Distributions in [formula omitted] with a thick curve

Abstract

A theory of distributions called “thick distributions” was developed to incorporate a point singularity in the test function space. In this present article we consider a more general situation where test functions are singular on a curve in R3. We construct a topological vector space of such test functions and, by duality, the space D⁎,C′(R3) of distributions that are thick on this curve. We study several operations, including partial differentiation. We introduce the notion of line thick delta functions which is a lifting of line delta functions to D⁎,C′(R3). These new distributions, in particular thick line delta functions, may have applications in providing more accurate models to some problems from physics, biology and engineering where a “line source” or a “tube source” is present. As an example of such applications, we propose a more refined model of a growth factor's reaction and diffusion of a very thin blood capillary in a bulk tumor, and give a solution of the corresponding PDE.