W.E. Fitzgibbon

Semilinear functional differential equations in Banach space

In this paper we prove existence and study the asymptotic behavior of mild solutions to a class of semi-linear abstract functional differentia1 equations which involve a nonlinear delay term. This class is characterized by the fact that the associated homogeneous linear differential equation generates a strongly continuous linear evolution system of compact operators. We also prove a regularity result by placing additional restrictions on our nonlinear delay term. Our approach is closely patterned on the recent treatment of abstract semilinear ordinary differential equations by Pazy [lo] and is similar to recent work on abstract functional differential equations by Travis and Webb [12]. More precisely we consider the nonlinear Banach space Volterra integral equation:

Semilinear integrodifferential equations in Banach space

Such equations have physical application; for example, they arise in problems concerned with heat flow in materials with memory. Linear versions of (1.2) are treated in [6], [ll], [24], [25], and [26] ; the nonlinear version is analysed in [14], [15], and [20] ; similar equations are also treated by [12] and [13]. If we replace u,(x, t) by u,,(x, t) we obtain an equation arising in the theory of viscoelasticity [4] and [5]. There is a growing recent literature concerning Volterra equations in abstract spaces. In [lo] Friedman and Shinbrot study a linear version of (1.1); they investigate the existence, boundedness and asymptotic behavior of solutions. R. Miller [23] associates a linear semigroup with solutions and he obtains information concerning the continuous dependence of solutions on initial data. Barbu [l] considers Volterra equations in Hilbert space; Crandall, Londen, and Nohel [2] and Crandall and Nohel [3] apply the theory of multivalued accretive operators to study such equations; Webb in [36] studies a version with A 3 0 and examines the semilinear equation in [33] and [34]. F or other related work the reader is referred to [7], [8], [30], [31], [32]. In this study we are motivated to apply the techniques of the theory of abstract quasi-linear equations of parabolic type found in Friedman [9] and Sobolevskii [29] to the class of semi-

A Mathematical Model of the Spread of Feline Leukemia Virus (FeLV) Through a Highly Heterogeneous Spatial Domain

We are concerned with a system of partial differential equations modeling the spread of feline leukemia virus (FeLV) through highly heterogeneous habitats or spatial domains. Our differential equations may feature discontinuities in the coefficients of divergence from differential operators and discontinuities in the coupling terms. Global well posedness, long term behavior, approximation, and homogenization results are provided.

Diffusion epidemic models with incubation and crisscross dynamics

A diffusion age-structured epidemic model is analyzed. The model describes an epidemic in a host-vector two-population system. Each population is diffusing in a spatial region. Each population is divided into susceptible, incubating, and infectious subclasses. The incubating and infectious subclasses in each population are determined by a structure variable corresponding to age since infection. The model consists of a system of nonlinear partial differential equations with crisscross dynamics. The existence, uniqueness, and asymptotic behavior of solutions are analyzed.

Stability for abstract nonlinear Volterra equations involving finite delay

ABSTRACT NONLINEAR VOLTERRA EQUATIONS 431 If IX(~) . ’ II v - + 1:~ It ~4s) exp (is 4~) d”) I/ X,(V) yE((II)!Ic ds 7 7 for t E [T, T]. (2.2) At this point we associate a nonlinear evolution operator with solutions to (2.1). Let p E C and let X(F) (t) denote the solution having prescribed initial history X,(F) = F. We define WY 4 v = %b) for t > 7. (2.3) Thus, for 0 < 7 < t we see that U(t, T): C 4 C; examination of the integral equation yields the observations that U(t, S) U(s, T) F = U(t, r) v for 0 < T < s < t, and that U(T, T) = 1 on C. Continuity properties of U(t, T) may be deduced directly from the integral equation.

Simple Models for the Transmission of Microparasites Between Host Populations Living on Noncoincident Spatial Domains

The goal of this chapter is to provide a simple deterministic mathe- matical approach to modeling the transmission of microparasites between two host populations living on distinct spatial domains. We shall consider two prototypi- cal situations: (1), a vector borne disease and, (2), an environmentally transmitted disease. In our models direct horizontal criss{cross transmission from infectious in- dividuals of one population to susceptibles of the other one does not occur. Instead parasite transmission takes place either through indirect criss{cross contacts be- tween infective vectors and susceptible individuals and vice{versa in case (1), and through indirect contacts between susceptible hosts and the contaminated part of the environment and vice{versa in case (2). We shall also assume the microparasite is benign in one of the host populations, a reservoir, that is it has no impact on demography and dispersal of individuals. Next we assume it is lethal to the second population. In applications we have in mind the second population is human while the ¯rst one is an animal { avian or rodent { population. Simple mathematical deter- ministic models with spatio{temporal heterogeneities are developed, ranging from basic systems of ODEs for unstructured populations to Reaction{Di®usion mod- els for spatially structured populations to handle heterogeneous environments and populations living in distinct habitats. Besides showing the resulting mathematical problems are well{posed we analyze the existence and stability of endemic states. Under some circumstances, persistence thresholds are given.

Nonlinear volterra equations with infinite delay

AbstractThis paper is concerned with the existence and stability of nonlinear Volterra equations which have infinite delay and are of the form:

Weakly continuous accretive operators

x (\varphi ) (t) = W (t, \tau ) \varphi (0) + \int\limits_\tau ^t {W (t, s)} F(s,x_s (\varphi )) ds, x_\tau (\varphi ) = \varphi \in C_u .