On well-posedness of the linear Cauchy problem with the distributional right-hand side and discontinuous coefficients

TL;DR

This paper establishes the well-posedness of a linear Cauchy problem involving distributional right-hand sides and discontinuous coefficients by working in a specialized distribution space that allows multiplication with discontinuous functions.

Contribution

It introduces a framework in the space of distributions with discontinuous test functions to prove existence and uniqueness of solutions for systems with distributional and discontinuous components.

Findings

Unique solution exists and depends continuously on the data.

The problem is ill-posed in classical distribution spaces but well-posed in the new framework.

The approach handles products of distributions and discontinuous functions effectively.

Abstract

We prove the well-posedness of the Cauchy problem for the linear differential system of the form $x^{\prime}-A(t)x=f$, where $f$ is a distribution and $A$ possesses at most first-kind discontinuities together with all its derivatives defined almost everywhere. The left-hand side of this system contains the product of a distribution and, in general, a discontinuous function, which is undefined in the classical space of the distributions with the smooth test functions $\mathcal D'$, so the Cauchy problem has no solution in $\mathcal D'$. In what follows, we cosider this system in the space of distributions with the discontinuous test functions, whose elements admit continuous and associative multiplication by functions possessing at most first-kind discontinuities (together with all their derivatives defined almost everywhere), and show that there exists the unique solution of the Cauchy problem which depends continuously on $f$.