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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Some abstract Cauchy problems in exceptional cases

Author: Louis R. Bragg
Journal: Proc. Amer. Math. Soc. 65 (1977), 105-112
MSC: Primary 34G05
MathSciNet review: 0492651
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Abstract: Let X be a Banach space and let $ A = {B^2}$ in which B is the infinitesimal generator of a strongly continuous group in X with dense domain $ \mathcal{D}(A)$. This paper develops solutions of the abstract Euler-Poisson-Darboux problem

\begin{displaymath}\begin{array}{*{20}{c}} {{u_{tt}}(t) + \frac{{1 - 2m}}{t}{u_t... ...i \in \mathcal{D}({A^r}),\quad r\; > m,} \hfill \\ \end{array} \end{displaymath}

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 = {\Delta _n}$, the Laplacian) in the exceptional cases.

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Keywords: Banach space, strongly continuous group, singular Cauchy problems, Euler-Poisson-Darboux equation, hypergometric equations, logarithmic solutions
Article copyright: © Copyright 1977 American Mathematical Society

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