The existence and uniqueness of the local solution for a Camassa-Holm type equation

A shallow water equation of Camassa-Holm type, containing nonlinear dissipative effect, is investigated. Using the techniques of the pseudoparabolic regularization and some prior estimates derived from the equation itself, we establish the existence and uniqueness of its local solution in Sobolev space H^s(R) with . Meanwhile, a new lemma and a sufficient condition which guarantee the existence of solutions of the equation in lower order Sobolev space H^s with are presented.     

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 .     

Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs(R), s?1, and global well-posedness in H¹(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation

     

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.     

Large solutions of semilinear elliptic equations under the Keller-Osserman condition

We consider the equation Δu=p(x)f(u) where p is a nonnegative nontrivial continuous function and f is continuous and nondecreasing on [0,∞), satisfies f(0)=0, f(s)>0 for s>0 and the Keller-Osserman condition where . We establish conditions on the function p that are necessary and sufficient for the existence of positive solutions, bounded and unbounded, of the given equation.     

The chain rule as a functional equation

We consider operators T from C¹(R) to C(R) satisfying the “chain rule”

     

MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR ELLIPTIC EQUATIONS INVOLVING NATURAL GROWTH TERM

In this article, under some conditions on the behaviors of the perturbed function f(x, s) or its primitive F(x,s) =∫so f(x,t)dt near infinity and near zero, a class of asymptotically linear elliptic equations involving natural growth term is studied. By computing the critical group, the existence of three nontrivial solutions is proved.     

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)).     

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.     

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.     

The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation

A generalization of the Camassa-Holm equation, a model for shallow water waves, is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with is established via a limiting procedure. In addition, a sufficient condition for the existence of weak solutions of the equation in lower order Sobolev space Hs with is developed.     

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.     

Linnik-type problems for automorphic L-functions

Let m?2 be an integer, and π an irreducible unitary cuspidal representation for GLm(AQ), whose attached automorphic L-function is denoted by L(s,π). Let be the sequence of coefficients in the Dirichlet series expression of L(s,π) in the half-plane Rs>1. It is proved in this paper that, if π is such that the sequence is real, then there are infinitely many sign changes in the sequence , and the first sign change occurs at some , where Qπ is the conductor of π, and the implied constant depends only on m and ε. This generalizes the previous results for GL₂. A result of the same quality is also established for , the sequence of coefficients in the Dirichlet series expression of in the half-plane Rs>1.     

Sharp local well-posedness for a fifth-order shallow water wave equation

In this paper we prove that the already-established local well-posedness in the range s>−5/4 of the Cauchy problem with an initial Hs(R) data for a fifth-order shallow water wave equation is extendable to s=−5/4 by using the space. This is sharp in the sense that the ill-posedness in the range s<−5/4 of this initial value problem is already known.     

Almost conservation law and global rough solutions to a nonlinear Davey-Stewartson equation

We prove an “almost conservation law” to obtain global-in-time well-posedness for the nonlinear Davey-Stewartson equation in Hs(R²), and .     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

Periodic solutions of Duffing equations with semi-quadratic potential and singularity

In this paper, we deal with the existence of periodic solutions of the second order differential equations x^″+g(x)=p(t) with singularity. We prove that the given equation has at least one periodic solution when g(x) has singularity at origin, satisfies

     

On estimates of the density of Feynman-Kac semigroups of α-stable-like processes

Suppose that α∈(0,2) and that X is an α-stable-like process on Rd. Let F be a function on Rd belonging to the class Jd,α (see Introduction) and be _∑s?tF(Xs−,Xs), t>0, a discontinuous additive functional of X. With neither F nor X being symmetric, under certain conditions, we show that the Feynman-Kac semigroup defined by