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.     

Shape optimization of stationary Navier-Stokes equation over classes of convex domains

In this work, we obtain some properties for the family of some convex domains. Based on these, we prove the existence of solutions of some shape optimization for stationary Navier-Stokes equations.     

On Gevrey solutions of threefold singular nonlinear partial differential equations

We study Gevrey asymptotics of the solutions to a family of threefold singular nonlinear partial differential equations in the complex domain. We deal with both Fuchsian and irregular singularities, and allow the presence of a singular perturbation parameter. By means of the Borel–Laplace summation method, we construct sectorial actual holomorphic solutions which turn out to share a same formal power series as their Gevrey asymptotic expansion in the perturbation parameter. This result rests on the Malgrange–Sibuya theorem, and it requires to prove that the difference between two neighboring solutions is exponentially small, what in this case involves an asymptotic estimate for a particular Dirichlet-like series.     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

Boundary value problems for systems of nonlinear integro-differential equations with deviating arguments

By establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique, we investigate the systems of nonlinear mixed integro-differential equations with deviating arguments and mixed boundary value conditions. Sufficient conditions are established for the existence of extremal system of solutions of the given problem.     

Asymptotic existence theorems for formal power series whose coefficients satisfy certain partial differential recursions

We study power series whose coefficients are holomorphic functions of another complex variable and a nonnegative real parameter s, and are given by a differential recursion equation. For positive integer s, series of this form naturally occur as formal solutions of some partial differential equations with constant coefficients, while for s=0 they satisfy certain perturbed linear ordinary differential equations. For arbitrary s?0, these series solve a differential-integral equation. Such power series, in general, are not multisummable. However, we shall prove existence of solutions of the same differential-integral equation that in sectors of, in general, maximal opening have the formal series as their asymptotic expansion. Furthermore, we shall indicate that the solutions so obtained can be related to one another in a fairly explicit manner, thus exhibiting a Stokes phenomenon.     

More on oscillation ofnth-order equations

In this paper we prove that a higher-order differential equation with one middle term has every bounded solution oscillatory. Moreover, the behavior of unbounded solutions is given. Two other results dealing with positive solutions are also given.     

Optima of dual integer linear programs

We consider dual pairs of packing and covering integer linear programs. Best possible bounds are found between their optimal values. Tight inequalities are obtained relating the integral optima and the optimal rational solutions.     

Loss of positivity in a nonlinear scalar heat equation

Using the theory of fixed point index, we discuss the existence of nontrivial (multiple) solutions of a nonlinear scalar heat equation with nonlocal boundary conditions depending on a positive parameter. Solutions lose positivity as the parameter decreases. For a certain parameter range, not all solutions can be positive but there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.     

Differential Equations for Singular Vectors of sp(2n)

ABSTRACT

Given a weight λ of sp(2n), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module. Moreover, we find a family of exact solutions of the system in a certain space of power series. The polynomial solutions correspond to the singular vectors in the Verma module. In particular, we find the explicit expression of a singular vector corresponding to the single condition that ? λ,α? + ht α is a nonnegative integer for some positive root α, whose existence was proven by Jantzen. In the case n = 2, we completely solved the system in a certain space of power series.     

Analytical construction of peaked solutions for the nonlinear evolution of an electromagnetic pulse propagating through a plasma

Abstract

We obtain analytical solutions, by way of the homotopy analysis method, to a nonlinear wave equation describing the nonlinear evolution of a vector potential of an electromagnetic pulse propagating in an arbitrary pair plasma with temperature asymmetry. As the method is analytical, we are able to construct peaked structures which propagate through the pair plasma, analogous to peakon solutions. These solutions are obtained through a novel matching of inner and outer homotopy solutions. In order to ensure that our analytical results are valid over the whole real line, we also discuss the convergence of the analytical results to the true solution, through minimization of the residual errors resulting from an approximate analytical solution. These results demonstrate the existence of peaked pulses propagating through a pair plasma. The algebraic decay rate of the pulses are determined analytically, as well. The method discussed here can be applied to approximate solutions to similar nonlinear partial differential equations of nonlinear Schr¨odinger type.     

Explicit solutions to quadratic BSDEs and applications to utility maximization in multivariate affine stochastic volatility models

Over the past few years quadratic Backward Stochastic Differential Equations (BSDEs) have been a popular field of research. However there are only very few examples where explicit solutions for these equations are known. In this paper we consider a class of quadratic BSDEs involving affine processes and show that their solution can be reduced to solving a system of generalized Riccati ordinary differential equations. In other words we introduce a rich and flexible class of quadratic BSDEs which are analytically tractable, i.e. explicit up to the solution of an ODE. Our results also provide analytically tractable solutions to the problem of utility maximization and indifference pricing in multivariate affine stochastic volatility models. This generalizes univariate results of Kallsen and Muhle-Karbe (2010) and some results in the multivariate setting of Leippold and Trojani (2010) by establishing the full picture in the multivariate affine jump-diffusion setting. In particular we calculate the interesting quantity of the power utility indifference value of change of numeraire. Explicit examples in the Heston, Barndorff-Nielsen–Shephard and multivariate Heston setting are calculated.     

Uniqueness and continuous dependence of weak solutions in compressible magnetohydrodynamics

We prove uniqueness and continuous dependence on initial data of weak solutions of the equations of compressible magnetohydrodynamics. The solutions we consider may exhibit discontinuities in density and in the gradients of velocity, temperature, and magnetic field. Continuous dependence is deduced by duality from existence and regularity of solutions of the adjoint of the first variation system. The analysis is complicated by the absence of strict parabolicity, the strong nonlinear coupling in the highest-order terms, and the lack of regularity in the coefficients of the adjoint system.Research supported in part by the NSF under Grant DMS-0305072.Received: May 5, 2004     

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces

In this paper, by using a monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under wide monotonicity conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Finite speed propagation of the solutions for the relativistic Vlasov–Maxwell system

In this paper, we investigate the continuous dependence with respect to the initial data of the solutions for the 1D and 1.5D relativistic Vlasov–Maxwell system. More precisely, we prove that these solutions propagate with finite speed. We formulate our results in the framework of mild solutions, i.e., the particle densities are solutions by characteristics and the electro-magnetic fields are Lipschitz continuous functions.     

Blow-up of solutions for a nonlinear beam equation with fractional feedback

A nonlinear beam equation describing the transversal vibrations of a beam with boundary feedback is considered. The boundary feedback involves a fractional derivative. We discuss the asymptotic behavior of solutions. In fact, we prove that solutions blow up in finite time under certain assumptions on the nonlinearity.     

A fixed-point theorem for S-type operators on Banach spaces and its applications to boundary-value problems

In this article we present a fixed-point theorem of operator equations defined in cones of Banach spaces, where the spectral radius of the operator is involved. The operator is an S-type operator, a meaning introduced here. As an application of our main theorem, we get the existence of positive solutions to two boundary-value problems, extending several results from the literature.     

The Wong–Zakai approximations of invariant manifolds and foliations for stochastic evolution equations

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise. We prove that the solutions of Wong–Zakai approximations almost surely converge to the solutions of the Stratonovich stochastic evolution equation. We also show that the invariant manifolds and stable foliations of the Wong–Zakai approximations converge to the invariant manifolds and stable foliations of the Stratonovich stochastic evolution equation, respectively.