|

Stabilization and Spectrum in Operator Differential Equations

Authors: Filinovsky A.V. Published: 17.06.2015
Published in issue: #3(60)/2015  
DOI: 10.18698/1812-3368-2015-3-3-19

 
Category: Mathematics | Chapter: Differential Equations, Dynamic Systems, and Optimal Control  
Keywords: operator differential equation, hyperbolic problem, Dirichlet boundary condition, stabilization, Laplace operator, spectrum

We investigate the Cauchy problem for a second order non-stationary linear operator differential equation in a Hilbert space. We consider the case of an unbounded self-adjoint positive operator with a special regard to the Laplace operator with Dirichlet boundary conditions. The corresponding problem is a mixed problem for a wave equation. Introducing the energy class solution we prove its representation by the Bochner-Stiltjes integral. We establish the connection between spectral properties of the Laplace operator and stabilization for large time values of the solutions to the mixed problem of the wave equation. We investigate the asymptotic behavior in time of the local energy function for the various types of spectrum. For bounded domains where the spectrum of the Laplace operator is purely discrete we any solution with a local energy tends to zero in time is identically zero. For arbitrary domains in the case of operator with non-empty point spectrum we prove that there are smooth and finite initial functions for which the local energy function does not decay. In the cases of continuous and absolutely continuous spectrum of the Laplace operator we prove the mean decay and the decay of the local energy function respectively.

References

[1] Lax P.D. Hyperbolic Partial Differential Equations. Amer. Math. Society, Providence, 2006.

[2] Filinovskiy A.V. Stabilization and spectrum in the problems of wave propagation. Qualitative properties of solutions to differential equations and related topics of spectral analysis, ed. by I.V. Astashova, Moscow, UNITY-DANA Publ., 2012, pp. 289-463, 647 p.

[3] Temnov A.N. Small vibrations of an ideal non-homogeneous self-gravitated fluid. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2002, no. 2, pp. 25-35 (in Russ.).

[4] Krein S.G. Linear differential equations in a Banach space. Moscow, Nauka Publ., 1967.

[5] Hille E., Phillips R.S. Functional analysis and semi-groups. Amer. Math. Soc. Coll. Publ., vol. 31. Providence, 1957.

[6] Nemytskiy V.V., Vainberg M.M., Gusarova R.S. Operator differential equations, Mat. analiz. Itogi nauki, Moscow, VINITI Publ., 1966, pp. 165-235 (in Russ.).

[7] Vishik M.I. Cauchy problem for the equations with operator coefficients, mixed boundary problem for the systems of differential equations and approximative method of its solving. Sb. Math., 1956, vol. 39, no. 1, pp. 51-148 (in Russ.).

[8] Vishik M.I., Ladyzhenskaya O.A. Boundary value problem for partial differential equations and some classes of operator equations. Uspekhi Mat. Nauk [Russian Mathematical Surveys], 1956, vol. 11, no. 6, pp. 41-97 (in Russ.).

[9] Ladyzhenskaya O.A. On the non-stationary operator equations and their applications to linear problems of mathematical physics. Sb. Math., 1958, vol. 45, no. 2, pp. 123-158 (in Russ.).

[10] Krein M.G. On some questions concerned with the Lyapounov ideas in the stability theory. Uspekhi Mat. Nauk [Russian Mathematical Surveys], 1948, vol. 3, no. 3, pp. 166-169 (in Russ.).

[11] Filinovskiy A.V. Stabilization of solutions of the wave equation in domains with non-compact boundaries. Sb. Math., 1998, vol. 189, no. 8, pp. 141-160 (in Russ.).

[12] Filinovskiy A.V. Stabilization of solutions of the first mixed problem for Helmholtz equation in the domains with star-shaped boundaries. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 1999, no. 2, pp. 22-33 (in Russ.).

[13] Filinovskiy A.V. Estimates of solutions of the first mixed problemfor the wave equation in domains with non-compact boundaries. Sb. Math., 2002, vol. 193, no. 9, pp. 107-138 (in Russ.).

[14] Gilbarg D., Trudinger N.S. Elliptic partial differential equations of second order. Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1983.

[15] Kato T. Perturbation theory for linear operators. Berlin-Heidelberg-New York-Tokyo, Springer-Verlag, 1995.

[16] Muckenhaupt C.S. Almost periodic functions and vibrating systems. J. Math. and Phys., 1929, vol. 8, pp. 163-198.

[17] Riesz F., Szokefalvy-Nagy B. Lecons d’analyse fonctionelle. Budapest, Akademiai Kiado, 1995.

[18] Sobolev S.L. On the almost-periodicity of solutions of the wave equation. P. I. Dokl. Akad. Nauk, vol. 48, no. 8, 1945, pp. 570-573 (in Russ.).

[19] Sobolev S.L. On the almost-periodicity of solutions of the wave equation. P. II. Dokl. Akad. Nauk, vol. 48, no. 9, 1945, pp. 646-648 (in Russ.).

[20] Wiener N. The Fourier integral and certain of its applications. Dover, 1933.

[21] Lustemik L.A., Sobolev V.I. Elements of functional analysis. Moscow, Nauka Publ., 1965 (in Russ.).