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evolution equations; stabilization of solutions; parabolic problem
The stabilization of solutions to an abstract differential equation is investigated. The initial value problem is considered in the form of an integral equation. The equation is solved by means of the Banach contraction mapping theorem or the Schauder fixed point theorem in the space of functions decreasing to zero at an appropriate rate. Stable manifolds for singular perturbation problems are compared with each other. A possible application is illustrated on an initial-boundary-value problem for a parabolic equation in several space variables.
[1] Appel J., Zabrejko P.P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge, 1990. MR 1066204
[2] Crandall M., Liggett T.: Generation of semigroups of nonlinear transformations on general Banach spaces. Amer. J. Math. 93 (1977), 265-298. MR 0287357
[3] Hale J.K.: Theory of Functional Differential Equations. Springer, New York, 1977. MR 0508721 | Zbl 1092.34500
[4] Hale J.K.: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs No. 25, Amer. Math. Soc., Providence, 1988. MR 0941371 | Zbl 0642.58013
[5] Iwamiya T., Takahashi T., Oharu T.: Characterization of nonlinearly perturbed semigroups. in: Functional Analysis and Related Topics, Proceedings, Kyoto 1991, H. Komatsu (ed.), Lecture Notes in Math. No. 1540, Springer, New York, 1993. Zbl 0819.47081
[6] Komatsu H.: Fractional powers of generators, II Interpolation spaces. Pacific J. Math. 21 (1967), 89-111. MR 0206716
[7] Krasnosel'skij M.A., Zabrejko P.P., Pustyl'nik E.I., Sobolevskij P.E.: Integral Operators in the Spaces of Integrable Functions (in Russian). Nauka, Moscow, 1966.
[8] Krein M., Dalecki J.: Stability of the Solutions of Differential Equations in Banach Spaces. Amer. Math. Soc., Providence, 1974. MR 0352639
[9] Lunardi A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995. MR 1329547 | Zbl 0816.35001
[10] Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York, 1983. MR 0710486 | Zbl 0516.47023
[11] Rauch J.: Stability of motion for semilinear equations. in: Boundary Value Problems for Linear Evolution Partial Differential Equations, Proceedings of the NATO Advanced Study Institute held in Liège, Belgium, September 6-17, 1976, H. G. Garnir (ed.), NATO Advanced Study Institute Series C, Mathematical and Physical Sciences, vol. 29, Reidel Publishing Co., Dordrecht, 1977. MR 0492677 | Zbl 0347.35015
[12] Tanabe H.: Equations of Evolution. Pitman, London, 1979. MR 0533824 | Zbl 0417.35003
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