Mild well-posedness of second order differential equation on the real line
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.
scientific article; zbMATH DE number 6157595