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Proceedings of the American Mathematical Society

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Uniqueness and norm convexity in the Cauchy problem for evolution equations with convolution operators


Author: Monty J. Strauss
Journal: Proc. Amer. Math. Soc. 35 (1972), 423-430
MSC: Primary 35S10
DOI: https://doi.org/10.1090/S0002-9939-1972-0310478-2
MathSciNet review: 0310478
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Abstract: Uniqueness in the Cauchy problem is shown under suitable conditions for evolution equations of the form $ {u_t}(x,t) - B(t,{D_x})u(x,t) = 0$ , where B is a pseudo-differential operator of order $ k \geqq 0$ in the x variables. This is proved as a corollary to a norm convexity relation. In the process of showing this, an extension to Hölder's inequality is derived.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0310478-2
Keywords: Uniqueness, Cauchy problem, evolution equation, Hölder's inequality, pseudo-differential operators
Article copyright: © Copyright 1972 American Mathematical Society