Spectral aspects of regularization of the Cauchy problem for a degenerate equation
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We study the Cauchy problem for an equation whose generating operator is degenerate on some subset of the coordinate space. To approximate a solution of the degenerate problem by solutions of well-posed problems, we define a class of regularizations of the degenerate operator in terms of conditions on the spectral properties of approximating operators. We show that the behavior (convergence, compactness, and the set of partial limits in some topology) of the sequence of solutions of regularized problems is determined by the deficiency indices of the degenerate operator. We define an approximative solution of the degenerate problem as the limit of the sequence of solutions of regularized problems and analyze the dependence of the approximative solution on the choice of an admissible regularization.
KeywordsBanach Space Cauchy Problem STEKLOV Institute Linear Manifold Contraction Semigroup
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- 3.A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1986) [in Russian].Google Scholar
- 4.O. A. Oleinik and E. V. Radkevich, Second Order Equations with Nonnegative Characteristic Form (VINITI, Moscow, 1971; Plenum, New York, 1973), Itogi Nauki, Ser.: Mat., Mat. Anal. 1969.Google Scholar
- 6.L. V. Korobenko and V. Zh. Sakbaev, “Solution of the Diffusion Equation with Discontinuous Coefficient on a Straight Line,” in Some Problems of Fundamental and Applied Mathematics (Moscow Inst. Phys. Technol., Moscow, 2006), pp. 71–85 [in Russian].Google Scholar
- 7.O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Embedding Theorems (Nauka, Moscow, 1996) [in Russian].Google Scholar
- 8.L. N. Slobodetskii, “Generalized Sobolev Spaces and Their Application to Boundary Problems for Partial Differential Equations,” Uch. Zap. Leningr. Gos. Pedagog. Inst. 197, 54–112 (1958) [Am. Math. Soc. Transl., Ser. 2, 57, 207–275 (1966)].Google Scholar
- 9.S. G. Krein, Linear Differential Equations in Banach Space (Nauka, Moscow, 1967; Am. Math. Soc., Providence, RI, 1972).Google Scholar
- 12.V. Zh. Sakbaev, “Degeneration and Regularization of the Operator in the Cauchy Problem for the Schrödinger Equation,” Sovrem. Mat. Prilozh. 38, 95–109 (2006) [J. Math. Sci. 147 (1), 6483–6497 (2007)].Google Scholar
- 14.M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (Academic, New York, 1972; Mir, Moscow, 1977).Google Scholar