Existence and nonexistence of global solutions for nonlinear evolution equation of fourth order

Abstract. The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation ua-a1uxx-a2uxxl-a3uxxu=ψ(ux)x are proved     

关于粘性Cahn-Hilliard方程的注记

In this note, we study the global existence of classical solutions for the viscous Cahn-Hilliard equation with spatial dimension n≤5. Based on the Schauder type estimates and energy estimates, we establish the global existence of classical solutions.     

一个具有非线性边界条件的抛物系统整体存在性和爆破

In this paper the nonnegative classical solutions of a parabolic system with nonlinear boundary conditions are discussed. The existence and uniqueness of a nonnegative classical solution are proved. And some sufficient conditions to ensure the global existence and nonexistence of nonnegative classical solution to this problem are given.     

GLOBAL SMOOTH SOLUTIONS TO CAUCHY PROBLEMS FOR MULTIDIMENSIONAL QUASILINEAR HYPERBOLIC SYSTEMS

§ 1. Introduction It is well-known that the smooth solutions to Cauchy problems for quasilinear hyperbolic systems, generally speaking, exist only locally in time and will occur singularities in finite time, even if the initial data are sufficiently smooch and small (see [1—2]). Therefore, the interesting problem is that what conditions can ensure the global existence of classical solutions for quasilinear hyperbolic systems Up to now, the most results on global existence of classical solutions are     

ON THE EXISTENCE AND UNIQUENESS OF CLASSICAL GLOBAL SOLUTION TO A TYPE OF B0USSINESQ EQUATIONS

The existence and uniqueness of classical global solutions to a type of Boussinesq equations withinitial and boundary values are studied in this paper. The existence of such solutions is proved bymeans of compactness theorem and Schauders fixed point theorem, and its uniqueness by the so calledenergy method.     

A Note on the Viscous Cahn-Hilliard Equation

In this note, we study the global existence of classical solutions for the viscous Cahn-Hilliard equation with spatial dimension n ≤ 5. Based on the Schauder type estimates and energy estimates, we establish the global existence of classical solutions.     

Time-periodic Solution to a Nonlinear Parabolic Type Equation of Higher Order

In this paper, the existence and uniqueness of time-periodic generalized solutions and time-periodic classical solutions to a class of parabolic type equation of higher order are proved by Galerkin method.     

Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy

In this paper, we study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with constant multiple and weakly linearly degenerate characteristic fields. Based on the existence of global classical solution proved by Zhou Yi et al., we show that, when t tends to infinity, the solution approaches a combination of C1 travelling wave solutions, provided that the total variation and the L1 norm of initial data are sufficiently small.     

Global Existence of Classical Solutions for Some Oldroyd-B Model via the Incompressible Limit

In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.     

On Korn’s Inequality

Two models based on the hydrostatic primitive equations are proposed. The first model is the primitive equations with partial viscosity only, and is oriented towards large-scale wave structures in the ocean and atmosphere. The second model is the viscous primitive equations with spectral eddy viscosity, and is oriented towards turbulent geophysical flows. For both models, the existence and uniqueness of global strong solutions are established. For the second model, the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.     

Global Existence and Asymptotic Behavior for the 3D Compressible Non–isentropic Euler Equations with Damping

We investigate the global existence and asymptotic behavior of classical solutions for the 3D compressible non-isentropic damped Euler equations on a periodic domain. The global existence and uniqueness of classical solutions are obtained when the initial data is near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.     

The 3D Non-isentropic Compressible Euler Equations with Damping in a Bounded Domain

The authors investigate the global existence and asymptotic behavior of classical solutions to the 3D non-isentropic compressible Euler equations with damping on a bounded domain with slip boundary condition. The global existence and uniqueness of classical solutions are obtained when the initial data are near an equilibrium. Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods.     

具浓度相关迁移率的对流扩散Cahn-Hilliard方程

In this paper, we study the global existence of classical solutions for the convective-diffusive Cahn-Hilliard equation with concentration dependent mobility. Based on the Schauder type estimates, we establish the global existence of classicalsolutions.     

On global classical solutions of the three dimensional non-relativistic Vlasov–Darwin System

We are concerned with the global well-posedness of the non-relativistic Vlasov–Darwin system with generalized variables approach in three dimensions. We obtain the global existence and uniqueness of classical solutions for the perturbation of global solutions with specified decay conditions. And generalizing the result of the quasi-spherical-symmetry case, we prove the existence and uniqueness of the global classical solution of the system when initial data sufficiently closes to a fixed spherically symmetric function. Moreover, we obtain asymptotic behavior for the Darwin potentials in both cases.     

CLASSICAL SOLUTIONS OF GENERAL GINZBURG-LANDAU EQUATIONS

In this paper,we prove the existence of global classical solutions to time-dependent Ginzburg-Landau(TDGL) equations.By the properties of Besov and Sobolev spaces,together with the energy method,we establish the global existence and uniqueness of classical solutions to the initial boundary value problem for time-dependent Ginzburg-Landau equations.     

On the solvability of a mathematical model for prion proliferation

We show that a model describing the interaction between normal and infectious prion proteins admits global solutions. More precisely, supposing the involved degradation rates to be bounded, we prove global existence and uniqueness of classical solutions. Based on this existence theory, we provide sufficient conditions for the existence of global weak solutions in the case of unbounded splitting rates. Moreover, we prove global stability of the disease-free steady state.     

一类Boussinesq型方程整体解的存在性和唯一性

证明一类6阶Boussinesq型方程Cauchy问题整体广义解和整体古典解的存在性和唯一性,给出解在有限时刻发生爆破的充分条件.     

Global classical solutions of viscous liquid–gas two‐phase flow model

In this paper, we consider the global existence and uniqueness of the classical solutions for the three‐dimensional where the existence of global classical solutions to the compressible Navier–Stokes equations was obtained by using the continuity methods under the assumption that the initial energy is sufficiently small. Copyright © 2012 John Wiley & Sons, Ltd.     

On the convective Cahn-Hilliard equation with degenerate mobility

In this paper, we study the existence of weak solutions for the convective Cahn-Hilliard equation with degenerate mobility. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed.     

Kuramoto-Sivashinsky方程解的定性分析

研究Kuramoto-Sivashinsky方程的两种初边值问题,运用Galerkin方法给出一系列先验估计结果,得到广义解和古典解的存在唯一性、正则性及某些条件下的渐近性质。