On the concatenability of solutions of partial differential equations

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 $p\in \mathbb{C}[ξ_1,\cdots, ξ_d,τ]$, 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 $p=a_0+a_1τ+\cdots+a_{d}τ^{d}\in \mathbb{C}[ξ_1,\cdots, ξ_d][τ]$, where $a_0,\cdots, a_{d}\in \mathbb{C}[ξ_1,\cdots, ξ_d]$ and $d\in \mathbb{N}$, $S_p$ has the concatenation property if and only if $d=1$.

Theorems & Definitions (7)

• Theorem 2.1
• proof
• Proposition 3.1: Jump Rule
• proof
• Theorem 4.1
• proof
• Remark 4.2