Blowup,Global Fast and Slow Solutions for a Semilinear Combustible System

In this paper, we investigate a semilinear combustible system $u_t-du_{xx}=v^p, v_t-dv_{xx}=u^q$ with double fronts free boundary, where p ≥ 1, q ≥ 1. For such a problem, we use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup and global existence property of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.     

Existence and nonexistence of global positive solutions for a weakly coupled P-Laplacian system

In this paper, we deal with a weakly coupled evolution P-Laplacian system with inhomogeneous terms. We obtain a critical criterion concerning existence and nonexistence of its global positive solutions. Such a criterion is different from that of the weakly coupled evolution P-Laplacian system with homogeneous terms. Further, we demonstrate existence and nonexistence of its global positive solutions.     

Some properties of solutions for a fourth‐order parabolic equation

This paper is concerned with a fourth‐order parabolic equation in one spatial dimension. On the basis of Leray–Schauder's fixed point theorem, we prove the existence and uniqueness of global weak solutions. Moreover, we also consider the regularity of solution and the existence of global attractor. Copyright © 2012 John Wiley & Sons, Ltd.     

Global existence,asymptotic behavior,and uniform attractor for a nonautonomous equation

In this paper, we establish the global existence, asymptotic behavior, and uniform attractor for a nonautonomous viscoelastic equation with a delay term. Under some suitable assumptions, we firstly prove the global well‐posedness of the problem by using the Faedo–Galerkin approximations together with some energy estimates and then obtain the general decay results of the energy via suitable Lyapunov functionals. Finally, we prove the existence of uniform attractors. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Cauchy problem for a doubly nonlinear degenerate parabolic equation with strongly nonlinear sources

In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||_f||| _h and 〈f〉_h, we give the sufficient and necessary conditions on the initial value to the existence of local solution of doubly nonlinear equation. Moreover some results on the global existence and nonexistence of solutions are considered. This work was supported by the National Natural Science Foundation of China (Grant No. 10531020)     

Global attractor for the regularized Bénard problem

In this paper, we study the asymptotic behaviour of weak solutions for the regularized Bénard problem. We establish the global existence and uniqueness of weak solutions of this problem and give a proof for the existence of global attractor in the three-dimensional case for this system.     

广义Davey-Stewartson方程组的整体解及自相似解

本文研究Dalvey-Stewartson方程组的整体解与自相似解的存在性.首先,运用Ba- nach不动点定理得到一个关于解整体存在性的一般性定理,然后把一类特殊的初始值用到该存在性结果上去从而得到自相似解存在的结论.     

Global attractors for the viscous hyperelastic‐rod wave equation

We consider the viscous hyperelastic‐rod wave equation subject to an external force, where the viscous term is given by second order differential operator in divergence form. Under some mild assumptions on the viscous term, first, we establish the global well‐posedness in both the periodic case and the case of the whole line, afterwards, we show the existence of global attractors for the two cases, respectively. Copyright © 2012 John Wiley & Sons, Ltd.     

The global existence of solutions for two classes of chemotaxis models

This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotaztic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotacticsensitivity terms, which extends the global existence results of [6] to other general cases.     

Some Notes on Reaction Diffusion Systems with Nonlinear Boundary Conditions

This paper deals with the existence and nonexistence of global positive solution to a semilinear reaction-diffusion system with nonlinear boundary conditions.For the heat diffusion case,the necessary and sufficient conditions on the global existence of all positive solutions are obtained.For the general fast diffusion case,we get some conditions on the global existence and nonexistence of positive solutions.The results of this paper fill the some gaps which were left in this field.     

GLOBAL EXISTENCE OF THE SOLUTION FOR THE LANDAU-LIFSHITZ EQUATION OF THE FERROMAGNETIC SPIN CHAIN

1IntroductionInthepresentpaper,wecollsidertileLandau-LifSllitzequationoftileferromagneticspinchainwhichtakestheform:where"x'ldenotesthevectorcrossproductinR',fi=(--D,D),D>0,u=(ul,u3,u3):fixR -R3istilespinvector,or>0isaGilbertdampingconstant,Pisarealparameter,J=diag(ri,acfac)isaconstantmatrix.TheaboveequationarisesinthedoublyanisotropicHeisenbergspinsystem.Thissystemisimpliedbytileconservationofenergyandmagnitudeofu.SeeDanielandLab.hmanan[1]please.Insomeferromagneticchainsystems,thematri…     

Hyperbolic Yamabe problem

In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the(1 + n)-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n = 2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem;while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.     

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.     

Global smooth solution of hydrodynamical equation for the Heisenberg paramagnet

In this note, we obtain the existence and uniqueness of global smooth solution for the Cauchy problem of multidimensional hydrodynamical equation for the Heisenberg paramagnet. Copyright © 2004 John Wiley & Sons, Ltd.     

The global weak solutions to the Cauchy problem of the generalized Novikov equation

In this paper, we study the Cauchy problem of the generalized Novikov equation. We first show that under suitable condition, the strong solution exists globally via some a priori estimates. Then, we prove the existence and uniqueness of global weak solutions by the approximation method. We also obtain the exact peaked solutions.     

The finite dimensional behaviour for the higher-order nonlinear Ginzburg-Landau system inn spatial dimensions

In this paper, we study the 2m-order nonlinear Ginzburg-Landau system inn spatial dimensions. We show the existence and uniqueness of the global generalized solution, and the existence of the global attractor for this system, and establish the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractor. This project is supported by the National Natural Science Foundation of China (No. 19571010).     

带有阻尼项和源项的波动方程的全局解和爆破解

讨论非线性波动方程(1.1)全局解的存在性和不存在性.当初值满足某些条件时,给出了新的判定全局解是否存在的条件.     

Global blow-up for a localized quasilinear parabolic system with nonlocal boundary conditions

In this article, we investigate the positive solution of a localized quasilinear parabolic system with nonlocal boundary conditions. Under certain conditions, the global existence and finite time blow-up criteria are established, and the global blow-up behaviour is also obtained.     

Exponential attractor of the 3D derivative Ginzburg-Landau equation

In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved.     

Fisher–KPP equation on the Heisenberg group

In this paper, we consider the Fisher–KPP equation on the Heisenberg group. We discuss the existence of global solutions, asymptotic behavior of global solutions and blow-up solutions. Moreover, we extend the obtained results to the time-fractional Fisher–KPP equation on the Heisenberg group.