Mildly degenerate Kirchhoff equations with weak dissipation: Global existence and time decay

We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t→+∞.We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t→+∞, with the same rate of the solution of the limit problem of parabolic type.     

Tight bounds for eternal dominating sets in graphs

The eternal domination number of a graph is the number of guards needed at vertices of the graph to defend the graph against any sequence of attacks at vertices. We consider the model in which at most one guard can move per attack and a guard can move across at most one edge to defend an attack. We prove that there are graphs G for which , where γ_∞(G) is the eternal domination number of G and α(G) is the independence number of G. This matches the upper bound proved by Klostermeyer and MacGillivray.     

Asymptotic behavior for nonlocal dispersal equations

This paper is concerned with the existence and asymptotic behavior of solutions of a nonlocal dispersal equation. By means of super-subsolution method and monotone iteration, we first study the existence and asymptotic behavior of solutions for a general nonlocal dispersal equation. Then, we apply these results to our equation and show that the nonnegative solution is unique, and the behavior of this solution depends on parameter λ in equation. For λ≤λ₁(Ω), the solution decays to zero as t→∞; while for λ>λ₁(Ω), the solution converges to the unique positive stationary solution as t→∞. In addition, we show that the solution blows up under some conditions.     

A non-local regularization of first order Hamilton-Jacobi equations

In this paper, we investigate the regularizing effect of a non-local operator on first-order Hamilton-Jacobi equations. We prove that there exists a unique solution that is C² in space and C¹ in time. In order to do so, we combine viscosity solution techniques and Green's function techniques. Viscosity solution theory provides the existence of a W^1,∞ solution as well as uniqueness and stability results. A Duhamel's integral representation of the equation involving the Green's function permits to prove further regularity. We also state the existence of C^∞ solutions (in space and time) under suitable assumptions on the Hamiltonian. We finally give an error estimate in L^∞ norm between the viscosity solution of the pure Hamilton-Jacobi equation and the solution of the integro-differential equation with a vanishing non-local part.     

Behaviors of solutions for a singular diffusion equation

This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L² decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L^∞ decay estimate is derived.     

The problem of harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on a (noncommutative) group is to decompose the most “natural” unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(∞) is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(∞) consists of.We deal with unitary representations of a reasonable class, which are in 1-1 correspondence with characters (central, positive definite, normalized functions on U(∞)). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.The main result of the present paper consists in explicitly constructing a 4-parameter family of “natural” representations and computing their characters. We view these representations as a substitute of the nonexisting regular representation of U(∞). We state the problem of harmonic analysis on U(∞) as the problem of computing the spectral measures for these “natural” representations. A solution to this problem is given in the next paper (Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, math/0109194, to appear in Ann. Math.), joint with Alexei Borodin.We also prove a few auxiliary general results. In particular, it is proved that the spectral measure of any character of U(∞) can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups U(N). This fact is a starting point for computing spectral measures.     

Semismall perturbations, semi-intrinsic ultracontractivity, and integral representations of nonnegative solutions for parabolic equations

We consider nonnegative solutions of a parabolic equation in a cylinder D×I, where D is a noncompact domain of a Riemannian manifold and I=(0,T) with 0<T?∞ or I=(−∞,0). Under the assumption [SSP] (i.e., the constant function 1 is a semismall perturbation of the associated elliptic operator on D), we establish an integral representation theorem of nonnegative solutions: In the case I=(0,T), any nonnegative solution is represented uniquely by an integral on (D×{0})∪(M_∂D×[0,T)), where M_∂D is the Martin boundary of D for the elliptic operator; and in the case I=(−∞,0), any nonnegative solution is represented uniquely by the sum of an integral on M_∂D×(−∞,0) and a constant multiple of a particular solution. We also show that [SSP] implies the condition [SIU] (i.e., the associated heat kernel is semi-intrinsically ultracontractive).     

The existence and uniqueness of the solution for stochastic functional differential equations with infinite delay

This paper is devoted to build the existence-and-uniqueness theorem of solutions to stochastic functional differential equations with infinite delay (short for ISFDEs) at phase space BC((−∞,0];Rd). Under the uniform Lipschitz condition, the linear growth condition is weaked to obtain the moment estimate of the solution for ISFDEs. Furthermore, the existence-and-uniqueness theorem of the solution for ISFDEs is derived, and the estimate for the error between approximate solution and accurate solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence-and-uniqueness theorem is also valid for ISFDEs on [t₀,T]. Moreover, the existence-and-uniqueness theorem still holds on interval [t₀,∞), where t₀∈R is an arbitrary real number.     

Existence and uniqueness of solutions to neutral stochastic functional differential equations with infinite delay

This paper investigates the existence and uniqueness theorem of solutions to neutral stochastic differential equations with infinite delay (short for INSFDEs) at a space BC((-∞,0];R^d). Under the uniform Lipschitz condition, linear growth condition is weaken to obtain the moment estimate of the solution for INSFDEs. Furthermore, the existence, uniqueness theorem of the solution for INSFDEs is derived, and the estimate for the error between approximate solution and exact solution is given. On the other hand, under the linear growth condition, the uniform Lipschitz condition is replaced by the local Lipschitz condition, the existence, uniqueness theorem is also valid for INSFDEs on [t₀,T]. Moreover, the existence, uniqueness theorem still holds on interval [t₀,∞), where t₀∈R is an arbitrary real number.     

Structure of boundary blow-up solutions for quasi-linear elliptic problems II: small and intermediate solutions

The structure of positive boundary blow-up solutions to quasi-linear elliptic problems of the form −Δ_pu=λf(u), u=∞ on ∂Ω, 1<p<∞, is studied in a bounded smooth domain , for a class of nonlinearities f∈C¹((0,∞)?{z₂})∩C⁰[0,∞) satisfying f(0)=f(z₁)=f(z₂)=0 with 0<z₁<z₂, f<0 in (0,z₁)∪(z₂,∞), f>0 in (z₁,z₂). Large, small and intermediate solutions are obtained for λ sufficiently large. It is known from Part I (see Structure of boundary blow-up solutions for quasilinear elliptic problems, part (I): large and small solutions, preprint), that the large solution is the unique large solution to the problem. We will see that the small solution is also the unique small solution to the problem while there are infinitely many intermediate solutions. Our results are new even for the case p=2.     

On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI

Painlevé's transcendental differential equation PVI may be expressed as the consistency condition for a pair of linear differential equations with 2×2 matrix coefficients with rational entries. By a construction due to Tracy and Widom, this linear system is associated with certain kernels which give trace class operators on Hilbert space. This paper expresses such operators in terms of Hankel operators Γ? of linear systems which are realised in terms of the Laurent coefficients of the solutions of the differential equations. Let P(t,∞):L²(0,∞)→L²(t,∞) be the orthogonal projection; then the Fredholm determinant τ(t)=det(I−P(t,∞)Γ?) defines the τ function, which is here expressed in terms of the solution of a matrix Gelfand-Levitan equation. For suitable values of the parameters, solutions of the hypergeometric equation give a linear system with similar properties. For meromorphic transfer functions that have poles on an arithmetic progression, the corresponding Hankel operator has a simple form with respect to an exponential basis in L²(0,∞); so det(I−Γ?P(t,∞)) can be expressed as a series of finite determinants. This applies to elliptic functions of the second kind, such as satisfy Lamé's equation with ?=1.     

Vanishing components in autonomous competitive Lotka-Volterra systems

Sufficient conditions are found for an n-dimensional autonomous competitive Lotka-Volterra system to have a component vanishing in an exponential rate as t→∞. These conditions incorporate a typical known result in the literature as a particular case. Moreover, if the n-dimensional system degenerates asymptotically to an m-dimensional subsystem as t→∞, then, under these conditions on the subsystem, the property that the ith component of every solution of the subsystem vanishes in an exponential rate is also preserved for the n-dimensional system.     

An existence result for a linear-superlinear elliptic system with Neumann boundary conditions

In this work, we consider an elliptic system of two equations in dimension one (with Neumann boundary conditions) where the nonlinearities are asymptotically linear at −∞ and superlinear at +∞. We obtain that, under suitable hypotheses, a solution exists for any couple of forcing terms in L².We also present a similar result in which the superlinearity is in only one of the two equations, and we discuss the resonant problem too.     

On the super fixed point property in product spaces

We prove that if F is a finite-dimensional Banach space and X has the super fixed point property for nonexpansive mappings, then F⊕X has the super fixed point property with respect to a large class of norms including all lp norms, 1?p<∞. This provides a solution to the “super-version” of the problem of Khamsi (1989).     

Properties of stationary solutions of a generalized Tjon-Wu equation

A generalized version of the Tjon-Wu equation is considered. Solutions of this equation are functions with values in the space of probability measures on [0,∞). We prove that the stationary solution μ_∗ of the equation has the following property: Either μ_∗ is supported at one point or suppμ_∗=[0,∞). Moreover we show that in the second case the distribution function of μ_∗ is continuous. Some open questions are discussed.     

Energy estimate and fundamental solution for degenerate hyperbolic Cauchy problems

The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the C^∞ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first-order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces (with a loss of derivatives) which gives the well-posedness of the Cauchy problem in C^∞. In the strictly hyperbolic case, we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.     

Modeling of solute transport into sub-aqueous sediments

A model for investigating the solute transport into a sub-aqueous sediment bed, under an imposed standing water surface wave, is developed. Under the assumption of Darcy flow in the bed, a model based on a two-dimensional, unsteady advection–diffusion equation is derived; the relative roles of the advective and diffusive transport are characterized by a Peclet number, Pe. Two solutions for the equation are developed. The first is a basic control volume method using the power-law scheme. The second is a smear-free, modified upwind solution for the special case of Pe → ∞. Results, at a given time step, are reported in terms of a laterally averaged solute verse depth profile. The main result of the paper is to demonstrate that the one-dimensional solute concentration verse depth profile is essentially independent of any numerical dissipation present in the solute field predictions. This demonstration is achieved by (i) using an extensive grid refinement study, and (ii) by comparing Pe → ∞ predictions obtained with the basic and smear-free solutions.     

Robust finite-time H∞ control for a class of uncertain switched neutral systems

This paper investigates the robust finite-time H_∞ control problem for a class of uncertain switched neutral systems with unknown time-varying disturbance. The uncertainties under consideration are norm bounded. By using the average dwell time approach, a sufficient condition for finite-time boundedness of switched neutral systems is derived. Then, finite-time H_∞ performance analysis for switched neutral systems is developed, and a robust finite-time H_∞ state feedback controller is proposed to guarantee that the closed-loop system is finite-time bounded with H_∞ disturbance attenuation level γ. All the results are given in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to show the effectiveness of the proposed method.     

Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system

In this paper, we study a scalar conservation law that models a highly re-entrant manufacturing system as encountered in semi-conductor production. As a generalization of Coron et al. (2010) [14], the velocity function possesses both the local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with initial and boundary data in L^∞. We also obtain the stability (continuous dependence) of both the solution and the out-flux with respect to the initial and boundary data. Finally, we prove the existence of an optimal control that minimizes, in the Lp-sense with 1?p?∞, the difference between the actual out-flux and a forecast demand over a fixed time period.