Abstract
This paper concerns the investigation of the stabilization of solutions of the Cauchy problem for a system of equations of the form σu/∂t = δu + fi(u, v); ∂v/∂t = δv + F2(u, v). It is proved that under certain assumptions the behavior of solutions as t → ∞ is determined by mutual arrangement of the set of initial conditions {(u, v): u = f1(x), v =f 2(x), xεRn} and the trajectories of the system of ordinary differential equations du/dt = F1(u, v), dv/dt = F2(u, v). The question of stabilization of the solutions of a single quasilinear parabolic equation is also considered.
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E. B. Dynkin, Markov Processes [in Russian], Moscow (1963).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Moscow (1967).
M. I. Freidlin, “Quasilinear parabolic equations and measures in function space,” Funktsional. Analiz i Ego Prilozhen.,1, No. 3, 74–82 (1967).
M. I. Freidlin, “Diffusion processes with a reflection and a problem with a skew derivative on a manifold with a boundary,” Teoriya Veroyat. i Ee Primen.,8, No. 1, 80–88 (1963).
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Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 85–92, January, 1968.
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Freidlin, M.I. Stabilization of solutions of certain parabolic equations and systems. Mathematical Notes of the Academy of Sciences of the USSR 3, 50–54 (1968). https://doi.org/10.1007/BF01386966
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DOI: https://doi.org/10.1007/BF01386966