The Laplace transform and the ascent method for abstract wave equations (Q1900947)

Let \(X\) be a Banach space and let \(A_i\), \(1 \leq i \leq N\), be the generators of uniformly bounded commuting cosine functions on \(X\). Let \(A\) be the closure of \(\sum^N_{i = 1} A_i\) and consider the second order Cauchy problem \(u''(t) = Au (t)\), \(t > 0\), \(u(0) = x\), \(u'(0) = y\). By methods based on the Laplace transform, it is shown that \(A\) is the generator of an \(\alpha\)-times integrated cosine function on \(X\) provided \(\alpha \geq {N - 1 \over 2}\). It follows in particular that the Laplacian \(\Delta\) generates an \(\alpha\)-times integrated cosine function on \(L^p (\mathbb{R}^N)\), \(1 < p < \infty\), provided \(\alpha \geq {N - 1 \over 2}\).