Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

Hamilton-Jacobi theory over time scales and applications to linear-quadratic problems

In this paper we first derive the verification theorem for nonlinear optimal control problems over time scales. That is, we show that the value function is the only solution of the Hamilton-Jacobi equation, in which the minimum is attained at an optimal feedback controller. Applications to the linear-quadratic regulator problem (LQR problem) gives a feedback optimal controller form in terms of the solution of a generalized time scale Riccati equation, and that every optimal solution of the LQR problem must take that form. A connection of the newly obtained Riccati equation with the traditional one is established. Problems with shift in the state variable are also considered. As an important tool for the latter theory we obtain a new formula for the chain rule on time scales. Finally, the corresponding LQR problem with shift in the state variable is analyzed and the results are related to previous ones.     

On the existence and asymptotic behaviour of solutions of an evolution equation and an application to the Feynman-Kac theorem

In this paper, we study the existence of mild solutions of a semilinear evolution equation on an unbounded interval. The rapidity of the growth of those solutions is characterized. Moreover, we investigate the local and global attractivity of solutions of an equation in question and we describe their asymptotic behaviour. Our investigations are conducted in a real separable Banach space.     

Dissipative Control Systems and Disturbance Attenuation for Nonlinear H ∞ Problems

We characterize functions satisfying a dissipative inequality associated with a control problem. Such a characterization is provided in terms of an epicontingent solution, or a viscosity supersolution to a partial differential equation called Isaacs' equation. Links between supersolutions and epicontingent solutions to Isaacs' equation are studied. Finally, we derive (possibly discontinuous) disturbance attenuation feedback of the H ^∞ problem from contingent formulation of Isaacs' equation. Accepted 20 January 1998     

DEVELOPABLE SURFACE PATCHES BOUNDED BY NURBS CURVES

In this paper we construct developable surface patches which are bounded by two rational or NURBS curves, though the resulting patch is not a rational or NURBS surface in general. This is accomplished by reparameterizing one of the boundary curves. The reparameterization function is the solution of an algebraic equation. For the relevant case of cubic or cubic spline curves, this equation is quartic at most, quadratic if the curves are Bézier or splines and lie on parallel planes, and hence it may be solved either by standard analytical or numerical methods.     

An integro-differential equation for surface ocean waves with finite depth

An explicit integro-differential equation formulation is derived for surface ocean waves with finite depth. The equation involves only 2D surface variables. For this equation, we establish the stability and existence of solutions, and explain the effect of depth on surface wave properties.     

Equivalence between Nonlinear H ∞ Control Problems and Existence of Viscosity Solutions of Hamilton—Jacobi—Isaacs Equations rid="*" id="*" This work was written while the author was visiting the University of California at Santa Barbara. He was partially supported by Consiglio Nazionale delle Ricerche of Italy.

In this paper we extend to completely general nonlinear systems the result stating that the suboptimal control problem is solved if and only if the corresponding Hamilton—Jacobi—Isaacs (HJI) equation has a nonnegative (super)solution. This is well known for linear systems, using the Riccati equation instead of the HJI equation. We do this using the theory of differential games and viscosity solutions. Accepted 14 February 1997     

A stochastic heat equation with the distributions of Lévy processes as its invariant measures

We consider a linear heat equation on a half line with an additive noise chosen properly in such a manner that its invariant measures are a class of distributions of Lévy processes. Our assumption on the corresponding Lévy measure is, in general, mild except that we need its integrability to show that the distributions of Lévy processes are the only invariant measures of the stochastic heat equation.     

Global solvability for Kirchhoff equation in special classes of non-analytic functions

In this paper we derive the following two properties: the first one is a precise representation of WKB solution to the Cauchy problem of a linear wave equation with a variable coefficient with respect to time, and the second one is the global solvability for Kirchhoff equation in some special classes of nonreal-analytic functions, which is proved by grace of the first property.     

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Under the condition that the coefficients are Lipschitz continuous, we study the infinitesimal generator of Markov semigroup corresponding to the multivalued stochastic equation. In order to provide a core of the infinitesimal generator, we investigate the associated multivalued elliptic equation and its viscosity solutions.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.     

Convergence of the increments of a stochastic integral associated to the stochastic wave equation

We study a type of one-dimensional wave equation on the plane with non-linear random forcing. We are interested in the almost sure behaviour of the normalized increments of the solution process associated to this type of wave equation. Also we study the behaviour of the normalized increments of some other stochastic integral equation.     

The contraction rate in Thompson's part metric of order-preserving flows on a cone – Application to generalized Riccati equations

We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation.     

Integrability and Self-Similarity in Transient Stimulated Raman Scattering

Summary. The phenomenon of stimulated Raman scattering (SRS) can be described by three coupled PDEs which define the pump electric field, the Stokes electric field, and the material excitation as functions of distance and time. In the transient limit these equations are integrable, i.e., they admit a Lax pair formulation. Here we study this transient limit. The relevant physical problem can be formulated as an initial-boundary value (IBV) problem where both independent variables are on a finite domain. A general method for solving IBV problems for integrable equations has been introduced recently. Using this method we show that the solution of the equations describing the transient SRS can be obtained by solving a certain linear integral equation. It is interesting that this equation is identical to the linear integral equation characterizing the solution of an IBV problem of the sine-Gordon equation in light-cone coordinates. This integral equation can be solved uniquely in terms of the values of the pump and Stokes fields at the entry of the Raman cell. The asymptotic analysis of this solution reveals that the long-distance behavior of the system is dominated by the underlying self-similar solution which satisfies a particular case of the third Painlevé transcendent. This result is consistent with both numerical simulations and experimental observations. We also discuss briefly the effect of frequency mismatch between the pump and the Stokes electric fields. Received December 10, 1996; second revision received October 10, 1997; final revision received January 20, 1998     

Regularity of Viscosity Solutions of the Biased Infinity Laplacian Equation

In this paper, we are interested in the regularity estimates of the nonnegative viscosity super solution of the $β$−biased infinity Laplacian equation $$∆^β_∞u = 0,$$ where $β ∈ \mathbb{R}$ is a fixed constant and $∆^β_∞u := ∆^N_∞u + β|Du|,$ which arises from the random game named biased tug-of-war. By studying directly the $β$−biased infinity Laplacian equation, we construct the appropriate exponential cones as barrier functions to establish a key estimate. Based on this estimate, we obtain the Harnack inequality, Hopf boundary point lemma, Lipschitz estimate and the Liouville property etc.     

带乘性噪声的空间分数阶随机非线性Schrödinger方程的广义多辛算法

带乘性噪声的空间分数阶随机非线性Schrödinger方程是一类重要的方程,可应用于描述开放非局部量子系统的演化过程.该方程为一个无穷维分数阶随机Hamilton系统,且具有广义多辛结构和质量守恒的性质.针对该方程的广义多辛形式,在空间上采用拟谱方法离散分数阶微分算子,在时间上则采用隐式中点格式,构造出一类保持全局质量的广义多辛格式.对行波解和平面波解等进行数值模拟,结果验证了所构造格式的有效性和保结构性质,时间均方收敛阶约在0.5到1之间.     

Two-Grid Finite Element Method with Crank-Nicolson Fully Discrete Scheme for the Time-Dependent Schrödinger Equation

In this paper, we study the Crank-Nicolson Galerkin finite element method and construct a two-grid algorithm for the general two-dimensional time-dependent Schrödinger equation. Firstly, we analyze the superconvergence error estimate of the finite element solution in $H^1$ norm by use of the elliptic projection operator. Secondly, we propose a fully discrete two-grid finite element algorithm with Crank-Nicolson scheme in time. With this method, the solution of the Schrödinger equation on a fine grid is reduced to the solution of original problem on a much coarser grid together with the solution of two Poisson equations on the fine grid. Finally, we also derive error estimates of the two-grid finite element solution with the exact solution in $H^1$ norm. It is shown that the solution of two-grid algorithm can achieve asymptotically optimal accuracy as long as mesh sizes satisfy $H = \mathcal{O}(h^{\frac{1}{2}})$.     

A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose L^∞$L^{\infty }$-norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation and we establish a relationship between the finite-time blow up of solutions and the negativity of their energy. The proof of this result is based on a Gamma-convergence technique.     

Painlev\'{e} Analysis and Auto-B\"{a}cklund Transformation for a General Variable Coefficient Burgers Equation with Linear Damping Term

This paper investigates a general variable coefficient (gVC) Burgers equation with linear damping term. We derive the Painlev\''{e} property of the equation under certain constraint condition of the coefficients. Then we obtain an auto-B\"{a}cklund transformation of this equation in terms of the Painlev\''{e} property. Finally, we find a large number of new explicit exact solutions of the equation. Especially, infinite explicit exact singular wave solutions are obtained for the first time. It is worth noting that these singular wave solutions will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of the gVC Burgers equation with linear damping term. It also reflects the complexity of nonlinear wave propagation in fluid from one aspect.     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.