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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Characterization for the solvability of nonlinear partial differential equations

Author: Elemer E. Rosinger
Journal: Trans. Amer. Math. Soc. 330 (1992), 203-225
MSC: Primary 35D05; Secondary 35A05, 46F10
MathSciNet review: 1028764
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Abstract: Within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been obtained. One of them has been the first global version of the Cauchy-Kovalevskaia theorem, which proves the existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear PDEs. These generalized solutions are analytic everywhere, except for closed, nowhere dense subsets which can be chosen to have zero Lebesgue measure.

This paper gives a certain extension of that result by establishing an algebraic necessary and sufficient condition for the existence of generalized solutions for arbitrary polynomial nonlinear PDEs with continuous coefficients. This algebraic characterization, given by the so-called neutrix or off diagonal condition, is proved to be equivalent to certain densely vanishing conditions, useful in the study of the solutions of general nonlinear PDEs.

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  • [1] H. A. Biagioni, Introduction to a nonlinear theory of generalized functions, Notas de Matematica, IMECC, Campinas, Brazil, 1988.
  • [2] J. F. Colombeau, New generalized functions and multiplication of distributions, North-Holland, Amsterdam, 1984. MR 738781 (86c:46042)
  • [3] -, Elementary introduction to new generalized functions, North-Holland, Amsterdam, 1985. MR 808961 (87f:46064)
  • [4] G. Köthe, Topological vector spaces. I, Springer, Heidelberg, 1960.
  • [5] M. Oberguggenberger, Weak limits of solutions to semilinear hyperbolic systems, Math. Ann. 274 (1986), 599-607. MR 848504 (87j:35242)
  • [6] -, Generalized solutions to semilinear hyperbolic systems, Mh. Math. 103 (1987), 133-144. MR 881719 (88e:35119)
  • [7] -, Semilinear wave equations with rough initial data: generalized solutions, P. Antosik and A. Kaminski (Eds.), Generalized Functions and Convergence, World Scientific Publ., London, 1990. MR 1085506 (92f:35104)
  • [8] E. E. Rosinger, Distributions and nonlinear partial differential equations, Springer, Berlin, 1978. MR 514014 (80j:46067)
  • [9] -, Nonlinear partial differential equations, sequential and weak solutions, North-Holland, Amsterdam, 1980. MR 590891 (83i:35002)
  • [10] E. E. Rosinger, Generalized solutions of nonlinear partial differential equations, North-Holland, Amsterdam, 1987. MR 918145 (89g:35001)
  • [11] -, Nonlinear partial differential equaations, an algebraic view of generalized solutions, North-Holland, Amsterdam, 1990.
  • [12] -, Global version of the Cauchy-Kovalevskaia theorem for nonlinear $ PDEs$, Acta Appl. Math. (to appear). MR 1096585 (93g:35006)
  • [13] J. A. Seebach, Jr., L. A. Seebach, and L. A. Steen, What is a sheaf?, Amer. Math. Monthly (1970), 681-703. MR 0263073 (41:7678)
  • [14] J. C. Van der Corput, Introduction to neutrix calculus, J. Analyse Math. 7 (1959), 281-398. MR 0124678 (23:A1989)
  • [15] R. C. Walker, The Stone-Čech compactification, Springer, Berlin, 1974. MR 0380698 (52:1595)

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Article copyright: © Copyright 1992 American Mathematical Society

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