Conditions for well-posedness in the Hadamard sense in spaces of generalized functions
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We have used expansions into orthogonal series. In the general case it is natural to use spectral expansions of linear selfadjoint operators. For this one may use the results of [13, 14].
Can the concept of weak well-posedness be carried over to nonlinear problems? Here, probably, J. F. Colombeau's method will turn out to be useful (see [13, 14]).
KeywordsGeneralize Function Nonlinear Problem Selfadjoint Operator Orthogonal Series Spectral Expansion
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- 1.S. L. Sobolev, “Methode nouvelle a resoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales,” Mat. Sb.,1 (43), 39–72 (1936).Google Scholar
- 2.A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems, Wiley, New York (1977).Google Scholar
- 3.M. M. Lavrent'ev, Conditionally Well-Posed Problems for Differential Equations [in Russian], Novosibirsk State Univ. (1973).Google Scholar
- 4.V. P. Maslov, “The existence of a solution of an ill-posed problem is equivalent to the convergence of a regularization process,” Usp. Mat. Nauk,23, No. 3, 183–184 (1968).Google Scholar
- 5.A. H. Zemanian, Generalized Integral Transformations, Interscience, New York (1968).Google Scholar
- 6.G. N. Mil'shtein, “The extension of semigroups of operators in locally convex spaces,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 2, 91–95 (1977).Google Scholar
- 7.J. de Graaf, “A theory of generalized functions based on holomorphic semigroups,” Nederl. Akad. Wetensch. Proc., A:86A, No. 4, 407–420 (1983); B:87A, No. 2, 155–171 (1984); C:87A, No. 2, 173–187 (1984).Google Scholar
- 8.S. Pilipovic, “Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions,” SIAM J. Math. Anal.,17, No. 2, 477–484 (1986).Google Scholar
- 9.V. Wrobel, “Generating Frechet-Montel spaces that are not Schwartz by closed linear operators,” Arch. Math. (Basel),46, No. 3, 257–260 (1986).Google Scholar
- 10.A. Szaz, “Periodic generalized functions,” Publ. Math. Debrecen,25, No. 3–4, 229–235 (1978).Google Scholar
- 11.A. Szaz, “Generalized periodic distributions,” Rev. Roumaine Math. Pures Appl.,23, No. 10, 1577–1582 (1978).Google Scholar
- 12.Yu. A. Dubinskii, “The algebra of pseudodifferential operators with analytic symbols and its applications to mathematical physics,” Usp. Mat. Nauk,37, No. 5, 97–137 (1982).Google Scholar
- 13.I. M. Gel'fand and A. K. Kostyuchenko, “Expansion in eigenfunctions of differential and other operators,” Dokl. Akad. Nauk SSSR,103, No. 3, 349–352 (1955).Google Scholar
- 14.I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 3: Theory of Differential Equations, Academic Press, New York (1967).Google Scholar
- 15.J. F. Colombeau, “A multiplication of distributions,” J. Math. Anal. Appl.,94, No. 1, 96–115 (1983).Google Scholar
- 16.J. F. Colombeau, Elementary Introduction to New Generalized Functions, North-Holland Math. Studies, Vol. 113, North-Holland, Amsterdam (1985).Google Scholar