一类双重退化抛物方程局部解的存在性

This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate p- Laplace equation(p＞2) u_t-div(b(x,t,u)|▽u|~(p-2)▽u)=f(x,u,t). The initial-boundary value problem in a bounded domain of R~N is considered under mixed boundary conditions.The existence of local-in-time weak solutions is obtained.     

Quenching Phenomenon for a Parabolic MEMS Equation

This paper deals with the electrostatic MEMS-device parabolic equation u_t-?u =λf(x)/(1-u)~p in a bounded domain ? of R~N,with Dirichlet boundary condition,an initial condition u0(x) ∈ [0,1) and a nonnegative profile f,where λ 0,p 1.The study is motivated by a simplified micro-electromechanical system(MEMS for short) device model.In this paper,the author first gives an asymptotic behavior of the quenching time T*for the solution u to the parabolic problem with zero initial data.Secondly,the author investigates when the solution u will quench,with general λ,u0(x).Finally,a global existence in the MEMS modeling is shown.     

Homogenization of a nonlinear degenerate parabolic differential equation

This paper is devoted to the homogenization of a nonlinear degenerate parabolic problem ɑtu∈-div(D(x/∈, u∈,▽u∈)+ K(x/∈, u∈))= f(x) with Dirichlet boundary condition. Here the operator D(y, s,s) is periodic in y and degenerated in ▽s. In the paper, under the two-scale convergence theory, we obtain the limit equation as ∈→ 0 and also prove the corrector results of ▽u∈ to strong convergence.     

LOCAL STABILITY OF TRAVELLING FRONTS FOR A DAMPED WAVE EQUATION

The paper is concerned with the long-time behaviour of the travelling fronts of the damped wave equation αutt+ut=uxx V’(u) on R.The long-time asymptotics of the solutions of this equation are quite similar to those of the corresponding reaction-diffusion equation ut=uxxV’(u).Whereas a lot is known about the local stability of travelling fronts in parabolic systems,for the hyperbolic equations it is only briefly discussed when the potential V is of bistable type.However,for the combustion or monostable type of V,the problem is much more complicated.In this paper,a local stability result for travelling fronts of this equation with combustion type of nonlinearity is established.And then,the result is extended to the damped wave equation with a case of monostable pushed front.     

VISCOSITY METHOD OF A NON－HOMOGENEOUS BURGERS EQUATION

Abstract In [1], Ding et al. studied the nonhomogeneous Burgers equation ut uux = μuxx 4x.(1.1) This paper will prove that when μ → 0 the solution of (1.1) will approach the generalized solution of ut uux = 4x.(1.2) The authors notice that the equation (1.2) is beyond the scope of investigations by Oleinik O. in [2]. The solutions here are unbounded in general. The paper also studies the δ-wave phenomenon when (1.2) is jointed with some other equation.     

Convergence to a Single Wave in the Fisher-KPP Equation(Dedicated to Haim Brezis,with admiration and respect)

The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t-(3/2 )log t+x∞, the solution of the equation converges as t → +∞ to a translate of the traveling wave corresponding to the minimal speed c_* = 2. The constant x∞ depends on the initial condition u(0, x). The proof is elaborate, and based on probabilistic arguments.The purpose of this paper is to provide a simple proof based on PDE arguments.     

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.     

Convergence to a Single Wave in the Fisher-KPP Equation

The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation,with an initial condition that is a compact perturbation of a step function.A well-known result of Bramson states that,in the reference frame moving as 2t-(3/2) log t+x∞,the solution of the equation converges as t-→ +o∞ to a translate of the traveling wave corresponding to the minimal speed c* =2.The constant x∞ depends on the initial condition u(0,x).The proof is elaborate,and based on probabilistic arguments.The purpose of this paper is to provide a simple proof based on PDE arguments.     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.     

Controllability of the Heat Equation with a Control Acting on a Measurable Set

The paper deals with the controllability of a heat equation.It is well-known that the heat equation yt y = uχE in(0,T)×Ω with homogeneous Dirichlet boundary conditions is null controllable for any T > 0 and any open nonempty subset E of Ω.In this note,the author studies the case that E is an arbitrary measurable set with positive measure.     

Long time dynamics of the 3D radial NLS with the combined terms

In this paper,we study the scattering and blow-up dichotomy result of the radial solution to nonlinear Schrodinger equation(NLS) with the combined terms iu_t+△u=-|u|~4u+|4|~(p-1)u,1+4/3p5 in energy space H~1(R~3).The threshold energy is the energy of the ground state W of the focusing,energy critical NLS,which means that the subcritical perturbation does not affect the determination of threshold,but affects the scattering and blow-up dichotomy result with subcritical threshold energy.This extends algebraic perturbation in a previous work of Miao,Xu and Zhao[Comm.Math.Phys.,318,767-808(2013)]to all mass supercritical,energy subcritical perturbation.     

Nonexistence Results for the Korteweg–de Vries and Kadomtsev–Petviashvili Equations

We study characteristic Cauchy problems for the Korteweg–de Vries (KdV) equation u_t = uu_x + u_xxx , and the Kadomtsev–Petviashvili (KP) equation u_yy =( u_xxx + uu_x + u_t ) _x with holomorphic initial data possessing non-negative Taylor coefficients around the origin. For the KdV equation with initial value u (0,  x )= u ₀( x ), we show that there is no solution holomorphic in any neighborhood of ( t ,  x )=(0, 0) in C² unless u ₀( x )= a ₀+ a ₁ x . This also furnishes a nonexistence result for a class of y -independent solutions of the KP equation. We extend this to y -dependent cases by considering initial values given at y =0, u ( t ,  x , 0)= u ₀( x ,  t ), u_y ( t ,  x , 0)= u ₁( x ,  t ), where the Taylor coefficients of u ₀ and u ₁ around t =0, x =0 are assumed non-negative. We prove that there is no holomorphic solution around the origin in C³, unless u ₀ and u ₁ are polynomials of degree 2 or lower. MSC 2000: 35Q53, 35B30, 35C10.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

Markoff数性质的研究

1913年,Frobenius对Markoff方程a~2+b~2+c~2=3abc提了一个著名猜想:若abc是Markoff方程的正整数解,则a,b的值由最大的数c唯一确定.此猜想仍未得到解决.本文证明了:任给定正整数s_i,t_i,w,u,v=1,2),若(a_i,b_i,c)是Markoff方程的两组不同的正整数解,且a_ib_ic(i=1,2),则gcd(s_1a_1+s_2a_2+t_1b_1+t_2b_2+w,uc+v)≤K(uc+v)~(13/14),其中K是仅与s_i,t_i,w,u,v(i=1,2)有关的正数.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Recursion Operators for a Class of Integrable Third-Order Evolution Equations

We consider   u_t = u ^α u_xxx + n ( u ) u_xu_xx + m ( u ) u ³ _x + r ( u ) u_xx + p ( u ) u ² _x + q ( u ) u_x + s ( u )  with  α= 0  and  α= 3  , for those functional forms of   m , n , p , q , r , s   for which the equation is integrable in the sense of an infinite number of Lie-Bäcklund symmetries. Recursion operators which are x - and t -independent that generate these infinite sets of (local) symmetries are obtained for the equations. A combination of potential forms, hodograph transformations, and x -generalized hodograph transformations are applied to the obtained equations.     

广义Davey-Stewartson系统驻波的强不稳定性

本文研究方程驻波的强不稳定性iu_t+△u+a|u|~(p-1)u+E_1(|u|~2)u=0,t≥0,x∈R~n,其中a0,1p(n+2)/(n+2)~+,n∈{2,3}.当1+4/n≤pn+2/(n-2)~+)时,文[Sharp threshold of global existence and instability of standing wave for a Davey-Stewartson system,Commun.Math.Phys.,2008,283:93-125]在驻波的频率满足一定假设条件下,证明了此方程驻波的强不稳定性.本文去掉这个假设,得到相同的结论.     

Shooting Method for Nonlinear Singularly Perturbed Boundary-Value Problems

Asymptotic formulas, as  ɛ→ 0⁺  , are derived for the solutions of the nonlinear differential equation  ɛ u" + Q ( u ) = 0  with boundary conditions   u (-1) = u (1) = 0  or   u '(-1) = u '(1) = 0  . The nonlinear term Q ( u ) behaves like a cubic; it vanishes at   s _-, 0, s ₊  and nowhere else in  [ s _-, s ₊]  , where   s _- < 0 < s ₊  . Furthermore,   Q '( s _±) < 0, Q '(0) > 0  and the integral of Q on the interval [ s _-, s ₊] is zero. Solutions to these boundary-value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.     

Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.     

Conditional Lie Bäcklund Symmetries and Sign-Invariants to Quasi-Linear Diffusion Equations

Consider the 1+1-dimensional quasi-linear diffusion equations with convection and source term u _t =[ u ^m ( u _x ) ⁿ ] _x + P ( u ) u _x + Q ( u ) , where P and Q are both smooth functions. We obtain conditions under which the equations admit the Lie Bäcklund conditional symmetry with characteristic η= u _xx + H ( u ) u ² _x + G ( u )( u _x )^{2− n} + F ( u ) u ^{1− n} _x and the Hamilton–Jacobi sign-invariant J = u _t + A ( u ) u ^{n +1} _x + B ( u ) u _x + C ( u ) which preserves both signs, ≥0 and ≤0, on the solution manifold. As a result, the corresponding solutions associated with the symmetries are obtained explicitly, or they are reduced to solve two-dimensional dynamical systems.