Mild well-posedness of second order differential equation on the real line
type: Item
instance of: scholarly article
title: Mild well-posedness of second order differential equation on the real line
zbMATH Open document ID: 1279.47064
author: Gang Cai
author: Shangquan Bu
published in: Taiwanese Journal of Mathematics
publication date: 2013-04-25T00:00:00Z
full work available at URL: http://journal.taiwanmathsoc.org.tw/index.php/TJM/article/view/1710
review text: This paper studies the incomplete second order differential equation in a Banach space \(X\) \[ u''=Au+f, \] where \(A\) is a densely defined closed operator on \(X\) with domain \(D(A)\). For \(1\leq p<\infty\), the authors show that the following two assertions are equivalent. (i) There exists a bounded linear operator \({\mathcal B}\) mapping \(L^p(\mathbb R,X)\) continuously into itself with range in \(W^{1,p}(\mathbb R,X)\) such that \({\mathcal B}(W^{1,p}(\mathbb R,D(A)))\subset W^{2,p}(\mathbb R,D(A))\) and \({\mathcal A}{\mathcal B}u={\mathcal B}{\mathcal A}u\) for \(u\in W^{2,p}(\mathbb R,D(A))\), where \({\mathcal A}u:=u''-Au\) for \(u\in W^{2,p}(\mathbb R,D(A))\). (ii) \((-\infty, 0]\subset \rho(A)\) and the functions \(m_1(x)=-(x^2+A)^{-1}\) and \(m_2(x)=-ix(x^2+A)^{-1}\), defined on \(\mathbb R\), are \(L^p\)-Fourier multipliers.
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 42A45
Mathematics Subject Classification ID: 34K30
zbMATH DE Number: 6157595
zbMATH Keywords: second order differential equations
zbMATH Keywords: mild well-posedness
zbMATH Keywords: \(L^p\)-Fourier multipliers
Wikidata QID: Q115243012
reviewed by: Ti-Jun Xiao
MaRDI profile type: Publication
OpenAlex ID: W2093130217
DOI: 10.11650/TJM.17.2013.1710
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description: scientific article; zbMATH DE number 6157595