Geodesic
On~the asymptotic behavior of the solution of the second boundary value problem for a~quasilinear parabolic system of chemical kinetics
Izvestiya. Mathematics , Tome 18 (1982) no. 1, pp. 195-204.

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The question of the stabilization of the solution of the second boundary value problem for a quasilinear parabolic system as the time increases without bound is studied. A theorem is proved on the stabilization of the components $u_i(t,x)$ to constants as $t\to\infty$ which are uniquely determined by the initial conditions and the functions $F_i$, and an exponential estimate for the rate of stabilization is also obtained. Bibliography: 5 titles. 
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A. N. Peregudov. On~the asymptotic behavior of the solution of the second boundary value problem for a~quasilinear parabolic system of chemical kinetics. Izvestiya. Mathematics , Tome 18 (1982) no. 1, pp. 195-204. http://geodesic.mathdoc.fr/item/IM2_1982_18_1_a10/

[1] Fridman A., Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968

[2] Peregudov A. N., “Vtoraya kraevaya zadacha dlya kvazilineinykh parabolicheskikh sistem tipa sistem khimicheskoi kinetiki”, ZhVM i MF, 1981, no. 1

[3] Kruzhkov S. N., Nelineinye uravneniya s chastnymi proizvodnymi, ch. 2, MGU, M., 1970

[4] Iosidzava T., “Funktsiya Lyapunova i ogranichennost reshenii”, Matematika, 9:5 (1965), 95–126

[5] Ilin A. M., Kalashnikov A. S., Oleinik O. A., “Lineinye uravneniya vtorogo poryadka parabolicheskogo tipa”, Uspekhi matem. nauk, 17:3 (1952), 3–146