一类半线性椭圆型方程正整体解的存在性

本文研究了一个半线性椭圆型方程正整体解的存在性问题.利用上下解方法克服了非齐次项出现带来的困难,获得了正整体解存在的必要条件,具体地给出了在此条件下正整体解所处的领域.     

Remarks on infinite energy solutions of nonlinear wave equations

We first consider the existence of a solution of the critical semilinear wave equation in Besov space which extends the results in [P. Germain, Global infinite energy solutions of critical semilinear wave equation, Revista Matematica Iberoamericana 24 (2) (2008) 463-497] to general dimensions. Next we derive the existence and uniqueness of global solutions for a semilinear wave equation in Marcinkiewicz space.     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

   Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

Bang-bang property for time optimal control of semilinear heat equation

This paper studies the bang-bang property for time optimal controls governed by semilinear heat equation in a bounded domain with control acting locally in a subset. Also, we present the null controllability cost for semilinear heat equation and an observability estimate from a positive measurable set in time for the linear heat equation with potential.     

带跳的分数倒向重随机微分方程及相应的随机积分偏微分方程

本文首次把Poisson随机测度引入分数倒向重随机微分方程,基于可料的Girsanov变换证明由Brown运动、Poisson随机测度和Hurst参数在(1/2,1)范围内的分数Brown运动共同驱动的半线性倒向重随机微分方程解的存在唯一性.在此基础上,本文定义一类半线性随机积分偏微分方程的随机黏性解,并证明该黏性解由带跳分数倒向重随机微分方程的解唯一地给出,对经典的黏性解理论作出有益的补充.     

The problem of start control for a class of semilinear distributed systems of Sobolev type

A result on the solvability of the Cauchy problem for a semilinear equation of Sobolev type in a Banach space is obtained with the help of the theory of degenerate operator semigroups. The result is used for investigating the problem of start control in the corresponding system. Abstract results are illustrated by the example of the semilinear Dzektser equation.     

Uniqueness of the ground state solutions of quasilinear Schrödinger equations

We are concerned with the uniqueness result of positive solutions for a class of quasilinear elliptic equation arising from plasma physics. We convert a quasilinear elliptic equation into a semilinear one and show the unique existence of positive radial solution for original equation under the suitable conditions on the power of nonlinearity and quasilinearity. We also investigate the non-degeneracy of a positive radial solution for a converted semilinear elliptic equation.     

Boundary controllability for the semilinear Schrödinger equations on Riemannian manifolds

We study the boundary exact controllability for the semilinear Schrödinger equation defined on an open, bounded, connected set Ω of a complete, n-dimensional, Riemannian manifold M with metric g. We prove the locally exact controllability around the equilibria under some checkable geometrical conditions. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the semilinear Schrödinger equation. We then establish the globally exact controllability in such a way that the state of the semilinear Schrödinger equation moves from an equilibrium in one location to an equilibrium in another location.     

Optimal Control Problem Governed by Semilinear Parabolic Equation and its Algorithm

In this paper, an optimal control problem governed by semilinear parabolic equation which involves the control variable acting on forcing term and coefficients appearing in the higher order derivative terms is formulated and analyzed. The strong variation method, due originally to Mayne et al to solve the optimal control problem of a lumped parameter system, is extended to solve an optimal control problem governed by semilinear parabolic equation, a necessary condition is obtained, the strong variation algorithm for this optimal control problem is presented, and the corresponding convergence result of the algorithm is verified.     

齐次群上二阶半线性偏微分方程的一类Picone型恒等式和Sturmian比较定理

本文给出了齐次群上的一类广义Picone型恒等式,由此证明了以下半线性方程组(其中 表示齐次群上的广义梯度)的Sturmian比较定理及一类振荡定理,并用于Heisenberg群上一类半线性方程.然后利用这里的广义Picone型恒等式证明了Heisenberg群上一类更一般的Hardv型不等式     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.     

Four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities

In this paper, we study the combined effect of concave and convex nonlinearities on the number of positive solutions for a semilinear elliptic equation. With the help of the Nehari manifold and the center mass function, we prove that there are at least four positive solutions for a semilinear elliptic equation in a finite strip with a hole.     

Null controllability for a semilinear parabolic equation with gradient quadratic growth

This article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term. The technique in this paper is a combination of Cole–Hopf transformation and some methods from [A.Y. Khapalov, Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach, SIAM J. Control Optim. 41 (2003) 1886–1900].     

Insensitizing controls for a class of nonlinear Ginzburg-Landau equations

This paper shows the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity, the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation is proved. Meanwhile, if the nonlinearity in the equation is only a smooth function without any additional growth condition, a local result on insensitizing controls is obtained. As usual, the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control. The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities

It is shown that a Lie point symmetry of the semilinear polyharmonic equations involving nonlinearities of power or exponential type is a variational/divergence symmetry if and only if the equation parameters assume critical values. The corresponding conservation laws for critical polyharmonic semilinear equations are established.     

关于二阶半线性椭圆方程的Dirichlet边值问题一个注记(英文)

本文研究二阶半线性椭圆方程的Dirichlet边值问题.利用山路引理和最小作用原理,获得了在新条件下具有Dirichlet边值问题的二阶半线性椭圆方程的弱解的存在性的结果.     

Approximate Controllability of a Semilinear System Involving a Fully Nonlinear Gradient Term

This paper concerns a control system governed by a semilinear degenerate equation involving a fully nonlinear gradient term. The equation may be weakly degenerate and strongly degenerate on a portion of the lateral boundary, and the gradient term can be controlled by the diffusion term. The linearized system is shown to be approximately controllable by constructing a control by means of its conjugate problem. By doing a series of precise compactness estimates, we prove that the semilinear system is approximately controllable.     

Numerical Solution of the Discrete BHH-Equation in the Normal Case

It is known that the solution of the semilinear matrix equation \(X - A\overline X B = C\) can be reduced to solving the classical Stein equation. The normal case means that the coefficients on the left-hand side of the resulting equation are normal matrices. We propose a method for solving the original semilinear equation in the normal case that permits to almost halve the execution time for equations of order n = 3000 compared to the library function dlyap, which solves Stein equations in Matlab.