Attractor dimension estimate for plane shear flow of micropolar fluid with free boundary

This research is motivated by a problem from lubrication theory. We consider a free boundary problem of a two‐dimensional boundary‐driven micropolar fluid flow. The existence of a unique global‐in‐time solution of the problem and the global attractor for the associated semigroup are known. In this paper we estimate the dimension of the global attractor in terms of the given data and the geometry of the domain of the flow by establishing a new version of the Lieb–Thirring inequality with constants depending explicitly on the geometry of the domain. We also obtain some new estimates for the Navier–Stokes shear flows. Copyright © 2005 John Wiley & Sons, Ltd.     

Beam evolution equation with variable coefficients

We investigate a initial‐boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping. We show the existence of a unique global weak solution and that the energy associated with this solution has a rate decay estimate. Besides, we prove the existence and uniqueness of non‐local strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

On the global existence and small dissipation limit for generalized dissipative Zakharov system

This paper deals with the existence and uniqueness of the global solutions to the initial boundary value problem for a generalized Zakharov system with direct self‐interaction of the dispersive waves and weak dissipation in the nondispersive subsystem. We prove the global existence of the generalized solution to the problem by a priori estimates and Galerkin method. We also establish the regularity of the global generalized solution and the existence and uniqueness of the global classical solution. Moreover, we obtain the convergence of the solutions of the generalized Zakharov system with dissipation as the dissipative coefficient approaches zero.     

A model of thermal dissipation for a one‐dimensional viscous reactive and radiative gas

We consider a model of thermal dissipation for a Stefan–Boltzmann model of viscous and reactive gas in a bounded interval. We prove the existence of a global‐in‐time solution, and we give the asymptotic behaviour for the corresponding Dirichlet problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Well‐posedness of nonlinear wave equation with combined power‐type nonlinearities

In this paper, we study the initial boundary‐value problem with combined power‐type nonlinearities by utilizing potential well method. We provide an algorithm to compute the depth of the potential well with the help of Mathematica, and derive the invariant subsets, global existence and blowup of solutions. Moreover, we obtain the invariant subsets, global existence and blowup of solutions for the critical case. Copyright © 2011 John Wiley & Sons, Ltd.     

Global existence of strong solutions to a time‐dependent 3D Ginzburg‐Landau model for superconductivity with partial viscous terms

We study an initial boundary value problem for a time‐dependent 3D Ginzburg‐Landau model of superconductivity with partial viscous terms. We prove the global existence of strong solutions.     

On the Cauchy problem for the nonlinear Schrodinger equations including fractional dissipation with variable coefficient

This paper is devoted to the Cauchy problem for the nonlinear Schrodinger equation with time‐dependent fractional damping term. We prove the local existence result, and we study the global existence and blow‐up solutions.     

On an incompressible model in radiation hydrodynamics

We consider a simplified model arising in radiation hydrodynamics based on the incompressible Navier–Stokes–Fourier system describing a macroscopic fluid motion coupled to a transport equation modeling the propagation of radiative intensity. We establish global‐in‐time existence for the associated initial‐boundary value problem in the framework of weak solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

The attractor of the dissipative coupled fractional Schrödinger equations

In the present paper, we consider the dissipative coupled fractional Schrödinger equations. The global well‐posedness by the contraction mapping principle is obtained. We study the long time behavior of solutions for the Cauchy problem. We prove the existence of global attractor. Copyright © 2013 John Wiley & Sons, Ltd.     

Global existence of weak solutions to a weakly dissipative modified two‐component Dullin–Gottwald–Holm system

We consider a weakly dissipative modified two‐component Dullin–Gottwald–Holm system. The existence of global weak solutions to the system is established. We first give the well‐posedness result of viscous approximate problem and obtain the basic energy estimates. Then, we show that the limit of the viscous approximation solutions is a global weak solution to the system. Copyright © 2014 John Wiley & Sons, Ltd.     

The stability of localized solutions of Landau‐Lifshitz equations

We study the Landau‐Lifshitz equation of ferromagnetism on ?², with an easy‐axis anisotropy. We establish the existence of topologically nontrivial, periodic solutions, and show they are stable against equivariant perturbations. Along the way, we establish the global well‐posedness of the Cauchy problem for a class of data with no size restriction. © 2002 Wiley Periodicals, Inc.     

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.     

Global existence and asymptotic behavior of solution for Rosenau equation with Stokes damped term

We study the small‐data Cauchy problem for n‐dimensional Stokes damped Rosenau equation. Under some assumptions, we prove the global existence and uniqueness of the small‐amplitude solution by utilizing the contraction mapping principle and study the asymptotic behavior of the solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Global solution of the Cauchy problem in nonlinear thermodi.usion in solid body

We prove a theorem about global existence (in time) of the solution to the initial‐value problem for a nonlinear hyperbolic parabolic system of coupled partial differential equation of second order describing the process of thermodiffusion in solid body. The corresponding global existence theorems has been proved using the L^p ‐ L^q time decay estimates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard (continuation argument and to continue the local solution to one de.ned for all t ∈ 〈0, ∞)).     

Global Existence and blowup phenomenon for a 1D radiation hydrodynamics model problem

We consider the global existence of classical solutions and blowup phenomena for a spatially one‐dimensional radiation hydrodynamics model problem, which consists of a scalar Burgers‐type equation coupled with a nonlocal advection‐reaction equation for radiation intensity. The model can be seen as an extension of the well‐known Hamer model that includes additionally the effects of scattering. It is well‐known that the initial value problem for Burgers' equation cannot be solved classically as soon as the derivative of the initial datum is negative somewhere. For our model problem, there is a critical negative number such that if the spatial derivative of the initial function is larger than this number, the associated initial‐value problem admits a global classical solution. However, when the spatial derivative of the initial data is below another negative threshold number, the initial value problem can also not be solved classically. Moreover, when there does not exist a global classical solution, it is shown that the first spatial derivative of solution must blow up in finite time. The results of the paper generalize the findings of Kawashima and Nishibata for the Hamer model. Copyright © 2012 John Wiley & Sons, Ltd.     

A Lagrangian Approach for the Incompressible Navier‐Stokes Equations with Variable Density

We investigate the Cauchy problem for the inhomogeneous Navier‐Stokes equations in the whole n‐dimensional space. Under some smallness assumption on the data, we show the existence of global‐in‐time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p‐1}_{p,1}({\Bbb R}^n)$ . In particular, piecewise‐constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.     

Global solutions for a semilinear,two‐dimensional Klein‐Gordon equation with exponential‐type nonlinearity

We prove the existence and uniqueness of global solutions for a Cauchy problem associated to a semilinear Klein‐Gordon equation in two space dimensions. Our result is based on an interpolation estimate with a sharp constant obtained by a standard variational method. © 2006 Wiley Periodicals, Inc.     

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

In this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19 :1263–1317; Nonlinear Anal. 1997; 28 :1299–1322; Chin. Ann. Math. 2004; 25B :37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12 (1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.     

On the weak solutions to a shallow water equation

We obtain the existence of global‐in‐time weak solutions to the Cauchy problem for a one‐dimensional shallow‐water equation that is formally integrable and can be obtained by approximating directly the Hamiltonian for Euler's equation in the shallow‐water regime. The solution is obtained as a limit of viscous approximation. The key elements in our analysis are some new a priori one‐sided supernorm and space‐time higher‐norm estimates on the first‐order derivatives. © 2000 John Wiley & Sons, Inc.     

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.