Integrated semigroups and the Cauchy problem for systems in \(L^ p\) spaces. (English) Zbl 0766.47014

Hieber, Matthias

J. Math. Anal. Appl. 162, No. 1, 300-308 (1991).

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.)

Keywords:

homogeneous differential operator; integrated semigroup; wave equation; Maxwell’s equations

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