On low regularity of the Ostrovsky, Stepanyams and Tsimring equation

In this paper we prove, by using the Fourier restriction norm method, that the initial value problem of the Ostrovsky, Stepanyams and Tsimring equation ut+uxxx+η(Hux+Huxxx)+uux=0 (x∈R, t?0), where η>0 and H denotes the usual Hilbert transformation, is locally well-posed in the Sobolev space Hs(R) for any , which improve our former result in Zhao and Cui (2009) [5].     

Local well-posedness for the dispersion generalized periodic KdV equation

In this paper, we set up the local well-posedness of the initial value problem for the dispersion generalized periodic KdV equation: t_∂u+x_∂α|Dx|u=x_∂u², u(0)=φ for α>2, and φ∈Hs(T). And we show that the is a lower endpoint to obtain the bilinear estimates (1.2) and (1.3) which are the crucial steps to obtain the local well-posedness by Picard iteration. The case α=2 was studied in Kenig et al. (1996) [10].     

A model containing both the Camassa-Holm and Degasperis-Procesi equations

A nonlinear dispersive partial differential equation, which includes the famous Camassa-Holm and Degasperis-Procesi equations as special cases, is investigated. Although the H¹-norm of the solutions to the nonlinear model does not remain constants, the existence of its weak solutions in lower order Sobolev space Hs with is established under the assumptions u₀∈Hs and ‖u0xL^∞_‖<∞. The local well-posedness of solutions for the equation in the Sobolev space Hs(R) with is also developed.     

Exponential stability of wave equations with potential and indefinite damping

First, we consider the linear wave equation utt−uxx+a(x)ut+b(x)u=0 on a bounded interval (0,L)⊂R. The damping function a is allowed to change its sign. If is positive and the spectrum of the operator (_∂xx−b) is negative, exponential stability is proved for small . Explicit estimates of the decay rate ω are given in terms of and the largest eigenvalue of (_∂xx−b). Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equation utt−σx(ux)+a(x)ut+b(x)u=0, if, additionally, the negative part of a is small enough compared with ω.     

Global solutions and blow-up phenomena to a shallow water equation

A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u₀∈Hs () and u₀∈L¹(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x)∈C([0,∞);Hs(R))∩C¹([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Convergence to strong nonlinear diffusion waves for solutions to p-system with damping on quadrant

In this paper, we consider the so-called p-system with linear damping on quadrant. We show that for a certain class of given large initial data (v₀(x),u₀(x)), the corresponding initial-boundary value problem admits a unique global smooth solution (v(x,t),u(x,t)) and such a solution tends time-asymptotically, at the Lp (2?p?∞) optimal decay rates, to the corresponding nonlinear diffusion wave which satisfies (1.9) provided the corresponding prescribed initial error function (V₀(x),U₀(x)) lies in (H³(R⁺)∩L¹(R⁺))×(H²(R⁺)∩L¹(R⁺)).     

Global existence for 2D nonlinear Schrödinger equations via high-low frequency decomposition method

We study global existence of solutions for the Cauchy problem of the nonlinear Schrödinger equation iut+Δu=|u|^2mu in the 2 dimension case, where m is a positive integer, m?2. Using the high-low frequency decomposition method, we prove that if then for any initial value φ∈Hs(R²), the Cauchy problem has a global solution in C(R,Hs(R²)), and it can be split into u(t)=eitΔφ+y(t), with y∈C(R,H¹(R²)) satisfying , where ? is an arbitrary sufficiently small positive number.     

Dirichlet inhomogeneous boundary value problem for the n+1 complex Ginzburg-Landau equation

We study the following complex Ginzburg-Landau equation with cubic nonlinearity on for under initial and Dirichlet boundary conditions u(x,0)=h(x) for x∈Ω, u(x,t)=Q(x,t) on ∂Ω where h,Q are given smooth functions. Under suitable conditions, we prove the existence of a global solution in H¹. Further, this solution approaches to the solution of the NLS limit under identical initial and boundary data as a,b→0⁺.     

Standing waves for supercritical nonlinear Schrödinger equations

Let V(x) be a non-negative, bounded potential in RN, N?3 and p supercritical, . We look for positive solutions of the standing-wave nonlinear Schrödinger equation Δu−V(x)u+up=0 in RN, with u(x)→0 as |x|→+∞. We prove that if V(x)=o(⁻²|x|) as |x|→+∞, then for N?4 and this problem admits a continuum of solutions. If in addition we have, for instance, V(x)=O(|x|^−μ) with μ>N, then this result still holds provided that N?3 and . Other conditions for solvability, involving behavior of V at ∞, are also provided.     

Well-posedness and blow-up phenomena for a higher order shallow water equation

This paper deals with the Cauchy problem for a higher order shallow water equation yt+auxy+buyx=0, where and k=2. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R) with s?7/2 is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered.     

Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control

We study the Schrödinger equation i∂_tu+Δu+V₀u+V₁u=0 on R³×(0,T), where V₀(x,t)=|x-a(t)|^-1, with a∈W^2,1(0,T;R³), is a coulombian potential, singular at finite distance, and V₁ is an electric potential, possibly unbounded. The initial condition u₀∈H²(R³) is such that . The potential V₁ is also real valued and may depend on space and time variables. We prove that if V₁ is regular enough and at most quadratic at infinity, this problem is well-posed and the regularity of the initial data is conserved for the solution. We also give an application to the bilinear optimal control of the solution through the electric potential.     

Heat flow, Brownian motion and Newtonian capacity

Let K be a compact, non-polar set in Rm(m?3) and let u be the unique weak solution of on Rm\K×(0,∞),u(x;0)=0 on Rm\K and u(x;t)=1 for all x on the boundary of K and for all t>0. The asymptotic behaviour of u(x;t) as t tends to infinity is obtained up to order O(t−m/2).     

Ground state solutions for a semilinear elliptic problem with a convection term

By a sub-supersolution method and a perturbed argument, we improve the earlier results concerning the existence of ground state solutions to a semilinear elliptic problem −Δu+p(x)q|∇u|=f(x,u), u>0, x∈RN, , where q∈(1,2], for some α∈(0,1), p(x)?0, ∀x∈RN, and f:RN×(0,∞)→[0,∞) is a locally Hölder continuous function which may be singular at zero.     

Ill-posedness of Kawahara equation and Kaup-Kupershmidt equation

In this paper we consider the Cauchy problems for the Kawahara equation and the Kaup-Kupershmidt equation. By using the general well-posedness principle introduced by I. Bejenaru and T. Tao (2006) [1], we prove that the Kawahara equation is ill-posed for the initial data in Hs(R) with and the Kaup-Kupershmidt equation is ill-posed for the initial data in Hs(R) with .     

Critical exponents and blow-up rate for a nonlinear diffusion equation with logarithmic boundary flux

In this paper, we establish the critical global existence exponent and the critical Fujita exponent for the nonlinear diffusion equation u_t=(log^σ(1+u)u_x)_x, in R₊×(0,+∞), subject to a logarithmic boundary flux , furthermore give the blow-up rate for the nonglobal solutions.     

Well-posedness of the Cauchy problem of Ostrovsky equation in anisotropic Sobolev spaces

We study the Cauchy problem of the Ostrovsky equation , with βγ<0. By establishing a bilinear estimate on the anisotropic Bourgain space Xs,ω,b, we prove that the Cauchy problem of this equation is locally well-posed in the anisotropic Sobolev space H(s,ω)(R) for any and some . Using this result and conservation laws of this equation, we also prove that the Cauchy problem of this equation is globally well-posed in H(s,ω)(R) for s?0.     

Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).