Some abstract Cauchy problems in exceptional cases

Abstract

Let X be a Banach space and let A = B 2 A = {B^2} in which B is the infinitesimal generator of a strongly continuous group in X with dense domain D ( A ) \mathcal {D}(A) . This paper develops solutions of the abstract Euler-Poisson-Darboux problem \[ u t t ( t ) + 1 − 2 m t u t ( t ) = A u ( t ) , a m p ; t > 0 , m = 1 , 2 , 3 , … , ‖ u ( t ) − ϕ ‖ → 0 as t → 0 , a m p ; ϕ ∈ D ( A r ) , r > m , \begin {array}{*{20}{c}} {{u_{tt}}(t) + \frac {{1 - 2m}}{t}{u_t}(t) = Au(t),} & {t > 0,\quad m = 1,2,3, \ldots ,} \hfill \\ {\left \| {u(t) - \phi } \right \|\to 0\;{\text {as}}\;t \to 0,} & {\phi \in \mathcal {D}({A^r}),\quad r\; > m,} \hfill \\ \end {array} \] and associated Cauchy problem in terms of solutions of related abstract wave problems. Connections between solutions of certain abstract hypergeometric equations play an important role in these developments. J. B. Diaz and H. Weinberger and E. K. Blum have obtained solutions of the standard Euler-Poisson-Darboux problem (i.e. A = Δ n A = {\Delta _n} , the Laplacian) in the exceptional cases.