Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
MathSciNet review: 0310478
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] A. Calderón, Uniqueness in the Cauchy problem for partial differential equations, Amer. J. Math. 80 (1958), 16-36. MR 21 #3675. MR 0104925 (21:3675)
  • [2] -, Existence and uniqueness theorems for systems of partial differential equations, Proc. Sympos. Fluid Dynamics and Appl. Math. (University of Maryland, 1961), Gordon and Breach, New York, 1962, pp. 147-195. MR 27 #6010. MR 0156078 (27:6010)
  • [3] B. Malgrange, Unicité du problème de Cauchy. Division des distributions, Séminaire Schwartz 1959/60, Faculté des Sci., Paris, 1960, §§8-11. MR 23 #A2275. MR 0151872 (27:1855)
  • [4] L. Nirenberg, Pseudo-differential operators, Proc. Sympos. Pure Math., vol. 16 Amer. Math. Soc., Providence, R.I., 1970. MR 42 #5108. MR 0270217 (42:5108)
  • [5] L. Nirenberg and F. Treves, On local solvability of linear partial differential equations. II. Sufficient conditions, Comm. Pure Appl. Math. 23 (1970), 459-509; correction, ibid. 24 (1971), 279-288. MR 41 #9064b. MR 0264471 (41:9064b)
  • [6] -, Remarks on the solvability of linear equations of evolution, Proc. Sympos. on Evolution Equations, Istituto di Alta Matematica, Rome 1970 (to appear).
  • [7] 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

Additional Information

Keywords: Uniqueness, Cauchy problem, evolution equation, Hölder's inequality, pseudo-differential operators
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society