Dispersive Estimates of Solutions to the Schrödinger Equation

We prove time decay L¹ → L^∞ estimates for the Schr?dinger group e^{it(−Δ + V)} for real-valued potentials satisfying V (x) = O (|x|^−δ), |x| ≫ 1, with δ > 5/2. Communicated by Bernard Helffer submitted 27/11/04, accepted 29/04/05     

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n≥4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.     

Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

On a finite segment [0, l], we consider the differential equation

Solitary Wave Solutions and Kink Wave Solutions for a Generalized PC Equation

We present in this paper a generalised PC (GPC) equation which includes several known models. The corresponding traveling wave system is derived and we show that the homoclinic orbits of the traveling wave system correspond to the solitary waves of GPC equation, and the heteroclnic orbits correspond to the kink waves. Under some parameter conditions, the existence of above two types of orbits is demonstrated and the explicit expressions of the two solutions are worked out.     

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.     

Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

Estimations of Solutions of the Sturm– Liouville Equation with Respect to a Spectral Parameter

This paper is concerned with estimations of solutions of the Sturm–Liouville equation $$\big(p(x)y'(x)\big)'+\Big(\mu^2 -2i\mu d(x)-q(x)\Big)\rho(x)y(x)=0, \ \ x\in[0,1],$$ ( p ( x ) y ' ( x ) ) ' + ( μ 2 - 2 i μ d ( x ) - q ( x ) ) ρ ( x ) y ( x ) = 0 , x ∈ [ 0 , 1 ] , where ${\mu\in\mathbb{C}}$ μ ∈ C is a spectral parameter. We assume that the strictly positive function ${\rho\in L_{\infty}[0,1]}$ ρ ∈ L ∞ [ 0 , 1 ] is of bounded variation, ${p\in W^1_1[0,1]}$ p ∈ W 1 1 [ 0 , 1 ] is also strictly positive, while ${d\in L_1[0,1]}$ d ∈ L 1 [ 0 , 1 ] and ${q\in L_1[0,1]}$ q ∈ L 1 [ 0 , 1 ] are real functions. The main result states that for any r > 0 there exists a constant c _r such that for any solution y of the Sturm–Liouville equation with μ satisfying ${|{\rm Im}\, \mu|\leq r}$ | Im μ | ≤ r , the inequality ${\|y(\cdot,\mu)\|_C\leq c_r\|y(\cdot,\mu)\|_{L_1}}$ ∥ y ( · , μ ) ∥ C ≤ c r ∥ y ( · , μ ) ∥ L 1 is true. We apply our results to a problem of vibrations of an inhomogeneous string of length one with damping, modulus of elasticity and potential, rewritten in an operator form. As a consequence, we obtain that the operator acting on a certain energy Hilbert space is the generator of an exponentially stable C ₀-semigroup.     

Sobolev Stability of Plane Wave Solutions to the Cubic Nonlinear Schrödinger Equation on a Torus

It is shown that plane wave solutions to the cubic nonlinear Schrödinger equation on a torus behave orbitally stable under generic perturbations of the initial data that are small in a high-order Sobolev norm, over long times that extend to arbitrary negative powers of the smallness parameter. The perturbation stays small in the same Sobolev norm over such long times. The proof uses a Hamiltonian reduction and transformation and, alternatively, Birkhoff normal forms or modulated Fourier expansions in time.     

Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap

Theoretical and Mathematical Physics - We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave...     

Boundary Behavior of Large Solutions to the Monge-Ampère Equation in a Borderline Case

This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD²u(x)=b(x)f(u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b ∈ C^∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.     

Regularity of Solutions of the Cahn-Hilliard Equation with Non-constant Mobility

In this paper, we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility. Basing on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions.     

Exact Solitary-wave Solutions and Periodic Wave Solutions for Generalized Modified Boussinesq Equation and the Effect of Wave Velocity on Wave Shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity＇s variation.     

Weak Solutions to the Cahn-Hilliard Equation with Degenerate Diffusion Mobility in ℝN

This paper is concerned with a popular form of Cahn-Hilliard equation which plays an important role in understanding the evolution of phase separation. We get the existence and regularity of a weak solution to nonlinear parabolic, fourth order Cahn-Hilliard equation with degenerate mobility M(u)=u^m(1-u)^m which is allowed to vanish at 0 and 1. The existence and regularity of weak solutions to the degenerate Cahn-Hilliard equation are obtained by getting the limits of Cahn-Hilliard equation with non-degenerate mobility. We explore the initial value problem with compact support and obtain the local non-negative result. Further, the above derivation process is also suitable for the viscous Cahn-Hilliard equation with degenerate mobility.     

Asymptotic Behavior of Solutions to the Generalized BBM-Burgers Equation

We investigate the asymptotic behavior of solutions of the initial-boundary value problem for the generalized BBM-Burgers equation u_t f(u)_x=u_(xx) u_(xx) on the half line with the conditions u(0, t)=, u-, u(∞,t)=u_ and, u_-相似文献   

Analytic Solutions of L∞ Optimal Control Problems for the Wave Equation

There are very few results about analytic solutions of problems of optimal control with minimal L norm. In this paper, we consider such a problem for the wave equation, where the derivative of the state is controlled at both boundaries. We start in the zero position and consider a problem of exact control, that is, we want to reach a given terminal state in a given finite time. Our aim is to find a control with minimal L norm that steers the system to the target.We give the analytic solution for certain classes of target points, for example, target points that are given by constant functions. For such targets with zero velocity, the analytic solution has been given by Bennighof and Boucher in Ref. 1.     

Analysis of the Cahn–Hilliard Equation with a Chemical Potential Dependent Mobility

The aim of this paper is to study the well-posedness and the existence of global attractors for a family of Cahn–Hilliard equations with a mobility depending on the chemical potential. Such models arise from generalizations of the (classical) Cahn–Hilliard equation due to Gurtin.