Wellposedness of abstract Cauchy problems for second order differential equations (Q756684)

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.