Wellposedness of abstract Cauchy problems for second order differential equations
type: Item
instance of: scholarly article
title: Wellposedness of abstract Cauchy problems for second order differential equations
zbMATH Open document ID: 0723.47035
DOI: 10.1007/BF02764775
author: T. Takenaka
author: Noboru Okazawa
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let A be a densely defined closed linear operator in a Banach space X. The first aim of the paper is to characterize exponential wellposedness of the abstract Cauchy problem for the second order equation \(u''(t)=Au(t)\) with initial values u(0), \(u'(0)\in D(A^{k+1})\). The characterization is in terms of the resolvent of A. Let A satisfy these conditions and let (C(t); \(t\in {\mathbb{R}})\) be the corresponding family of generalized solution operators, C(t): D(A\({}^ k)\to X\). A new family (T(t); Re t\(>0)\) of operators in B(X) is defined by the abstract Weierstrass formula \[ T(t)x=\frac{1}{\sqrt{\pi t}}\int^{\infty}_{0}e^{-s^ 2/4t}C(s)x dx\quad (x\in D(A^ k)). \] It is shown that (T(t); \(t\geq 0)\) is a holomorphic semigroup of class \((H_ k)\) with complete infinitesimal generator A.
Mathematics Subject Classification ID: 47D09
Mathematics Subject Classification ID: 47E05
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 4192476
zbMATH Keywords: densely defined closed linear operator in a Banach space
zbMATH Keywords: exponential wellposedness of the abstract Cauchy problem
zbMATH Keywords: second order equation
zbMATH Keywords: generalized solution operators
zbMATH Keywords: Weierstrass formula
zbMATH Keywords: holomorphic semigroup
zbMATH Keywords: infinitesimal generator
reviewed by: Jürgen Voigt
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02764775
OpenAlex ID: W1991300557
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description: scientific article; zbMATH DE number 4192476