Periodic and almost periodic solutions of volterra integral differential equations with infinite memory

Conditions are given which guarantee that if T > 0 is sufficiently small, then x(t) = ∝₀^∞ [dE(s)] x(t — s)+ f(t) has a unique T-periodic solution x for each continuous T-periodic function f. The vectors x and f are n-dimensional; the matrix function E(s) is n × n with bounded total variation. The proof adapts readily to provide an analogous result when x and f are almost periodic functions whose non-zero Fourier frequencies are bounded away from zero. The results are applied to study certain perturbations of the above equation.     

Global solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity and discontinuous initial data

In this paper, we study the evolutions of the interfaces between the gas and the vacuum for viscous one-dimensional isentropic gas motions. We prove the global existence and uniqueness for discontinuous solutions of the Navier-Stokes equations for compressible flow with density-dependent viscosity coefficient. Precisely, the viscosity coefficient μ is proportional to ρ^θ with 0<θ<1. Specifically, we require that the initial density be piecewise smooth with arbitrarily large jump discontinuities, bounded above and below away from zero, in the interior of gas. We show that the discontinuities in the density persist for all time, and give a decay result for the density as t→+∞.     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

Pseudo-atoms, atoms and a Jordan type decomposition in effect algebras

The aim of this paper is to introduce and investigate the concept of pseudo-atoms of a real-valued function m defined on an effect algebra L; a few examples of pseudo-atoms and atoms are given in the context of null-additive, null-null-additive and pseudo-null-additive functions and also, some fundamental results for pseudo-atoms under the assumption of null-null-additivity are established. The notions of total variation |m|, positive variation m⁺ and negative variation m⁻ of a real-valued function m on L are studied elaborately and it is proved for a modular measure m (which is of bounded total variation) defined on a D-lattice L that, m is pseudo-atomic (or atomic) if and only if its total variation |m| is pseudo-atomic (or atomic). Finally, a Jordan type decomposition theorem for an extended real-valued function m of bounded total variation defined on an effect algebra L is proved and some properties on decomposed parts of m such as continuity from below, pseudo-atomicity (or atomicity) and being measure, are discussed. A characterization for the function m to be of bounded total variation is established here and used in proving above-mentioned Jordan type decomposition theorem.     

Asymptotic behavior of solutions to the damped quasilinear equation ∂2∂t2 u(x,t) + γ∂u∂t (x,t) − ∂∂xσ ∂u∂x (x,t)) = 0

Asymptotic lower bounds for the L² norms of solutions of initial-boundary value problems associated with the equation of the title are derived for a simple case in which the equation fails to exhibit strict hyperbolicity. It is shown that in such cases it can be expected that the norm of a solution will be bounded away from zero as t → +∞ even as the damping factor γ becomes infinitely large.     

An O(n) algorithm for discrete n-point convex approximation with applications to continuous case

Given a bounded real function ? defined on a closed bounded real interval I, the problem is to find a convex function g so as to minimize the supremum of $\text{¦f(t) ? g(t)¦}$ for all t in I, over the class of all convex functions on I. The usual approach is to consider a discrete version of the problem on a grid of (n + 1) points in I, apply a conventional linear program to obtain an optimal solution, and let the grid size go to zero. This paper presents an alternative algorithm of complexity O(n), which is based on the concept of the greatest convex minorant of a function, for computation of a special “maximal” optimal solution to the discrete problem. It establishes the rate of convergence of this optimal solution to a solution of the original problem as the grid size goes to zero. It presents an alternative efficient linear program that generates the maximal optimal solution to the discrete problem. It also gives an O(n) algorithm for the discrete n-point monotone approximation problem.     

Conversion estimates for gas-solid reactions

Many chemical engineering processes involve a reaction between a diffusing substance and an immobile solid phase. The usual treatments are based on the assumptions that either diffusion or reaction dominates. Instead, we shall model an isothermal process in which reaction and diffusion are of the same order as would occur, for instance, in low temperature coke burning. Mass balances then lead to a parabolic system for the concentrations of the two phases. The system can be reduced to a scalar parabolic problem for the cumulative gas concentration. The popular pseudo-steady-state approximation is then obtained by setting the porosity ∈ equal to zero. This pseudo-steady-state problem is an elliptic problem in which time appears only as a parameter in the boundary condition. In previous work, we have shown that the pseudo-steady-state solution provides an O(∈) approximation to the exact concentration, uniformly in space and time. The present paper is concerned with estimates for the conversion, that is, the fraction of solid that has been converted to products by time t. We obtain bounds for the conversion in terms of a similar quantity (explicity calculable in some cases) for the pseudo-steady-state problem.     

Locally Lipschitz solutions of a single conservation law in several space variables

We consider the Cauchy problem for a single conservation law in several space variables. Letting u(x, t) denote the solution with initial data u₀, we state necessary and sufficient conditions on u₀ so that u(x, t) is locally Lipschitz continuous in the half space {t > 0}. These conditions allow for the preservation of smoothness of u₀ as well as for the smooth resolution of discontinuities in u₀. One consequence of our result is that u(x, t) cannot be locally Lipschitz unless u₀ has locally bounded variation. Another is that solutions which are bounded and locally Lipschitz continuous in {t > 0} automatically have boundary values u₀ at t = 0 in the sense that u(·, t) → u₀ in L_loc¹. Finally, we give an elementary proof that locally Lipschitz solutions satisfy Kruzkov's uniqueness condition.     

Stabilization of a locally damped incompressible wave equation

We show that the solutions of an incompressible vector wave equation with a locally distributed nonlinear damping decay in an algebraic rate to zero, that is, denoting by E(t) the total energy associated to the system, there exist positive constants C (which depends on E(0)) and γ satisfying, for t?0: E(t)?C(1+t)^−γ.     

Symplectic groups and the Klein-Gordon field

In this paper a Cohen factorization theorem x = a^t · x_t (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? a^t is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L¹($\text{R}$⁺) into B such that ∥ θ ∥ = 1 and θ(L¹($\text{R}$⁺)). B = B = B · θ(L¹($\text{R}$⁺)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.     

A self-correcting point process

Suppose a point process is attempting to operate as closely as possible to a deterministic rate ρ, in the sense of aiming to produce ρt points during the interval (0,t] for all t. This can be modelled by making the instantaneous rate of t of the process a suitable function of n-ρt, n being the number of points in [0, t]. This paper studies such a self-correcting point process in two cases: when the point process is Markovian and the rate function is very general, and when the point process is arbitrary and the rate function is exponential. In each case it is shown that as t→∞ the mean number of points occuring in (0, t] is ρt+O(1) while the variance is bounded further, in the Markov case all the absolute central moments are bounded. An application to the outputs of stationary D/M/s queues is given.     

An abstract Szeg? theorem

Given a semi-group U(t) of bounded linear operators with bounded self-adjoint generator A we estimate the logarithm of the section determinants of U(t) in terms of A. When A is subject to an additional condition, which is related to so-called Følner sequences of orthogonal projections, this estimate implies a Szeg? type theorem for bounded, self-adjoint, and strictly positive operators. We show that the condition mentioned is satisfied when A is a Toeplitz operator or a compact operator.     

Trace Estimates for Relativistic Stable Processes

In this paper, we study the asymptotic behavior, as the time t goes to zero, of the trace of the semigroup of a killed relativistic α-stable process in bounded C ^1,1 open sets and bounded Lipschitz open sets. More precisely, we establish the asymptotic expansion in terms of t of the trace with an error bound of order t ^2/α t ^?d/α for C ^1,1 open sets and of order t ^1/α t ^?d/α for Lipschitz open sets. Compared with the corresponding expansions for stable processes, there are more terms between the orders t ^?d/α and t ^(2?d)/α for C ^1,1 open sets, and, when α∈(0,1], between the orders t ^?d/α and t ^(1?d)/α for Lipschitz open sets.     

Generalized Feynman–Kac Semigroups,Associated Quadratic Forms and Asymptotic Properties

In this paper, we study the Feynman–Kac semigroup T _t f(x)=E _x[f(X _t)exp(N _t)],where X _t is a symmetric Levy process and N _t is a continuous additive functional of zero energy which is not necessarily of bounded variation. We identify the corresponding quadratic form and obtain large time asymptotics of the semigroup. The Dirichlet form theory plays an important role in the whole paper.     

Regularity and long time behavior of the Boltzmann equation for granular gases

This paper investigates regularity of solutions of the Boltzmann equation with dissipative collisions in a thermal bath. In the case of pseudo-Maxwellian approximation, we prove that for any initial datum f₀(ξ) in the set of probability density with zero bulk velocity and finite temperature, the unique solution of the equation satisfies f(ξ,t)∈H^∞(R³) for all t>0. Furthermore, for any t₀>0 and s?0 the Hs norm of f(ξ,t) is bounded for t?t₀. As a consequence, the exponential convergence to the unique steady state is also established under the same initial condition.     

Necessary and sufficient conditions for the existence and ɛ-uniqueness of bounded solutions of the equation x′ = f(x) − h(t)

We introduce the notion of ?-unique bounded solution to the nonlinear differential equation x′ = f(x) ? h(t), where f: ? → ? is a continuous function and h(t) is an arbitrary continuous function bounded on ?. We derive necessary and sufficient conditions for the existence and ?-uniqueness of bounded solutions to this equation.     

On the almost automorphy of bounded solutions of differential equations with piecewise constant argument

In this paper we give sufficient spectral conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=Ax([t])+f(t), t∈R, where A is a bounded linear operator in X and f is an X-valued almost automorphic function.     

Some properties of nonoscillating solutions of functional-differential equations ofn-th order

A functional-differential equation ofn-th order is considered, wheren≥2,m≥1 are integers andA _t/^t: C([t₀, ∞), R)→ R, i=1,2,...,m are functionals defined for everyt∈[t ₀, ∞). Sufficient conditions have been found for which all bounded non-oscillatory solutions and all non-oscillatory solutions of the functional-differntial equation tend to zero fort→∞.     

Refined distributional approximations for the uncovered set in the Johnson–Mehl model

Let Φ^z be the uncovered set (i.e., the complement of the union of intervals) at time z in the one-dimensional Johnson–Mehl model. We derive a bound for the total variation distance between the distribution of the number of components of Φ^z∩(0,t] and a compound Poisson-geometric distribution, which is sharper and simpler than an earlier bound obtained by Erhardsson. We also derive a previously unavailable bound for the total variation distance between the distribution of the Lebesgue measure of Φ^z∩(0,t] and a compound Poisson-exponential distribution. Both bounds are O(z_β(t)/t) as t→∞, where z_β(t) is defined so that the expected number of components of Φ^z_β(t)∩(0,t] converges to β>0 as t→∞, and the parameters of the approximating distributions are explicitly calculated.     

Nonlinear reaction-diffusion models for interacting populations

This paper proves that several initial-boundary value problems for a wide class of nonlinear reaction-diffusion equations have solutions c_i(x, t), 1 ? i ? N (with c_i(x, t) representing the concentration of the ith species at position x in a set $\text{Ω}$ at time t ? 0), which exist for all t ? 0 and are unique, smooth, nonnegative, and strictly positive for t > 0. The Volterra-Lotka predator-prey model with diffusion (to which the results above are proved to apply) is then studied in more detail. It is proved that any bounded solution of this model loses its spatial dependence and behaves like a periodic function of time alone as t → ∞. It is proved that if the spatial dimension is one or if the diffusion coefficients of the two species are equal, then all solutions are bounded.