On the Cauchy problem for convolution equations

Jan Kisyński¹

¹

Colloquium Mathematicum, Tome 133 (2013) no. 1, pp. 115-132.

Résumé

We consider one-parameter $(C_{0})$-semigroups of operators in the space $\mathcal S'({\mathbb R}^n;{\mathbb C}^m)$ with infinitesimal generator of the form $(G\,*)|_{\mathcal S'({\mathbb R}^n;{\mathbb C}^m)}$ where $G$ is an $M_{m\times m}$-valued rapidly decreasing distribution on ${\mathbb R}^n$. It is proved that the Petrovskiĭ condition for forward evolution ensures not only the existence and uniqueness of the above semigroup but also its nice behaviour after restriction to whichever of the function spaces $\mathcal S({\mathbb R}^n;{\mathbb C}^m)$, $\mathcal D_{L^{p}}({\mathbb R}^n;{\mathbb C}^m)$, $p\in [1,\infty ]$, $(\mathcal O_{a})({\mathbb R}^n;{\mathbb C}^m)$, $a\in \mathopen ]0,\infty \mathclose [$, or the spaces $\mathcal D'_{L^{q}}({\mathbb R}^n;{\mathbb C}^m)$, $q\in \mathopen ]1,\infty ]$, of bounded distributions.

Détail

DOI : 10.4064/cm133-1-8

Keywords: consider one parameter semigroups operators space mathcal mathbb mathbb infinitesimal generator form * mathcal mathbb mathbb where times valued rapidly decreasing distribution nbsp mathbb proved petrovski condition forward evolution ensures only existence uniqueness above semigroup its nice behaviour after restriction whichever function spaces mathcal mathbb mathbb mathcal mathbb mathbb infty mathcal mathbb mathbb mathopen infty mathclose spaces mathcal mathbb mathbb mathopen infty bounded distributions