stability properties of linear volterra equations where a t and b t t s are almost periodic in t when eq e is of convolution type that is a t a a constant and b t s b t s miller 121 and grossman and miller 7 have shown that the null solution of e with b e l 0 co is uniformly asymptotically stable if and only if the resolvent r t s r t s for e is of class l 0 cc in section 3 we show theorem 3 1 that the null solution of e is uniformly asymptotically stable if and only if sup a r a o 1 z t a j iz t s l ds co where z t s is the principal matrix solution of e introduced by becker 11 as will be noted later z t s is identical with the resolvent r t s therefore our theorem 3 1 is an extension of 12

on the existence of solutions of an equation arising in the theory of laminar flow in a uniformly porous channel with injection we establish the existence of concave solutions of berman s equation which describes the laminar flow in channels with injection through porous walls it is found that the unique concave solutions exist for all injection reynolds number r 0

asymptotic constancy of solutions of linear parabolic volterra difference equations abstract in this paper we obtain some sufficient conditions under which every solution of a class of linear parabolic volterra difference equations of neutral type tends to a constant as n we also consider the asymptotic constancy of solutions for a class of linear ordinary volterra difference equations of neutral type the results obtained improve and generalize some known results

generalized cahn hilliard equation for biological applications recently we considered a stochastic discrete model which describes fronts of cells invading a wound e khain l m sander and c m schneider mizell j stat phys 128 209 2007 in the model cells can move proliferate and experience cell cell adhesion in this work we focus on a continuum description of this phenomenon by means of a generalized cahn hilliard equation gch with a proliferation term as in the discrete model there are two interesting regimes for subcritical adhesion there are propagating pulled fronts similar to those of the fisher kolmogorov equation the problem of front velocity selection is examined and our theoretical predictions are in the good agreement with a numerical solution of the gch equation for supercritical adhesion there is a nontrivial transient behavior where density profile exhibits a secondary peak to analyze this regime we investigated relaxation dynamics for the cahn hilliard equation without proliferation we found that the relaxation process exhibits self similar behavior the results of continuum and discrete models are in good agreement with each other for the different regimes we analyzed

sharp interface limits of the cahn hilliard equation with degenerate mobility in this work sharp interface limits for the degenerate cahn hilliard equation with a polynomial double well free energy and a mobility that vanishes at the minima of the double well are derived for the choice of a quadratic mobility the leading order sharp interface motion is not governed by pure surface diffusion as has been previously claimed in the literature but contains a contribution from nonlinear porous medium type bulk diffusion at the same order our analysis reveals that there are two subcases one where the solution for the order parameter is bounded between the minima proven to exist for the first mobility by elliott and garcke siam j math anal 27 1996 pp 404 423 and one where it converges to the classical stationary solution of the cahn hilliard equation consistent treatment of the bulk diffusion requires the matching of exponentially large and small terms in combination with multiple inner layers moreover the leading order sharp interface motion depends sensitively on the choice of mobility the asymptotic analysis shows that for example with a biquadratic mobility the leading order sharp interface motion is driven only by surface diffusion the sharp interface models are corroborated by comparing relaxation rates of perturbations to a radially symmetric stationary state with those obtained by the phase field model

vortical and acoustical mode coupling inside a two dimensional cavity with transpiring walls in a long low aspect ratio two dimensional cavity where gaseous motion is permitted along transpiring walls a time dependent field is established when low amplitude sinusoidal pressure oscillations with nonzero mean are introduced an accurate solution is extracted here for the time dependent field by way of small parameter perturbations contingent upon small pressure wave amplitudes navier stokes equations are linearized to the order of the mean flow mach number to furnish interaction equations governing the unsteady field the latter is decomposed into acoustic and solenoidal fields coupled through dirichlet type boundary conditions solving for the solenoidal field from the momentum equation employs separation of variables and multiple scale expansions based on a careful choice of an inner scale in fact the unique inner scale used in the two variable derivative expansion method is original in the sense that it stems from an unconventional nonlinear variable transformation a uniformly valid s

spectral method study of domain coarsening the self organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as ostwald ripening the late stage of ostwald ripening is studied here through the use of a mesoscopic model in conjunction with a special configuration that allows for the direct measurement of system characteristics such as domain size the mesoscopic model is a stochastic partial integrodifferential equation and is studied through the use of recently developed and benchmarked spectral schemes the size of the growing region is not observed to increase as a power law in this model during the late stages of self organization in contrast to the power law growth observed in simulations of the earlier stages of self organization the results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system

invariant and attracting sets of volterra difference equations with delays will be concerned with the invariant and asymptotic properties of volterra differ ence equations with delays sufficient conditions for determining the invariant and attracting sets of the equations are obtained examples are given to illustrate the obtained results 2003 elsevier science ltd all rights reserved

reaction diffusion systems with time delays monotonicity invariance comparison and convergence in a recent paper we have obtained differential inequality results developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a banach space the primary application for these results is to reaction diffusion systems in which the reaction nonlinearity contains time delays in the present paper we apply these results to reaction diffusion systems with time delays of the form u t d u f x u t x x u u n 0 x where is a bounded open set in r n with smooth boundary u r n d diag d 1 d n with d i 0

notes and corrections on asymptotic constancy criteria of solution of linear parabolic volterra difference equations abstract in this paper we make some notes and corrections on asymptotic constancy criteria of solutions of linear parabolic volterra difference equations 1

asymptotic behavior of nonlinear difference equations with delays abstract this paper deals with asymptotic behavior of nonlinear difference equations with delays sufficient conditions for attracting set and basin of attraction of the equations are obtained examples are given to show the applicability of the proposed approach

asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients abstract in this paper we obtain sufficient conditions under which every solution of the neutral differential equation with positive and negative coefficients x t c t x t r p t x t q t x t 0 t t 0 tends to a constant as t

the oscillatory channel flow with large wall injection in the presence of smallamplitude pressure oscillations the linearized navierstokes equations are solved to obtain an accurate description of the timedependent field in a channel having a rectangu

computational studies of coarsening rates for the cahn hilliard equation with phase dependent diffusion mobility we study computationally coarsening rates of the cahn hilliard equation with a smooth double well potential and with phase dependent diffusion mobilities the latter is a feature of many materials systems and makes accurate numerical simulations challenging our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis we demonstrate that the numerical solutions are consistent with the physical gibbs thomson effect even if the mobility is degenerate in one or both phases for the two sided degenerate mobility we report computational results showing that the coarsening rate is on the order of l c t 1 4 independent of the volume fraction of each phase for the one sided degenerate mobility that is non degenerate in the positive phase but degenerate in the negative phase we illustrate that the coarsening rate depends on the volume fraction of the positive phase for large positive volume fractions the coarsening rate is on the order of l c t 1 3 and for small positive volume fractions the coarsening rate becomes l c t 1 4

weak solutions for the cahn hilliard equation with degenerate mobility in this paper we study the well posedness of cahn hilliard equations with degenerate phase dependent diffusion mobility we consider a popular form of the equations which has been used in phase field simulations of phase separation and microstructure evolution in binary systems we define a notion of weak solutions for the nonlinear equation the existence of such solutions is obtained by considering the limits of cahn hilliard equations with non degenerate mobilities

phase separation kinetics in a model with order parameter dependent mobility we present extensive results from two dimensional simulations of phase separation kinetics in a model with order parameter dependent mobility we find that the time dependent structure factor exhibits dynamical scaling and the scaling function is numerically indistinguishable from that for the cahn hilliard ch equation even in the limit where surface diffusion is the mechanism for domain growth this supports the view that the scaling form of the structure factor is universal and leads us to question the conventional wisdom that an accurate representation of the scaled structure factor for the ch equation can only be obtained from a theory which correctly models bulk diffusion

periodicity and stability of linear volterra difference systems abstract we give sufficient conditions for a system of volterra difference equations to have a unique periodic solution then a criterion for uniform asymptotic stability is established

the oscillatory pipe flow with arbitrary wall injection the linearized navierstokes equations play a central role in describing the unsteady motion of a viscous fluid inside a porous tube asymptotic solutions of these equations have been found and here

vortical and acoustical mode coupling inside a porous tube with uniform wall suction this paper considers the oscillatory motion of gases inside a long porous tube of the closed open type in particular the focus is placed on describing an analytical solution for the internal acoustico vortical coupling that arises in the presence of appreciable wall suction this unsteady field is driven by longitudinal oscillatory waves that are triggered by small unavoidable fluctuations in the wall suction speed under the assumption of small amplitude oscillations the time dependent governing equations are linearized through a regular perturbation of the dependent variables further application of the helmholtz vector decomposition theorem enables us to discriminate between acoustical and vortical equations after solving the wave equation for the acoustical contribution the boundary driven vortical field is considered the method of matched asymptotic expansions is then used to obtain a closed form solution for the unsteady momentum equation developing from flow decomposition an exact series expansion is also derived and shown to coincide with the numerical solution for the problem the numerically verified end results suggest that the asymptotic scheme is capable of providing a sufficiently accurate solution this is due to the error associated with the matched asymptotic expansion being smaller than the error introduced in the navier stokes linearization a basis for comparison is established by examining the evolution of the oscillatory field in both space and time the corresponding boundary layer behavior is also characterized over a range of oscillation frequencies and wall suction velocities in general the current solution is found to exhibit features that are consistent with the laminar theory of periodic flows by comparison to the sexl profile in nonporous tubes the critically damped solution obtained here exhibits a slightly smaller overshoot and depth of penetration these features may be attributed to the suction effect that tends to attract the shear layers closer the wall