Some Remarks on Uniqueness for a Class of Singular Abstract Cauchy Problems

Abstract

Of concern is the Cauchy problem for equations of the form $u''(t) + \alpha (t)u’(t) + {S^2}u(t) = 0(’ = d/dt)$ on a complex Hilbert space $X$. $S$ is a selfadjoint operator on $X$ while $\alpha$ is a continuous function on $(0,\infty )$ which can be unbounded at $t = 0$. Uniqueness results are obtained for these equations by applying a uniqueness theorem for nonlinear equations. Furthermore, nonuniqueness examples for the linear abstract Euler-Poisson-Darboux equation, which is contained in this class, show that the uniqueness theorem is best possible.