Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Uniqueness and norm convexity in the Cauchy problem for evolution equations with convolution operators
HTML articles powered by AMS MathViewer

by Monty J. Strauss
Proc. Amer. Math. Soc. 35 (1972), 423-430
DOI: https://doi.org/10.1090/S0002-9939-1972-0310478-2

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.
References
  • A.-P. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16–36. MR 104925, DOI 10.2307/2372819
  • A.-P. Calderón, Existence and uniqueness theorems for systems of partial differential equations, Fluid Dynamics and Applied Mathematics (Proc. Sympos., Univ. of Maryland, 1961) Gordon and Breach, New York, 1962, pp. 147–195. MR 0156078
  • Bernard Malgrange, Noyaux valeurs principales, Séminaire Schwartz (1959/60), Faculté des Sciences de Paris, Paris, 1959/1960, pp. Exp. 5–6, 11 (French). MR 0151872
  • L. Nirenberg, Pseudo-differential operators, Global Analysis (Proc. Sympos. Pure Math., Vols. XIV, XV, XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 149–167. MR 0270217
  • Louis Nirenberg and François Trèves, On local solvability of linear partial differential equations. I. Necessary conditions, Comm. Pure Appl. Math. 23 (1970), 1–38. MR 264470, DOI 10.1002/cpa.3160230102
  • —, Remarks on the solvability of linear equations of evolution, Proc. Sympos. on Evolution Equations, Istituto di Alta Matematica, Rome 1970 (to appear). M. Strauss, Uniqueness and norm convexity for the Cauchy problem, Thesis, New York University, 1971.
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35S10
  • Retrieve articles in all journals with MSC: 35S10
Bibliographic Information
  • © Copyright 1972 American Mathematical Society
  • 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