On the concatenability of solutions of partial differential equations

Let ${\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)$ denote the space of distributions on ${\mathbb{R}}^d$. For a linear partial different equation $p(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1},\cdots ,\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_d},\frac{\mathrm{\partial }}{\mathrm{\partial }t})u=0$ (briefly $D_{p}u=0$) corresponding to a polynomial $p\in \mathbb{C}[{\xi }_1,\cdots ,{\xi }_d,\tau ]$, let $S_p:=\{u\in C(\mathbb{R},{\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d)):D_{p}u=0\}$. The set $S_p$ has the `concatenability property’ if whenever $u_1,u_2\in S_p\cap C^1(\mathbb{R},{\mathcal{D}}^{\mathrm{\prime }}({\mathbb{R}}^d))$ are such that $u_1(0)=u_2(0)$, their concatenation $u_1\mathrm{\&}{u}_2$ (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 $d=1$.