Second order incomplete expiring Cauchy problems (Q1824842)

On a Banach space let A be a linear operator with nonempty resolvent and consider the problem \[ u''(t)=A(u(t)),\quad u(0)=x,\quad \lim_{t\to \infty}\| u^{(k)}(t)\| =0,\quad k=0,1,2. \] This problem has a unique solution continuously depending on \(x\in {\mathcal D}(A)\) iff A has a square root that generates a \(C_ 0\) semigroup, that strongly converges to zero as t goes to infinity. Interesting applications to some partial differential equations close this well written paper.