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

Strauss, Monty J.

doi:10.1090/S0002-9939-1972-0310478-2

Uniqueness and norm convexity in the Cauchy problem for evolution equations with convolution operators
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Proc. Amer. Math. Soc. 35 (1972), 423-430 Request permission

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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Remarks on the solvability of linear equations of evolution

Uniqueness and norm convexity for the Cauchy problem