Hieber, Matthias Integrated semigroups and the Cauchy problem for systems in \(L^ p\) spaces. (English) Zbl 0766.47014 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 \((n-1)| 1/2-1/p|\). Cited in 6 Documents 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 PDF BibTeX XML Cite \textit{M. Hieber}, J. Math. Anal. Appl. 162, No. 1, 300--308 (1991; Zbl 0766.47014) Full Text: DOI References: [1] Arendt, W, Vector valued Laplace transforms and Cauchy problems, Israel J. math., 59, 327-352, (1987) · Zbl 0637.44001 [2] Arendt, W, Sobolev imbeddings and integrated semigroups, (1990), preprint · Zbl 0762.47013 [3] Arendt, W; Kellermann, H, Integrated solutions of Volterra integro-differential equations and applications, () · Zbl 0675.45017 [4] Brenner, P, The Cauchy problem for symmetric hyperbolic systems in Lp, Math. scand., 19, 27-37, (1966) · Zbl 0154.11304 [5] Brenner, P, The Cauchy problem for systems in Lp and lp,α, Ark. mat., 11, 75-101, (1973) [6] Gilliam, D; Schullenberger, J.R, A class of symmetric hyperbolic systems with special properties, Comm. partial differential equations, 4, No. 5, 509-536, (1979) · Zbl 0448.35061 [7] Goldstein, J.A; Sandefur, J.T, Equipartion of energy for symmetric hyperbolic systems, () [8] \scM. Hieber, Laplace transforms and α-times integrated semigroups, Forum Math., to appear. · Zbl 0766.47013 [9] \scM. Hieber, Integrated semigroups and differential operators on Lp spaces, Math. Ann., to appear. · Zbl 0724.34067 [10] Hörmander, L, Estimates for translation invariant operators in Lp spaces, Acta math., 104, 93-140, (1960) · Zbl 0093.11402 [11] Kato, T, A short introduction to perturbation theory for linear operators, (1982), Springer-Verlag New York/Heidelberg/Berlin · Zbl 0493.47008 [12] Kellermann, H; Hieber, M, Integrated semigroups, J. funct. anal., 84, 160-180, (1989) · Zbl 0689.47014 [13] Kreiss, H.O, Über matrizen die beschränkte halbgruppen erzeugen, Math. scand., 7, 71-80, (1959) · Zbl 0090.09801 [14] Kreiss, H.O, Über sachgemäβe cauchyprobleme, Math. scand., 13, 109-128, (1963) · Zbl 0145.13303 [15] Littman, W, The wave operator and Lp-norms, J. math. mech., 12, 55-68, (1963) · Zbl 0127.31705 [16] Neubrander, F, Integrated semigroups and their applications to the abstract Cauchy problem, Pacific J. math., 135, 111-155, (1988) · Zbl 0675.47030 [17] Stein, E.M, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton, NJ · Zbl 0207.13501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.