Geodesic
On the stability of stationary solutions of nonlinear parabolic systems
Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 465-484 
V. S. Belonosov; M. P. Vishnevskii. On the stability of stationary solutions of nonlinear parabolic systems. Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 465-484. http://geodesic.mathdoc.fr/item/SM_1977_33_4_a1/
@article{SM_1977_33_4_a1,
     author = {V. S. Belonosov and M. P. Vishnevskii},
     title = {On~the stability of stationary solutions of nonlinear parabolic systems},
     journal = {Sbornik. Mathematics},
     pages = {465--484},
     year = {1977},
     volume = {33},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1977_33_4_a1/}
}
TY  - JOUR
AU  - V. S. Belonosov
AU  - M. P. Vishnevskii
TI  - On the stability of stationary solutions of nonlinear parabolic systems
JO  - Sbornik. Mathematics
PY  - 1977
SP  - 465
EP  - 484
VL  - 33
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_1977_33_4_a1/
LA  - en
ID  - SM_1977_33_4_a1
ER  - 
%0 Journal Article
%A V. S. Belonosov
%A M. P. Vishnevskii
%T On the stability of stationary solutions of nonlinear parabolic systems
%J Sbornik. Mathematics
%D 1977
%P 465-484
%V 33
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1977_33_4_a1/
%G en
%F SM_1977_33_4_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper the linearization method is developed for the problem of stability of stationary solutions of nonlinear autonomous parabolic systems of general form. Estimates, uniform with respect to time, are established for Hölder norms of the solutions of the linearized systems. These estimates are then used in the study of the nonlinear problems. Theorems are proved concerning the stability of the first approximation and the conditional stability, generalizing the well-known Lyapunov theorems from the theory of ordinary differential equations. Bibliography: 11 titles.

[1] V. S. Belonosov, M. P. Vishnevskii, “Nekotorye voprosy kachestvennoi teorii kraevykh zadach dlya nelineinykh parabolicheskikh sistem”, Matem. problemy khimii, ch. I, Novosibirsk, 1975, 132–138

[2] T. I. Zelenyak, “O stabilizatsii reshenii kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka s odnoi prostranstvennoi peremennoi”, Diff. uravneniya, IV:1 (1968), 34–45

[3] V. S. Belonosov, T. I. Zelenyak, Nelokalnye problemy v teorii kvazilineinykh parabolicheskikh uravnenii, izd-vo NGU, Novosibirsk, 1975 | MR

[4] M. I. Freedman, P. L. Falb, I. Anton, “Transform theoretic approach to the stability of a class of nonlinear partial differential equations”, J. Diff. Equat., 9:1 (1971), 25–45 | DOI | MR | Zbl

[5] Hansjorg Kielhöfer, “On the Lyapunov stability of stationary solutions of semilinear parabolic differential equations”, J. Diff. Equat., 22:1 (1976), 193–208 | DOI | MR

[6] V. A. Solonnikov, “O kraevykh zadachakh dlya lineinykh parabolicheskikh sistem differentsialnykh uravnenii obschego vida”, Trudy matem. in-ta imeni V. A. Steklova, LXXXIII, 1965, 3–163 | MR

[7] M. P. Vishnevskii, “Periodicheskie i blizkie k nim resheniya parabolicheskikh sistem”, Dinamika sploshnoi sredy, no. 16, Novosibirsk, 1974, 84–90

[8] N. V. Nikolenko, “Ustoichivost semeistva tsiklov dinamicheskikh sistem”, Izv. VUZov, Matematika, 1973, no. 7, 63–69 | MR | Zbl

[9] E. Khille, R. Fillips, Funktsionalnyi analiz i polugruppy, IL, Moskva, 1962

[10] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equation satisfying general boundary conditions, II”, Comm. Pure Appl. Math., 17:1 (1964), 35–92 | DOI | MR | Zbl

[11] F. Riss, B. Sekefalvi-Nad, Lektsii po funktsionalnomu analizu, IL, Moskva, 1954