NON-UNIFORM DEPENDENCE ON INITIAL DATA FOR THE MODIFIED CAMASSA-HOLM EQUATION ON THE LINE

In this paper, we study the Cauchy problem for the modified Camassa-Holm equation mt + umx + 2ux m = 0, m =(1- δ2x)2u,u(x, 0) = u0(x) ∈ Hs(R), x ∈ R, t 0,and show that the solution map is not uniformly continuous in Sobolev spaces Hs(R) for s 7/2. Compared with the periodic problem, the non-periodic problem is more difficult,e.g., it depends on the conservation law. Our proof is based on the estimates for the actual solutions and the approximate solutions, which consist of a low frequency and a high frequency part.     

EXISTENCE RESULT FOR A CLASS OF N-LAPLACIAN EQUATIONS INVOLVING CRITICAL GROWTH

In this paper, we consider a class of N-Laplacian equations involving critical growth{-?_N u = λ|u|~(N-2) u + f(x, u), x ∈ ?,u ∈ W_0~(1,N)(?), u(x) ≥ 0, x ∈ ?,where ? is a bounded domain with smooth boundary in R~N(N 2), f(x, u) is of critical growth. Based on the Trudinger-Moser inequality and a nonstandard linking theorem introduced by Degiovanni and Lancelotti, we prove the existence of a nontrivial solution for any λ λ_1, λ = λ_?(? = 2, 3, ···), and λ_? is the eigenvalues of the operator(-?_N, W_0~(1,N)(?)),which is defined by the Z_2-cohomological index.     

On Reducibility of Beam Equation with Quasi-p erio dic Forcing Potential

In this paper, the Dirichlet boundary value problems of the nonlinear beam equation u_(tt) + ?_u~2 + αu + ∈Φ(t)(u + u~3) = 0, α 0 in the dimension one is considered, where u(t, x) and Φ(t) are analytic quasi-periodic functions in t, and∈ is a small positive real-number parameter. It is proved that the above equation admits a small-amplitude quasi-periodic solution. The proof is based on an infinite dimensional KAM iteration procedure.     

DECAY ESTIMATES OF PLANAR STATIONARY WAVES FOR DAMED WAVE EQUATIONS WITH NONLINEAR CONVECTION IN MULTI-DIMENSIONAL HALF SPACE

This paper is concerned with the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space R n + : u tt u + u t + divf (u) = 0, t > 0, x = (x 1 , x ′ ) ∈ R n + := R + × R n 1 , u(0, x) = u 0 (x) → u + , as x 1 → + ∞ , u t (0, x) = u 1 (x), u(t, 0, x ′ ) = u b , x ′ = (x 2 , x 3 , ··· , x n ) ∈ R n 1 . (I) For the non-degenerate case f ′ 1 (u + ) < 0, it was shown in [10] that the above initialboundary value problem (I) admits a unique global solution u(t, x) which converges to the corresponding planar stationary wave φ(x 1 ) uniformly in x 1 ∈ R + as time tends to infinity provided that the initial perturbation and/or the strength of the stationary wave are sufficiently small. And in [10] Ueda, Nakamura, and Kawashima proved the algebraic decay estimates of the tangential derivatives of the solution u(t, x) for t → + ∞ by using the space-time weighted energy method initiated by Kawashima and Matsumura [5] and improved by Nishihkawa [7]. Moreover, by using the same weighted energy method, an additional algebraic convergence rate in the normal direction was obtained by assuming that the initial perturbation decays algebraically. We note, however, that the analysis in [10] relies heavily on the assumption that f ′ (u) < 0. The main purpose of this paper isdevoted to discussing the case of f ′ 1 (u b ) ≥ 0 and we show that similar results still hold for such a case. Our analysis is based on some delicate energy estimates.     

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f’(u) > 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.     

Dimension of divergence sets for dispersive equation

Consider the generalized dispersive equation defined by{iδtu+Ф(√-△)u=0,(x,t)∈R^n×R,u(x,0)=f(x),F∈F(R^n),(*)whereФ(√-△)is a pseudo-differential operator with symbolФ(|ζ|).In the present paper,assuming thatФsatisfies suitable growth conditions and the initial data in H^s(R^n),we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations(*)fails.These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.     

REGULARIZATION OF AN ILL-POSED HYPERBOLIC PROBLEM AND THE UNIQUENESS OF THE SOLUTION BY A WAVELET GALERKIN METHOD

We consider the problem K(x)u xx = u tt , 0 < x < 1, t ≥ 0, with the boundary condition u(0,t) = g(t) ∈ L 2 (R) and u x (0, t ) = 0, where K(x) is continuous and 0 < α≤ K (x) < +∞. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on g. Considering the existence of a solution u(x, ) ∈ H 2 (R) and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore we prove the uniqueness of the solution for this problem.     

On a Class of Infinite-Dimensional Hamiltonian Systems with Asymptotically Periodic Nonlinearities

The authors study the existence of homoclinic type solutions for the following system of diffusion equations on R × RN:{■tu-xu + b ·▽xu + au + V(t,x)v = Hv(t,x,u,v),-■tv-xv-b·▽xv + av + V(t,x)u = Hu(t,x,u,v),where z =(u,v):R × RN → Rm × Rm,a > 0,b =(b1,···,bN) is a constant vector and V ∈ C(R × RN,R),H ∈ C1(R × RN × R2m,R).Under suitable conditions on V(t,x) and the nonlinearity for H(t,x,z),at least one non-stationary homoclinic solution with least energy is obtained.     

THE EXISTENCE OF NONTRIVIAL SOLUTIONS TO A SEMILINEAR ELLIPTIC SYSTEM ON RN WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

In this paper, we prove the existence of at least one positive solution pair (u, v) ∈ H 1 (R N ) × H 1 (R N ) to the following semilinear elliptic system{-u + u = f(x, v), x ∈RN ,-v + v = g(x,u), x ∈ R N ,(0.1) by using a linking theorem and the concentration-compactness principle. The main con-ditions we imposed on the nonnegative functions f, g ∈ C 0 (R N × R 1 ) are that, f (x, t) and g(x, t) are superlinear at t = 0 as well as at t = +∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245(2008), 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem{-u + u = f(x, u), x ∈Ω,u ∈H10(Ω)where ΩRN is bounded and a result of Li and Yang [G. Li and J. Yang: Communications in P.D.E. Vol. 29(2004) Nos.5 6.pp.925–954, 2004] concerning (0.1) when f and g are asymptotically linear.     

MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRDINGER-POISSON EQUATIONS WITH SIGN-CHANGING POTENTIAL

In this article, we study the following nonhomogeneous Schr¨odinger-Poisson equations -?u + λV(x)u + K(x)φu = f(x, u) + g(x), x ∈ R~3,?-?φ = K(x)u~2, x ∈ R~3,where λ 0 is a parameter. Under some suitable assumptions on V, K, f and g, the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory. In particular, the potential V is allowed to be signchanging.     

Quasi-periodic solutions with prescribed frequency in a nonlinear Schrdinger equation

In this paper, one-dimensional (1D) nonlinear Schrdinger equation iut-uxx + Mσ u + f ( | u | 2 )u = 0, t, x ∈ R , subject to periodic boundary conditions is considered, where the nonlinearity f is a real analytic function near u = 0 with f (0) = 0, f (0) = 0, and the Floquet multiplier Mσ is defined as Mσe inx = σne inx , with σn = σ, when n 0, otherwise, σn = 0. It is proved that for each given 0 σ 1, and each given integer b 1, the above equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with b-dimensional Diophantine frequencies, corresponding to b-dimensional invariant tori of an associated infinite-dimensional Hamiltonian system. Moreover, these b-dimensional Diophantine frequencies are the small dilation of a prescribed Diophantine vector. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method.     

一阶离散周期问题的多解性

Let T 1 be an integer, T = {0, 1, 2,..., T- 1}. This paper is concerned with the existence of periodic solutions of the discrete first-order periodic boundary value problems△u(t)- a(t)u(t) = λu(t) + f(u(t- τ(t)))- h(t), t ∈ T,u(0) = u(T),where △u(t) = u(t + 1)- u(t), a : T → R and satisfies∏T-1t=0(1 + a(t)) = 1, τ : T → Z t- τ(t) ∈ T for t ∈ T, f : R → R is continuous and satisfies Landesman-Lazer type condition and h : T → R. The proofs of our main results are based on the Rabinowitz's global bifurcation theorem and Leray-Schauder degree.     

CHINESE ANNALS OF MATHEMATICS Vol.6 Ser.A No.4 CONTENTS AND ABSTRACTS

Approximation of Nonlinear Dirichlet Problem by Finite Element MethodsLi Likang(李立康)The author obtains approximate solution of the nonlinear Dirichlet problem-▽·(a(x,u)▽u)=f(x),x∈Ω,u(x)=g(x),x∈Γby finite element method.In this paper finite element spaces V_h is defined by the followingV_h={v_h|v_h∈C~0(Ω),v_h is a linear function on each K_i},     

MULTIPLICITY RESULTS FOR FOURTH ORDER ELLIPTIC EQUATIONS OF KIRCHHOFF-TYPE

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {△2u- a+b∫R3|▽u|2dx △u+V(x)u=f(x, u), x∈ R3,u∈H2(R3),wherea, b 0 are constants and the primitive of the nonlinearityfis of superlinear growth near infinity inuand is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the AmbrosettiRabinowitz type condition.     

A Note on First Order Differential Equation

In this note, a theorem and its three corollaries on solution of the first order ordinary differential equation are given. Theorem Suppose that b, F∈C,a∈C~1,b(y)≠0. If a(t) and b(t) satisfy the equality a′(t)b(t)=1, (1) then the first order differential equation y′=b(y)F(x,a(y)) (2) has a solution y=f(u) (3) where u=u(x) is a solution of the equatien     

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u0∈ L1(Rn) ∩ L2(Rn) and the external force f ∈ L1(Rn× R+) ∩ L1(R+, L2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t 0, where the dimension n ≥ 2, C 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.     

GLOBAL EXISTENCE AND BLOW-UP OF SOLUTIONS FOR GENERALIZED POCHHAMMER-CHREE EQUATIONS

In this article,we study the initial boundary value problem of generalized Pochhammer-Chree equation u_(tt)-u_(xx)-u_(xxt)-u_(xxtt)=f(u) xx,x ∈Ω,t 0,u(x,0) = u0(x),u t(x,0)=u1(x),x ∈Ω,u(0,t) = u(1,t) = 0,t≥0,where Ω=(0,1).First,we obtain the existence of local W k,p solutions.Then,we prove that,if f(s) ∈ΩC k+1(R) is nondecreasing,f(0) = 0 and |f(u)|≤C1|u| u 0 f(s)ds+C2,u 0(x),u 1(x) ∈ΩW k,p(Ω) ∩ W 1,p 0(Ω),k ≥ 1,1 p ≤∞,then for any T 0 the problem admits a unique solution u(x,t) ∈ W 2,∞(0,T;W k,p(Ω) ∩ W 1,p 0(Ω)).Finally,the finite time blow-up of solutions and global W k,p solution of generalized IMBq equations are discussed.     

On the existence of solutions for an elliptic system of equations with arbitrary order growth

Let BR be the ball centered at the origin with radius R in RN ( N ≥2). In this paper we study the existence of solution for the following elliptic systemu -△u+λu=p/(p + q)κ(| x |)) u(p-1)vq1,x ∈BR1,-△u+λu=p/(p + q)κ(| x |)) upv(q-1)1,x ∈BR1,u > 01,v > 01,x ∈ BR1,(u)/(v)=01,(v)/(v)=01,x ∈BRwhereλ > 0 , μ > 0 p ≥ 2, q ≥ 2,ν is the unit outward normal at the boundary BR . Under certainassumptions on κ ( | x | ), using variational methods, we prove the existence of a positive and radially increasing solution for this problem without growth conditions on the nonlinearity.     

Global blow-up for a heat system with localized sources and absorptions

In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic systemut=Δu vp(x0, t)-aur, x∈Ω, t＞0,vt=Δv uq(x0,t)-bvs, x∈Ω, t＞0subject to homogeneous Dirichlet conditions and nonnegative initial data, where x0 ∈Ω is a fixed point, p, q, r, s ≥ 1 and a, b ＞ 0 are constants. In the situation when nonnegative solution (u, v) of the above problem blows up in finite time, it is showed that the blow-up is global and this differs from the local sources case. Moreover, for the special case r = s = 1,are obtained uniformly on compact subsets of Ω, where T* is the blow-up time.     

Notes on Automorphisms of Prime Rings

Let R be a prime ring, L a noncentral Lie ideal and σ a nontrivial automorphism of R such that usσ(u)ut= 0 for all u ∈ L, where s, t are fixed non-negative integers. If either char R s + t or char R = 0, then R satisfies s4, the standard identity in four variables. We also examine the identity(σ([x, y])-[x, y])n=0 for all x, y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. If either char R n or char R = 0, then R is commutative.