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Some remarks on uniqueness for a class of singular abstract Cauchy problems


Authors: James A. Donaldson and Jerome A. Goldstein
Journal: Proc. Amer. Math. Soc. 54 (1976), 149-153
DOI: https://doi.org/10.1090/S0002-9939-1976-0390408-1
MathSciNet review: 0390408
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0390408-1
Keywords: Singular Cauchy problem, uniqueness, dissipative operators, EPD equation
Article copyright: © Copyright 1976 American Mathematical Society

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