A RIEMANN-HILBERT APPROACH TO THE INITIAL-BOUNDARY PROBLEM FOR DERIVATIVE NONLINEAR SCHRODINGER EQUATION

We use the Fokas method to analyze the derivative nonlinear Schrodinger （DNLS） equation iqt （x, t） = -qxx （x, t）＋（rq^2）x on the interval [0, L]. Assuming that the solution q（x, t） exists, we show that it can be represented in terms of the solution of a matrix Riemann- Hilbert problem formulated in the plane of the complex spectral parameter ξ. This problem has explicit （x, t） dependence, and it has jumps across {ξ∈C｜Imξ^4 = 0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a（ξ）, b（ξ）}, {A（ξ）, B（ξ）}, and {A（ξ）, B（ξ）}, which in turn are defined in terms of the initial data q0（x） = q（x, 0）, the bound- ary data g0（t）= q（0, t）, g1（t） = qx（0, t）, and another boundary values f0（t） = q（L, t）, f1（t） = qx（L, t）. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.     

The Initial-Boundary-Value Problems for the Hirota Equation on the Half-Line

An initial boundary-value problem for the Hirota equation on the half-line,00, is analysed by expressing the solution q(x, t) in terms of the solution of a matrix Riemann-Hilbert(RH) problem in the complex k-plane. This RH problem has explicit(x, t) dependence and it involves certain functions of k referred to as the spectral functions. Some of these functions are defined in terms of the initial condition q(x,0) = q₀(x), while the remaining spectral functions are defi...     

一类二阶方程的极限边值问题

In this paper both the necessary and sufficient conditions for the existence of the solution of the boundary value problem x=X(t,x,x),px(0)+qx(0)=r,x(∞)=const.and the continuous dependence of the solution on the boundary value are investigated.     

Boundedness and convergence for the non-Liénard type differential equation

In this article, the author studies the boundedness and convergence for the {(x) = a(y) - f(x),(y) = b(y)β(x) - g(x) e(t),where a(y), b(y), f(x),g(x),β(x) are real continuous functions in y ∈ R or x ∈ R,β(x) ≥ 0 for all x and e(t) is a real continuous function on R = {t: t ≥ 0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.     

INITIAL VALUE PROBLEM FOR A CLASS OF NONLINEAR PARABOLIC SYSTEM OF FOURTH-ORDER

In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u_1=-A(x, t)u_(x4)+B(x, t)u_(x2)+(g(u))_(x2)+(grad h(u))_x+f(u)are studied, where u(x, t)=(u_1(x, t).…, u_J(x, t) is a J-dimensional unknown vector valued function, f(u) and g(u) are the J-dimensional vector valued function of u(x, t), h(u) is a scalar function of u, A(x, t) and B(x, t) are J×J matrices of functions. The existent, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J=1, h(u)=0, g(u)=au~3, A=a_1, B=a_2, where a_1, a_2 a are constants, the system is a generalized diffusion model equation in population problem.     

BOUNDEDNESS AND CONVERGENCE FOR THE NON-LINARD TYPE DIFFERENTIAL EQUATION

In this article, the author studies the boundedness and convergence for the non-Lienard type differential equation (x|·)=a(y)-f(x) (y|·)=b(y)β(x)-g(x) e(t) where a(y),b(y),f(x),g(x),β(x) are real continuous functions in y∈R or x∈R,β(x)≥0 for all x and e(t) is a real continuous function on R = {t: t≥0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.     

Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form(P_(a,b)){D~αu(x) + f(x, u(x)) = 0, x ∈(0, 1),u(0) = u(1) = 0, D~(α-3)u(0) = a, u(1) =-b,where 3 α≤ 4, Dαis the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) =-p(x)t~σ, with σ∈(-1, 1)and p being a nonnegative continuous function that may be singular at x = 0 or x = 1and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch¨auder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem(P_(0,0)).Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in(0, 1)×[0, ∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem(P_(a,b)), which behaves like the unique solution of the homogeneous problem corresponding to(P_(a,b)). Some examples are given to illustrate the existence results.     

Long-time asymptotics for the initial-boundary value problem of coupled Hirota equation on the half-line

The object of this work is to investigate the initial-boundary value problem for coupled Hirota equation on the half-line.We show that the solution of the coupled Hirota equation can be expressed in terms of the solution of a 3×3 matrix Riemann-Hilbert problem formulated in the complex k-plane.The relevant jump matrices are explicitly given in terms of the matrix-valued spectral functions s(k)and S(k)that depend on the initial data and boundary values,respectively.Then,applying nonlinear steepest descent techniques to the associated 3×3 matrix-valued Riemann-Hilbert problem,we can give the precise leading-order asymptotic formulas and uniform error estimates for the solution of the coupled Hirota equation.     

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.     

A uniqueness theorem for indefinite Sturm-Liouville operators

In this paper, we discuss the inverse problem for indefinite Sturm-Liouville operators on the finite interval [a, b]. For a fixed index n(n = 0, 1, 2, ··· ), given the weight function ω(x), we will show that the spectral sets {λ n (q, h a , h k )} +∞ k=1 and {λ-n (q, h b , h k )} +∞ k=1 for distinct h k are sufficient to determine the potential q(x) on the finite interval [a, b] and coefficients h a and h b of the boundary conditions.     

WEAK TRAVELLING WAVE FRONT SOLUTIONS OF GENERALIZED DIFFUSION EQUATIONS WITH REACTION

The author demonstrate that the two-point boundary value problem {p′(s)=f′(s)-λp^β(s)for s∈(0,1);β∈(0,1),p(0)=p(1)=0,p(s)>0 if s∈(0,1),has a solution(λ^-，p^-（s）),where |λ^-| is the smallest parameter,under the minimal stringent restrictions on f(s), by applying the shooting and regularization methods. In a classic paper, Kohmogorov et.al.studied in 1937 a problem which can be converted into a special case of the above problem. The author also use the solution(λ^-,p^-(s)) to construct a weak travelling wave front solution u(x,t)=y(ξ),ξ=x-Ct,C=λ^-N/(N+1),of the generalized diffusion equation with reaction δ/δx（k（u）|δu/δx|^n-1 δu/δx)-δu/δt=g（u），where N>0,k(s)>0 a.e.on(0,1),and f(a):=n+1/N∫0ag(t)k^1/N(t)dt is absolutely continuous ou[0,1],while y(ξ) is increasing and absolutely continuous on (-∞,+∞) and (k(y(ξ))|y′(ξ)|^N)′=g(y(ξ))-Cy′(ξ)a.e.on(-∞，+∞）,y(-∞)=0,y(+∞)=1.     

Positive solution of singularly perturbed Dirichlet problem with singularities

A class of singularly perturbed boundary value problem with singularities is considered. Introducing the stretched variables, the boundary layer corrective terms near x = 0 and x = 1 are constructed. Under suitable conditions, by using the theory of differential inequalities the existence and asymptotic behavior of solution for boundary value problem are proved, uniformly valid asymptotic expansion of solution with boundary layers are obtained,     

A BOUNDARY VALUE PROBLEM FOR A NONLINEAR ORDINARY DIFFERENTIAL EQUATION INVOLVING A SMALL PARAMETER—The Riemann problem for a Generalized Diffusion Equatiois

This paper studies the boundary value problem involving a small parameter $$((k(V(t))+s)|V'(s)|^{N-1}V'(s))'+(sg(V(s))+f(V(s)))V'(s)=0 for s\in R$$, $$V(-\infty)=A,V(+\infty)=B;A0$$, $$U(x,0)=A for x<0,U(x,0)=B for x>0$$ under the hypotheses H1—H4 . The author's aim is not only to determine explicitly the discontinuous solution ,to the reduced problem；and the form and the number of its curves of discontinuity, but also to present, in an extremely natural way, the jump conditions which it must satisfy on each of its curves of diseontinuity. It is proved that the problem has a unique solution $U_{\varepsilon}(x,t)=V_{\varepsilon}(s),s=x/p(t),s\geq0,V_{\varepsilon}$pointwise converges to $V_{0}(s)$ as $s\downarrow0,V_{0}(s)$ has at least one jump point if and only if k(y) possesses at least one interval of degeneracy in [A-B], and there exists a one-to-one correspondence between the collection of all intervals of degeneracy of k(y) in [A-B] and the set of all jump points of $V_{0}(s)$     

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 UNIFORMLY VALID ESTIMATE OF SOLUTIONS TO SINGULAR PERTURBATION ROBIN BOUNDARY VALUE PROBLEM

In this paper we study the Robin boundary value problem with a small parameterεy″=f(t, y, ω(ε)y′, ε),a_0y(0) +b_0y′(0)=(ε), a_1y(1)+b_1y′(1)=η(ε),where the function ω(ε) is continuous on ε≥0 with ω(0)=0. Assuming all known functions are suitably smooth, f satisfies Nagumo's condition, f_y>0, a_t~2-b_t~2≠0, (-1)~ia_ib_i≤0 (i=0, 1) and the reduced equation 0=f(t, y, 0, 0) has a solution y(t) (0≤t≤1), we prove the existence and the uniqueness of the solution for the boundary value problem and givo an asymptotic expansion of the solution in the power ε~(1/2) which is uniformly valid on 0≤t≤1.     

Long-time Asymptotic Behavior for the Derivative Schr¨odinger Equation with Finite Density Type Initial Data*

In this paper, the authors apply ? steepest descent method to study the Cauchy problem for the derivative nonlinear Schr¨odinger equation with finite density type initial data iq_t + q_xx + i(|q|²q)x = 0,q(x, 0) = q0(x),where lim/x→±∞ q0(x) = q± and |q±| = 1. Based on the spectral analysis of the Lax pair,they express the solution of the derivative Schr¨odinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region ξ =x/t with |ξ + 2| < 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) + O(t^?3/4 ),in which the leading term is N(I) solitons, the second term is a residual error from a ? equation. For the region |ξ + 2| > 1, the long time asymptotic is given by q(x, t) = T (∞)^?2q^r_Λ(x, t) ? t^?1/2 if¹¹ + O(t^?3/4 ),in which the leading term is N(I) solitons, the second t^?1/2 order term is soliton-radiation interactions and the third term is a residual error from a ? equation. These results are verification of the soliton resolution conjecture for the derivative Schr¨odinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively.     

AN IMPROVEMENT OF A RESULT OF IVOCHKINA AND LADYZHENSKAYA ON A TYPE OF PARABOLIC MONGE-AMP\`ERE EQUATION

For the initial-hotmdary value problem about a type of parabolic Monge-Ampere equation of the form (IBVP): {-Dtu + (deD^2xu)^1/n = f(x,t), (x,t) ∈ Q = Ω&#215;(0,T)}, u(x,t) =Ф(x,t)(x,t) ∈δpQ}, where Ω is a bounded convex domain in R^n, the result in [4] by Ivochkina and Ladyzheokaya is improved in the sense that, under assumptions that the data of the problempossess lower regularity and satisfy lower order compatibility conditions than than in [4], the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This cannot be reallzed by only using the method in [4]. The main additional effort the authors have done is a kind of nonlinear perturbation.     

The Existence of Solutions of Elliptic Equations with Neumann Boundary Condition for Superlinear Problems

In this paper, we study and discuss the existence of multiple solutions of a class of non-linear elliptic equations with Neumann boundary condition, and obtain at least seven non-trivial solutions in which two are positive, two are negative and three are sign-changing. The study of problem (1.1):{-△u αu=f(u),x∈Ω, x∈Ω,δu/δr=0,x∈δΩ,is based on the variational methods and critical point theory. We form our conclusion by using the sub-sup solution method, Mountain Pass Theorem in order intervals, Leray-Schauder degree theory and the invariance of decreasing flow.     

On the Initial-Boundary Value Problems for Quasilinear Symmetric Hyperbolic System with Characteristic Boundary

In this paper we discuss the initial-boundary value problems for quasilinear symmetric hyperbolic system with characteristic boundary.Suppose \Omega is a bounded domain,its boundary \partial \Omega is sufficiently smooth.We consider the quasilinear symmetric hyperbolic system $[\sum\limits_{i = 0}^n {{\alpha _i}(x,u)\frac{{\partial u}}{{\partial {x_i}}}} = f(x,u)\]$ in the domain [0, h]*\Omega. The initial-boundary conditions are $u|_x_0=0$(2) $Mu|_[0,h]*\partial \Omega=0$(3) (No loss of generality, the initial condition may be considered as homogeneous one) . We assume the coefficients of (1), (3) are sufficiently smooth, the compatibility condition and the following conditions are satisfied. 1) when t = 0, u=0, the $\alpha_0(x,u)$ is a positive definite matrix. 2) If [\tilde u\] denotes any vector function satisfying the condition (3), the boundary [0,h]*\partial \Omega is non-characteristic or regular oliarabterigitic for the operator $[\sum\limits_{i = 0}^n {{\alpha _i}(x,\tilde u) \times \frac{\partial }{{\partial {x_i}}}} \]$. and if $v(0,v_1,\cdots,v_n)$ denotes the normal direction to the boundary, the matrix $\beta(x,\tilde u\)=\sum\limits_{i=0}^n v_i \alpha_i(x,\tilde u\)$ is equal to \beta_0, which only depends on x, and Mu=0 is a maximum non-negatiye subspace of quahratio form u\beta_0u. 3) There exists a non-singular matrix Q (x), such that the matrix $\tilde =beta(x,v)=Q'(x)\beta(x,Q^-1v)Q(x)$ may be reduced to a block diagonal matrix $[\left( {\begin{array}{*{20}{c}} {{B_1}}&0\0&{{B_2}} \end{array}} \right)\]$ and the boundary condition may be reduced to $v_1=\cdots=v_L=0$(4) when (t,x) lies on the boundary [0,h] \times \partial \Omega, and v satisfies (4),the block B_1 will be equal to a non-singular matrix B_10 and B_2 will vanish. Under these assumptions, we have proved: Theorem I. There exists a sufficiently small number \delta, such that if h \leq \delta, the local smooth solution of the initial-boundary value problem (1)—(3) uniquely exists. This theorem has been applied to gas dynamics. For both steady flow and unsteady flow in three dimentional space we can use Theorem I to obtain the result of the unique existance of local smooth solution for the correponding system of equations, if there isn;t any shook wave.     

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R2 of the form 0 〈 y 〈p（x,t）, where p（x,t） is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little H61der spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient y is larger than a threshold value y〉0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.