EXISTENCE OF PERIODIC SOLUTIONS FOR 3-D COMPLEX GINZBERG-LANDAU EQUATION

In this paper, the authors consider complex Ginzburg-Landau equation(CGL) in three spatial dimensions ut=ρu+(1+iγ）△u-(1+iμ）|u|^2σu+f,where u is an unknown complex-value function defined in 3+ 1 dimensional space-time R^3+1，△ is a Laplacian in R^3, ρ > 0, γ μ are real parameters, Ω∈R^3 is a bounded domain. By using the method of Galeerkin and Faedo-Schauder fix point theorem we prove the existence of approximate solution uN of the problem. By establishing the uniform boundedness of the norm ||uN|| and the standard compactness arguments, the convergence of the approximate solutions is considered. Moreover, the existence of the periodic solution is obtained.     

BUBBLES OF LANDAU-LIFSHITZ EQUATIONS WITH APPLIED FIELDS

In this paper, we discuss the Landau-Lifshitz equations with applied magnetic fields. The equations describing the bubbles in the ferromagnets and the behaviors of the solutions near the singularities are given. We found that the applied fields do not affect the bubbles and we have the same conclusions as in reference [1].     

THE CLASSICAL SOLUTIONS TO A CAUCHY PROBLEM OF PARABOLIC TYPE COUPLED WITH OPERATORS

We consider the equation ut=Tf[B(x,t,Du,Φu)D^2u]+F(x,t,u,Du,Φu,Ψu) where Φand Ψ are vector-valued mappings.We obtain the existence anduniqueness of classical solution to the equation for a ε-periodic initial data.The problem is naturally arisen from image denoising.     

LOCAL WELL-POSEDNESS OF INTERACTION EQUATIONS FOR SHORT AND LONG DISPERSIVE WAVES

The well-posedness of the Cauchy problem for the system{iδtu+δx^2u=uv+|u|^2u,t,x∈IR，δtv+δxHδxv=δx|u|^2,u(0,x)=u0(x),v(0,x)=v0(x),is considered. It is proved that there exists a unique local solution (u(x,t), v(x,t))∈C（[0,T);H^s)×C([0,T);Hs^-1/2) for any initial data (u0,v0)∈H^s(IR)×H^s-1/2(IR)(s≥1/4) and the solution depends continuously on the initial data.