In this paper, a spectral method to analyze the generalized Benjamin Bona Mahony equations is used. The existence and uniqueness of global smooth solution of these equations are proved. The large time error estimati...In this paper, a spectral method to analyze the generalized Benjamin Bona Mahony equations is used. The existence and uniqueness of global smooth solution of these equations are proved. The large time error estimation between the spectral approximate solution and the exact solution is obtained.展开更多
The long time uniform stability of solutions to the initial value problems for 2 dimen sional Magnetohydrodynamics equations is studied. The decay estimates are given.
The Boussinesq approximation ,where the viscosity depends polynomially on the shear rate ,finds more and more frequent use in geological practice,In this paper ,we consider the periodic initial value problem and inita...The Boussinesq approximation ,where the viscosity depends polynomially on the shear rate ,finds more and more frequent use in geological practice,In this paper ,we consider the periodic initial value problem and inital value problem for this modified Boussinesq approximation with the viscous part of the stress tensor T^v=τ(e)-μ1△e,where the nonlinear function τ(e) satisfies τij(e)eij≥C|e|^p or τij(e)eij ≥C(|e|^2+|e|^p).The existence,uniqueness and regulartiy of the weak solution is proved for p> 2n/(n+2).展开更多
In this paper, we consider the asymptotic behavior of solutions for a class of nonclassical diffusion equation. We show the squeezing property and the existence of exponential attractor for this equation. We also make...In this paper, we consider the asymptotic behavior of solutions for a class of nonclassical diffusion equation. We show the squeezing property and the existence of exponential attractor for this equation. We also make the estimates on its fractal dimension and exponential attraction.展开更多
In this paper ,the existence of homoclinic orbits,for a perturbed cubic-quintic nonlinear Schroedinger equation with even periodic boundary conditions ,under the geralized parameters conditions is established.More spe...In this paper ,the existence of homoclinic orbits,for a perturbed cubic-quintic nonlinear Schroedinger equation with even periodic boundary conditions ,under the geralized parameters conditions is established.More specifically,we combine geometric singular perturbation theory,with ,Melnikov analysis and integrable theory to prove the persistences of homoclinic orbits.展开更多
This paper deals with the asymptotic behaviour of solutions for thegeneralized symmetric regularized long wave equations with dissipation term. We first show theexistence of global weak attractors for the periodic ini...This paper deals with the asymptotic behaviour of solutions for thegeneralized symmetric regularized long wave equations with dissipation term. We first show theexistence of global weak attractors for the periodic initial value problem of this equations in H^2x H^1. And then by an energy equation and an idea of Ghidaglia and Guo, we conclude that the globalweak attractor is actually the global strong attractor for S(t) in H^2 (Ω) x H^1 (Ω). The finitedimensionality of the global attractor is also established.展开更多
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 dat...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.展开更多
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...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.展开更多
The existence, partial regularity and uniqueness of weak solution to the initial boundary value problem for the unsaturated Landau-Lifschitz systems are given.
In this paper,the perturbed higher-order NLS equation with periodic boundary condition is considered. The existence of the homoclinic orbits for the truncation equation is established by Melnikov analysis and geometri...In this paper,the perturbed higher-order NLS equation with periodic boundary condition is considered. The existence of the homoclinic orbits for the truncation equation is established by Melnikov analysis and geometric singular perturbation theory.展开更多
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 spac...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.展开更多
In this paper,we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions ut=ρu+(1+iγ)△u-1+iμ—|u|^2σu,(1) u(0,x)=u0(x),(2) where u is an unknown complex-value function defined in 3+1 di...In this paper,we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions ut=ρu+(1+iγ)△u-1+iμ—|u|^2σu,(1) u(0,x)=u0(x),(2) 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,We show that the semigroup S(t) associated with the problem(1),(2) satisfies Lipschitz continuity and the squeezing property for the initial-value problem(1),(2) with the initial-value condition belonging to H^2(Ω）,therefore we obtain the existence of exponential attractor.展开更多
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 fo...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].展开更多
文摘In this paper, a spectral method to analyze the generalized Benjamin Bona Mahony equations is used. The existence and uniqueness of global smooth solution of these equations are proved. The large time error estimation between the spectral approximate solution and the exact solution is obtained.
文摘The long time uniform stability of solutions to the initial value problems for 2 dimen sional Magnetohydrodynamics equations is studied. The decay estimates are given.
文摘The Boussinesq approximation ,where the viscosity depends polynomially on the shear rate ,finds more and more frequent use in geological practice,In this paper ,we consider the periodic initial value problem and inital value problem for this modified Boussinesq approximation with the viscous part of the stress tensor T^v=τ(e)-μ1△e,where the nonlinear function τ(e) satisfies τij(e)eij≥C|e|^p or τij(e)eij ≥C(|e|^2+|e|^p).The existence,uniqueness and regulartiy of the weak solution is proved for p> 2n/(n+2).
文摘In this paper, we consider the asymptotic behavior of solutions for a class of nonclassical diffusion equation. We show the squeezing property and the existence of exponential attractor for this equation. We also make the estimates on its fractal dimension and exponential attraction.
文摘In this paper ,the existence of homoclinic orbits,for a perturbed cubic-quintic nonlinear Schroedinger equation with even periodic boundary conditions ,under the geralized parameters conditions is established.More specifically,we combine geometric singular perturbation theory,with ,Melnikov analysis and integrable theory to prove the persistences of homoclinic orbits.
基金This research is supported by the National Natural Science Foundation of China(Grant 10271034).
文摘This paper deals with the asymptotic behaviour of solutions for thegeneralized symmetric regularized long wave equations with dissipation term. We first show theexistence of global weak attractors for the periodic initial value problem of this equations in H^2x H^1. And then by an energy equation and an idea of Ghidaglia and Guo, we conclude that the globalweak attractor is actually the global strong attractor for S(t) in H^2 (Ω) x H^1 (Ω). The finitedimensionality of the global attractor is also established.
文摘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.
文摘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.
基金the National Natural Science Foundation of China!(No. 19971030)
文摘The existence, partial regularity and uniqueness of weak solution to the initial boundary value problem for the unsaturated Landau-Lifschitz systems are given.
文摘In this paper,the perturbed higher-order NLS equation with periodic boundary condition is considered. The existence of the homoclinic orbits for the truncation equation is established by Melnikov analysis and geometric singular perturbation theory.
文摘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.
文摘In this paper,we consider the complex Ginzburg-Landau equation (CGL) in three spatial dimensions ut=ρu+(1+iγ)△u-1+iμ—|u|^2σu,(1) u(0,x)=u0(x),(2) 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,We show that the semigroup S(t) associated with the problem(1),(2) satisfies Lipschitz continuity and the squeezing property for the initial-value problem(1),(2) with the initial-value condition belonging to H^2(Ω）,therefore we obtain the existence of exponential attractor.
文摘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].
