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 ) + α ( t ) u ′ ( t ) + S 2 u ( t ) = 0 ( ′ = d / d t ) u(t) + \alpha (t)u’(t) + {S^2}u(t) = 0(’ = d/dt) on a complex Hilbert space X X . S S is a selfadjoint operator on X X while α \alpha is a continuous function on ( 0 , ∞ ) (0,\infty ) which can be unbounded at t = 0 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.