Uniqueness for a nonlinear abstract Cauchy problem (Q1120719)

Let H be a complex Hilbert space and let A be a linear (in general, unbounded) operator defined on a domain D in H. For the operator \(L=d^ n/dt^ n-A,\) the author shows under rather general conditions on the operator A, that the only solution to the differential inequality \[ \| Lu(t)\|^ 2\leq c[\omega (t)+\int^{t}_{0}\omega (s)ds], \] where \(\omega (t)=\| u(t)\|^ 2+\sum^{n-1}_{j=1}\| d^ ju(t)/dt^ j\|^ 2\) for which \(\omega (0)=0\) is the trivial one. A similar result is given for \[ \| Lu(t)\|^ 2\leq c[\mu (t)+\int^{t}_{0}\mu (s)ds], \] where \(\mu (t)=\omega (t)+| (Mu(t),u(t))|\) and the operator M is the symmetric part of A. These results extend earlier theorems (for \(n=1,2\) but A dependent on t) by \textit{G. N. Hile} and \textit{M. H. Protter} [Pac. J. Math. 99, 57-88 (1982; Zbl 0525.34007)].