On the concatenability of solutions of partial differential equations

TL;DR

This paper investigates when solutions of linear partial differential equations can be concatenated while remaining solutions, concluding that this property holds only for equations of first order.

Contribution

It establishes a precise criterion showing that the concatenability property of solutions occurs exclusively for first-order linear PDEs.

Findings

Concatenability holds only for first-order PDEs.

Higher-order PDE solution sets lack the concatenation property.

The result characterizes the structure of solution spaces based on order.

Abstract

Let ${\mathcal{D}}'({\mathbb{R}}^d)$ denote the space of distributions on ${\mathbb{R}}^d$. For a linear partial different equation $p(\frac{\partial}{\partial x_1},\cdots, \frac{\partial}{\partial x_d}, \frac{\partial}{\partial t}) u=0$ (briefly $D_pu=0$) corresponding to a polynomial $p\in \mathbb{C}[\xi_1,\cdots, \xi_d,\tau]$, let $S_p:=\{u\in C(\mathbb{R}, {\mathcal{D}}'({\mathbb{R}}^d)):D_pu=0\}$. The set $S_p$ has the `concatenability property' if whenever $u_1,u_2\in S_p\cap C^1(\mathbb{R}, {\mathcal{D}}'({\mathbb{R}}^d))$ are such that $u_1(0)=u_2(0)$, their concatenation $u_1\& u_2$ (defined to be $u_1(t)$ for $t\le 0$, and $u_2(t)$ for $t\ge 0$) belongs to $S_p$. It is shown that for $p=a_0+a_1\tau+\cdots+a_{d}\tau^{d}\in \mathbb{C}[\xi_1,\cdots, \xi_d][\tau]$, where $a_0,\cdots, a_{d}\in \mathbb{C}[\xi_1,\cdots, \xi_d]$ and $d\in \mathbb{N}$, $S_p$ has the concatenation property if and only if $d=1$.