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Integrated semigroups and the Cauchy problem for systems in $$L^ p$$ spaces. (English) Zbl 0766.47014
Summary: We prove that (under suitable hypotheses) every homogeneous differential operator on $$L^ p(\mathbb{R}^ n)^ N$$, corresponding to a system which is well-posed in $$L^ 2(\mathbb{R}^ n)^ N$$, generates an $$\alpha$$-times integrated semigroup on $$L^ p(\mathbb{R}^ n)^ N$$ $$(1<p<\infty)$$ whenever $$\alpha>n| 1/2-1/p|$$. For some special systems of mathematical physics, such as the wave equation or Maxwell’s equations this constant can be improved to be $$(n-1)| 1/2-1/p|$$.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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