English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we present a general method for generation of uniformlycontinuous groups on abstract Fr\'{e}chet spaces (without appealing to spectraltheory) and apply it to a such space of distributions, namely ${\mathscrF}L^{2}_{loc}(\mathbb{R}^{n})$, so that the linear evolution problem\begin{equation*} \left\{\begin{array}{l} u_{t} = a(D)u, t \in \mathbb{R} \\u(0) = u_0 \end{array} \right. \end{equation*}always has a unique solution insuch a space, for every pseudodifferential operator $a(D)$ with constantcoefficients. We also provide necessary and sufficient conditions so that thespaces $L^{2}$ and ${\mathscr E}'$ are left invariant by this group; and weconclude that the solution of the heat equation on ${\mathscrF}L^{2}_{loc}(\mathbb{R}^{n})$ for all $t \in \mathbb{R}$ extends the standardsolution on Hilbert spaces for $t \geqslant 0$.

On the generation of groups of bounded linear operators on Fr\'{e}chet spaces