- nisms and functional behaviors, thus recognizing and inferring the network structure is of both theoretical and practical signif- icance. Although various approaches have been proposed to estimate network topologies, many are not well established to the noisy nature of network dynamics and ubiquity of transmission delay among network individuals. This paper focuses on to- pology inference of uncertain complex dynamical networks. An auxiliary network is constructed and an adaptive scheme is proposed to track topological parameters. It is noteworthy that the considered network model is supposed to contain practical stochastic perturbations, and noisy observations are taken as control inputs of the constructed auxiliary network. In particular, the control technique can be further employed to locate hidden sources （or latent variables） in networks. Numerical examples are provided to illustrate the effectiveness of the proposed scheme. In addition, the impact of coupling strength and coupling delay on identification performance is assessed. The proposed scheme provides engineers with a convenient approach to infer topologies of general complex dynamical networks and locate hidden sources, and the detailed performance evaluation can further facilitate practical circuit design.

The Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.

The non-cutoff Vlasov-Maxwell-Boltzmann system with weak angular singularity

We establish the global existence of small-amplitude solutions near a global Maxwellian to the Cauchy problem of the Vlasov-Maxwell-Boltzmann system for non-cutoff soft potentials with weak angular singularity. This extends the work of Duan et al.(2013), in which the case of strong angular singularity is considered, to the case of weak angular singularity.

Fundamental Solutions of Nonlocal Hörmander’s Operators

Consider the following nonlocal integro-differential operator:forα∈(O,2),L_(σ,b)^(a)f(x)=p.v.∫_(R^(d))-{0}f(x+σ(x)z-f(x)/|z|^(d+α)dz+b(x).△f(x),whereσ:R^(d)×R^(d)and b:R^(d)→R^(d)are smooth and have bounded first-order derivatives,and p.v.stands for the Cauchy principal value.Let B1(x):=σ(x)and Bj+1(x):=b(x).△Bj(x)-△b(x)·Bj(x)for j∈N.Under the followingHormander's type condition:for any x∈R^(d)and some n=n(x)∈N,Rank[B1(x),B2(K),...,Bn(x)]=d,by using the Malliavin calculus,we prove the existence of the heat kernelρt(x,y)to the operator L_(σ,b)^(α)as well as the continuity of x→ρt(x,.)in L^(1)(R^(d))as a densityfunction for each t>0.Moreover,whenσ(x)=σis constant and Bj∈C_(b)^(∞)for eachj∈N,under the following uniform Hormander's type condition:for some j0∈N,we also show the smoothness of(t,x,y)→ρt(x,y)withρt(,.)∈b^(∞)(R^(d)×R^(d))for each t>0.

Spectral gap, isoperimetry and concentration on trees

We consider the nearest-neighbor model on the finite tree T with generator L. We obtain a twosided estimate of the spectral gap by factor 2. We also identify explicitly the Lipschitzian norm of the operator(-L)^(-1) in propriate functional space. This leads to the identification of the best constant in the generalized Cheeger isoperimetric inequality on the tree, and to transportation-information inequalities.

The Cauchy problem for a class of nonlinear degenerate parabolic-hyperbolic equations

In this article, we prove a general existence theorem for a class of nonlinear degenerate parabolichyperbolic equations. Since the regions of parabolicity and hyperbolicity are coupled in a way that depends on the solution itself, there is almost no hope of decoupling the regions and then taking into account the parabolic and the hyperbolic features separately. The existence of solutions can be obtained by ?nding the limit of solutions for the regularized equation of strictly parabolic type. We use the energy methods and vanishing viscosity methods to prove the local existence and uniqueness of solution.

Viscous Shock Wave to an Inflow Problem for Compressible Viscous Gas with Large Density Oscillations

This paper is concerned with the inflow problem for one-dimensional compressible Navier-Stokes equations. For such a problem, Huang, Matsumura, and Shi showed in [4] that there exists viscous shock wave solution to the inflow problem and both the boundary layer solution, the viscous shock wave, and their superposition are time-asymptotically nonlinear stable provided that both the initial perturbation and the boundary velocity are assumed to be sufficiently small. The main purpose of this paper is to show that similar stability results still hold for a class of large initial perturbation which can allow the initial density to have large oscillations. The proofs are given by an elementary energy method and our main idea is to use the smallness of the strength of the viscous shock wave and the boundary velocity to control the possible growth of the solutions induced by the nonlinearity of the compressible Navier-Stokes equations and the inflow boundary condition.The key point in our analysis is to deduce the desired uniform positive lower and upper bounds on the density.

Numerical Integrators for Dispersion-Managed KdV Equation

In this paper,we consider the numerics of the dispersion-managed Kortewegde Vries(DM-KdV)equation for describingwave propagations in inhomogeneous media.The DM-KdV equation contains a variable dispersion map with discontinuity,which makes the solution non-smooth in time.We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation,where a necessary constraint on the time step has been identified.Then,two exponential-type dispersionmap integrators up to second order accuracy are derived,which are efficiently incorporatedwith the Fourier pseudospectral discretization in space,and they can converge regardless the discontinuity and the step size.Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast&strong dispersion-management regime.

The Discontinuous Galerkin Method by Divergence-Free Patch Reconstruction for Stokes Eigenvalue Problems

The discontinuous Galerkin method by divergence-free patch reconstruction is proposed for Stokes eigenvalue problems.It utilizes the mixed finite element framework.The patch reconstruction technique constructs two categories of approximation spaces.Namely,the local divergence-free space is employed to discretize the velocity space,and the pressure space is approximated by standard reconstruction space simultaneously.Benefit from the divergence-free constraint;the identical element patch serves two approximation spaces while using the element pair Pm+1/Pm.The optimal error estimate is derived under the inf-sup condition framework.Numerical examples are carried out to validate the inf-sup test and the theoretical results.

Exact synchronization and asymptotic synchronization for linear ODEs

This paper studies exact synchronization and asymptotic synchronization problems for a controlled linear system of ordinary differential equations. In this paper, we build up necessary and sufficient conditions for exact synchronization and asymptotic synchronization problems. When a system is not controllable but exactly synchronizable, it can be asymptotically synchronized in any given rate and the state of exact synchronization is given. However, when a system is not controllable and can be asymptotically synchronized in any given rate,it may not be exactly synchronizable.