Conditions for well-posedness in the Hadamard sense in spaces of generalized functions
- 129 Downloads
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
Unable to display preview. Download preview PDF.
- 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