Asymptotic integration of the Cauchy problem with countably multiple spectrum. (English. Russian original) Zbl 0566.35004
Math. Notes 35, 35-47 (1984); translation from Mat. Zametki 35, No. 1, 63-82 (1984).
The authors consider the singularly perturbed Cauchy problem of the form \(L_{\epsilon}u\equiv \epsilon (du/dt)-Au=h(t),\) \(u_{t=0}=\psi\), by the regularization method, where A is a constant non-diagonalizable operator acting in an infinite-dimensional Hilbert space, and has a countable number of multiple spectral points. Under certain conditions, the uniformly valid asymptotic expansion of the solution has been constructed, and the remainder term has been estimated.
Reviewer: Furu Jiang
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35P05 | General topics in linear spectral theory for PDEs |
| 35G10 | Initial value problems for linear higher-order PDEs |
| 35K25 | Higher-order parabolic equations |
Keywords:
singularly perturbed; Cauchy problem; regularization method; Hilbert space; multiple spectral points; asymptotic expansionReferences:
| [1] | S. A. Lomov, Introduction to the General Theory of Singular Perturbations [in Russian], Nauka, Moscow (1981). · Zbl 0514.34049 |
| [2] | A. G. Eliseev and S. A. Iomov, ?The perturbation method in a Banach space,? Dokl. Akad. Nauk SSSR,264, No. 1, 34-38 (1982). |
| [3] | M. P. Myagkova, ?The regularized asymptotic of the Cauchy problem in the case of identically multiple spectral points,? in: All-Union Conference on Asymptotic Methods [in Russian], ILIM, Frunze (1975), pp. 327-329. |
| [4] | I. S. Lomov, ?Regularization of singular perturbations in terms of the spectrum of the lmit operator,? Vestn. Mosk. Gos. Univ., Ser. Mekh., Mat., No. 3, 6-13 (1975). |
| [5] | N. I. Kabatsii, ?Asymptotic solutions a mixed boundary-value problem for a parabolic equation,? in: All-Union Conference on Asymptotic Methods [in Russian], Part 1, Nauka, Alma-Ata (1979), pp. 108-109. |
| [6] | N. N. Prilepin, ?The method of regularization for the solution of certain parabolic problems under the existence of the zero point of the spectrum,? in: All-Union Conference on Asymptotic Methods [in Russian], Part 1, Nauka, Alma-Ata (1979), pp. 106-108. |
| [7] | Kh. L. Territin, ?Asymptotic expansions of solutions of ordinary differential equations that contain a small parameter,? Matematika,1, No. 2, 29-59 (1957). |
| [8] | S. F. Feshchenko, N. I. Shkil’, and L. D. Nikolenko, Asymptotic Methods in the Theory of Linear Differential Equations [in Russian], Naukova Dumka, Kiev (1966). · Zbl 0141.28004 |
| [9] | A. G. Eliseev, ?The theory of perturbations in a finite-dimensional Banach space in the case of multiple spectrum of the limit operator,? in: Methods of Small Parameter and Their Application [in Russian], Abstracts of Lectures at the All-Union School Seminar, Minsk (1982), pp. 78-100. |
| [10] | V. A. Al’in, ?Necessary and sufficient conditions for the basicity of a subsystem of the eigenfunctions and associated eigenfunctions of M. V. Keldysh’s pencil of ordinary differential operators,? Dokl. Akad. Nauk SSSR,227, No. 4, 796-799 (1976). |
| [11] | N. I. Ionkin, ?Solution of a boundary-value problem of the theory of heat conduction with a nonclassical boundary condition,? Differents. Uravn.,13, No. 2, 294-304 (1977). |
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