On the concatenability of solutions of partial differential equations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914380175638528 |
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| author | Sasane, Sara Maad Sasane, Amol |
| author_facet | Sasane, Sara Maad Sasane, Amol |
| contents | 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 $d=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08608 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the concatenability of solutions of partial differential equations Sasane, Sara Maad Sasane, Amol Analysis of PDEs 35E20, 46F10 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 $d=1$. |
| title | On the concatenability of solutions of partial differential equations |
| topic | Analysis of PDEs 35E20, 46F10 |
| url | https://arxiv.org/abs/2603.08608 |