Global existence and asymptotic behavior for the compressible Navier–Stokes equations with a non‐autonomous external force and a heat source

In this paper, we prove the global existence and asymptotic behavior, as time tends to infinity, of solutions in Hⁱ (i=1, 2) to the initial boundary value problem of the compressible Navier–Stokes equations of one‐dimensional motion of a viscous heat‐conducting gas in a bounded region with a non‐autonomous external force and a heat source. Some new ideas and more delicate estimates are used to prove these results. Copyright © 2008 John Wiley & Sons, Ltd.     

Global existence of solutions to a hyperbolic-parabolic system

In this paper, we investigate the global existence of solutions to a hyperbolic-parabolic model of chemotaxis arising in the theory of reinforced random walks. To get -estimates of solutions, we construct a nonnegative convex entropy of the corresponding hyperbolic system. The higher energy estimates are obtained by the energy method and a priori assumptions.

     

Asymptotic Behavior for Global Smooth Solution to a One-dimensional Nonlinear Thermoviscoelastic System

This paper is concerned with asymptotic behavior, as time tends to infinity, of globally defined smooth (large) solutions to the system in one-dimensional nonlinear thermoviscoelasticity. Our results show that the global smooth solution approaches to the solution in the H¹ norm to the corresponding stationary problem, as time tends to infinity.     

Global existence to a three‐dimensional non‐linear thermoelasticity system arising in shape memory materials

This paper is concerned with the unique global solvability of a three‐dimensional (3‐D) non‐linear thermoelasticity system arising from the study of shape memory materials. The system consists of the coupled evolutionary problems of viscoelasticity with non‐convex elastic energy and non‐linear heat conduction with mechanical dissipation. The present paper extends the previous 2‐D existence result of the authors Reference [1] to 3‐D case. This goal is achieved by means of the Leray–Schauder fixed point theorem using technique based on energy arguments and DeGiorgi method. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space

Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity

We consider the Cauchy problem for the nonlinear dissipative evolution system with ellipticity on one dimensional space

Global existence and asymptotic behavior of classical solutions for a 3D two‐species chemotaxis‐Stokes system with competitive kinetics

This paper considers the 2‐species chemotaxis‐Stokes system with competitive kinetics under homogeneous Neumann boundary conditions in a 3‐dimensional bounded domain with smooth boundary. Both chemotaxis‐fluid systems and 2‐species chemotaxis systems with competitive terms were studied by many mathematicians. However, there have not been rich results on coupled 2‐species–fluid systems. Recently, global existence and asymptotic stability in the above problem with (u·∇)u in the fluid equation were established in the 2‐dimensional case. The purpose of this paper is to give results for global existence, boundedness, and stabilization of solutions to the above system in the 3‐dimensional case when is sufficiently small.     

Global non‐existence of solutions of a class of wave equations with non‐linear damping and source terms

In this paper we consider the non‐linear wave equation a,b>0, associated with initial and Dirichlet boundary conditions. We prove, under suitable conditions on α,β,m,p and for negative initial energy, a global non‐existence theorem. This improves a result by Yang (Math. Meth. Appl. Sci. 2002; 25 :825–833), who requires that the initial energy be sufficiently negative and relates the global non‐existence of solutions to the size of Ω. Copyright © 2004 John Wiley & Sons, Ltd.     

The Global Existence of Solutions for a Cross-diffusion System

This paper is concerned with the global existence of solutions for a class of quasilinear cross-diffusion system describing two species competition under self and cross population pressure. By establishing and using more detailed interpolation results between several different Banach spaces, the global existence of solutions are proved when the self and cross diffusion rates for the first species are positive and there is no self or cross-diffusion for the second species.     

Existence and asymptotic behavior of global smooth solution for p‐system with nonlinear damping and fixed boundary effect

This paper is concerned with the initial boundary value problem for the p‐system with nonlinear damping and fixed boundary condition. We show that the corresponding problem admits a unique global solution, and such a solution tends time asymptotically to the corresponding nonlinear diffusion wave governed by the classical Darcy's law provided that the corresponding prescribed initial error function is sufficiently small. Copyright © 2013 John Wiley & Sons, Ltd.     

半导体模型中退化漂移-扩散问题解的存在性

研究半导体物理中出现的漂移—扩散模型,这是一个关于带电粒子浓度n,p,静电位ψ的抛物—椭圆耦合方程组,并带有混合初边值条件.对初值在L2+(Ω)时弱解的存在性展开讨论,利用正则化方法和适当的函数变换,使抛物型方程的解具有正下界n,p≥ε>0,同时得到一系列先验估计.然后通过紧性引理和Schauder不动点定理,得到初值在L2+(Ω)时弱解的存在性.     

On global existence of solutions to a cross-diffusion system

We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in n-dimensional domains (n?1). We prove the global existence of classical solutions to the system for n<10.     

Global Smooth Solutions and the Asymptotic Behavior of the Motion of a Viscous,Heat-conductive,One-dimensional Real Gas

The system of balance laws of mass, momentum and energy for a viscous, heal-conductive, one-dimensional real gas is considered. The existence of globally defined smooth solution to an initial boundary value problem is established. Because of the boundary conditions' effect, vacuum will be developped as time tends to infinity.     

Global existence of solutions for non‐small data to non‐linear spherically symmetric thermoviscoelasticity

We consider some initial–boundary value problems for non‐linear equations of thermoviscoelasticity in the three‐dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd.     

Positive solutions for p‐Kirchhoff type problems on

In this paper, we establish the existence and non‐existence of positive solutions for p‐Kirchhoff type problems with a parameter on without assuming the usual compactness conditions. We show that the p‐Kirchhoff type problems have at least one positive solution when the parameter is small, while the p‐Kirchhoff type problems have no positive solutions when the parameter is large. Our argument is based on variational methods, monotonicity methods, cut‐off functional techniques, and a priori estimates techniques. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic behaviour of a two‐dimensional differential system with non‐constant delay

The asymptotic behaviour and stability properties are studied for a real two‐dimensional system x^′(t) = A(t)x (t) + B(t)x (θ (t)) + h (t, x (t), x (θ (t))), with a nonconstant delay t ‐ θ (t) ≥ 0. It is supposed that A,B and h are matrix functions and a vector function, respectively. The method of investigation is based on the transformation of the considered real system to one equation with complex‐valued coefficients. Stability and asymptotic properties of this equation are studied by means of a suitable Lyapunov‐Krasovskii functional. The results generalize the great part of the results of J. Kalas and L. Baráková [J. Math. Anal. Appl. 269 , No. 1, 278–300 (2002)] for two‐dimensional systems with a constant delay (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Navier–Stokes equations with degenerate viscosity,vacuum and gravitational force

The global existence of weak solutions to the compressible Navier–Stokes equations with vacuum attracts many research interests nowadays. For the isentropic gas, the viscosity coefficient depends on density function from physical point of view. When the density function connects to vacuum continuously, the vacuum degeneracy gives some analytic difficulties in proving global existence. In this paper, we consider this case with gravitational force and fixed boundary condition. By giving a series of a priori estimates on the solution coping with the degeneracy of vacuum, gravitational force and boundary effect, we give global existence and uniqueness results similar to the case without force and boundary. Copyright © 2006 John Wiley & Sons, Ltd.     

Self‐similar asymptotic profile of the solution to a nonlinear evolution equation with critical dissipation

In this paper, we obtain optimal decay estimates for the solutions to an evolution equation with critical, structural, dissipation, and absorbing power nonlinearity: with μ>0, θ is a positive integer, and p>1+4θ/n, in space dimension n∈(2θ,4θ). We use these estimates to find the self‐similar asymptotic profile of the solutions, when μ≥1.     

On the cauchy problem for a one‐dimensional full compressible non‐Newtonian fluids

This paper is concerned with global existence and asymptotic behavior of H¹ solutions to the Cauchy problem of one‐dimensional full non‐Newtonian fluids with the weighted small initial data. We then obtain the global existence of Hⁱ(i = 2,4) solutions and their asymptotic behavior for the system. Copyright © 2015 John Wiley & Sons, Ltd.