On the Cauchy problem for convolution equations

Colloquium Mathematicae, Tome 131 (2013), p. 115-132 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider one-parameter (C₀)-semigroups of operators in the space $^{'} (ℝ ⁿ; ℂ^{m})$ with infinitesimal generator of the form ${(G *) |}_{^{'} (ℝ ⁿ; ℂ^{m})}$ where G is an $M_{m \times m}$ -valued rapidly decreasing distribution on ℝⁿ. 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 $(ℝ ⁿ; ℂ^{m})$ , $_{L^{p}} (ℝ ⁿ; ℂ^{m})$ , p ∈ [1,∞], $(_{a}) (ℝ ⁿ; ℂ^{m})$ , a ∈ ]0,∞[, or the spaces $_{L^{q}}^{'} (ℝ ⁿ; ℂ^{m})$ , q ∈ ]1,∞], of bounded distributions.

Publié le : 2013-01-01

Zbl 1295.35374

EUDML-ID : urn:eudml:doc:286490

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     volume = {131},
     year = {2013},
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 (éd.). On the Cauchy problem for convolution equations. Colloquium Mathematicae, Tome 131 (2013) pp. 115-132. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm133-1-8/