Convergence in gradient systems with branching of

V. A. Galaktionov   ; S. I. Pokhozhaev   ; A. E. Shishkov

Sbornik. Mathematics, Tome 198 (2007) no. 6, pp. 817-838

Résumé

The basic model is a semilinear elliptic equation with coercive $C^1$ non-linearity: $\Delta\psi+f(\psi)=0$ in $\Omega$, $\psi=0$ on $\partial\Omega$, where $\Omega\subset\mathbb R^N$ is a bounded smooth domain. The main hypothesis $(H_R)$ about resonance branching is as follows: if a branching of equilibria occurs at a point $\psi$ with $k$-dimensional kernel of the linearized operator $\Delta+f'(\psi)I$, then the branching subset $S_k$ at $\psi$ is a locally smooth $k$-dimensional manifold. For $N=1$ the first result on the stabilization to a single equilibrium is due to Zelenyak (1968). It is shown that Zelenyak's approach, which is based on the analysis of Lyapunov functions, can be extended to general gradient systems in Hilbert spaces with smooth resonance branching. The case of asymptotically small non-autonomous perturbations of such systems is also considered. The approach developed here represents an alternative to Hale's stabilization method (1992) and other similar techniques in the theory of gradient systems. Bibliography: 32 titles.

Bibliographie

[1] M. A. Krasnosel'skii, P. P. Zabreǐko, Geometrical methods of nonlinear analysis, Grundlehren Math. Wiss., 263, Springer-Verlag, Berlin, 1984 ; M. A. Krasnoselskii, P. P. Zabreiko, Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR | Zbl | MR | Zbl

[2] L. Lyusternik, “Ob odnom klasse nelineinykh operatorov v gilbertovom prostranstve”, Izv. AN SSSR. Ser. matem., 3:3 (1939), 257–264 | Zbl

[3] L. Lusternik, L. Schnirelmann, “Sur le problème de trois géodésiques fermées sur les surfaces de genre 0”, C. R. Math. Acad. Sci. Paris, 189 (1929), 269–271 | Zbl

[4] L. Lusternik, L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels. I. Pt. Espaces à un nombre fini de dimensions, Hermann, Paris, 1934 ; L. A. Lyusternik, L. G. Shnirelman, Topologicheskie metody v variatsionnykh zadachakh, MGU, M., 1930 | Zbl | Zbl

[5] D. Clark, “A variant of Lusternik–Schnirelman theory”, Indiana Univ. Math. J., 22 (1973), 65–74 | DOI | MR | Zbl

[6] S. I. Pokhozhaev, “Ob odnom podkhode k nelineinym uravneniyam”, Dokl. AN SSSR, 247:6 (1979), 1327–1331 | MR | Zbl

[7] S. I. Pohozaev, “The fibering method in nonlinear variational problems”, Topological and variational methods for nonlinear boundary value problems (Proceedings of the 20th seminar in partial differential equations, Cholin, 1995), Pitman Res. Notes Math. Ser., 365, Longman, Harlow, 1997, 35–88 | MR | Zbl

[8] A. Friedman, Partial differential equations, Holt, Rinehart Winston, Inc., New York–Montreal, QC–London, 1969 | MR | Zbl

[9] J. K. Hale, Asymptotic behavior of dissipative systems, Math. Surveys Monogr., 25, Amer. Math. Soc., Providence, RI, 1988 | MR | Zbl

[10] G. R. Sell, Y. You, Dynamics of evolutionary equations, Appl. Math. Sci., 143, Springer-Verlag, New York, 2002 | MR | Zbl

[11] V. A. Galaktionov, “Evolution completeness of separable solutions of nonlinear diffusion equations in bounded domains”, Math. Methods Appl. Sci., 27:15 (2004), 1755–1770 | DOI | MR | Zbl

[12] H. Kielhöfer, Bifurcation theory. An introduction with applications to PDEs, Appl. Math. Sci., 156, Springer-Verlag, New York, 2004 | MR | Zbl

[13] T. I. Zelenyak, “O stabilizatsii reshenii kraevykh zadach dlya parabolicheskikh uravnenii vtorogo poryadka s odnoi prostranstvennoi peremennoi”, Differents. uravneniya, 4 (1968), 34–45 | MR | Zbl

[14] H. Matano, Convergence of solutions of one-dimensional semilinear parabolic equations, 18:2 (1978), 221–227 | MR | Zbl

[15] R. Chill, “On the Łojasiewicz–Simon gradient inequality”, J. Funct. Anal., 201:2 (2003), 572–601 | DOI | MR | Zbl

[16] A. Haraux, M. A. Jendoubi, “Decay estimates to equilibrium for some evolution equations with an analytic nonlinearity”, Asymptot. Anal., 26:1 (2001), 21–36 | MR | Zbl

[17] J. Palis, W. de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York–Heidelberg–Berlin, 1982 | MR | Zbl

[18] P. Poláčik, F. Simondon, “Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains”, J. Differential Equations, 186:2 (2002), 586–610 | DOI | MR | Zbl

[19] J. K. Hale, G. Raugel, “Convergence in gradient-like systems with applications to PDE”, Z. Angew. Math. Phys., 43:1 (1992), 63–124 | DOI | MR | Zbl

[20] J. Busca, M. A. Jendoubi, P. Poláčik, “Convergence to equilibrium for semilinear parabolic problems in $\mathbb R^N$”, Comm. Partial Differential Equations, 27:9 (2002), 1793–1814 | DOI | MR | Zbl

[21] M. Sh. Birman, M. Z. Solomyak, Spectral theory of self-adjoint operators in Hilbert space, Math. Appl. (Soviet Ser.), 5, D. Reidel Publ., Dordrecht, 1987 ; M. Sh. Birman, M. Z. Solomyak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, LGU, L., 1980 | MR | Zbl | MR

[22] M. M. Vaǐnberg, V. A. Trenogin, Theory of branching of solutions of non-linear equations, Noordhoff International Publ., Leyden, 1974 ; M. M. Vaienberg, V. A. Trenogin, Teoriya vetvleniya reshenii nelineinykh uravnenii, Nauka, M., 1969 | MR | Zbl | MR | Zbl

[23] M. S. Berger, Nonlinearity and functional analysis, Lectures on nonlinear problems in mathematical analysis, Academic Press, New York–San Francisco–London, 1977 | MR | Zbl

[24] K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985 | MR | Zbl

[25] V. A. Galaktionov, J. L. Vázquez, A stability technique for evolution partial differential equations., A dynamical systems approach, Progr. Nonlinear Differential Equations Appl., 56, Birkhäuser, Boston, MA, 2004 | MR | Zbl

[26] E. Feireisl, F. Issard-Roch, H. Petzeltová, “A non-smooth version of the Lojasiewicz–Simon theorem with applications to non-local phase-field systems”, J. Differential Equations, 199:1 (2004), 1–21 | DOI | MR | Zbl

[27] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., 840, Springer-Verlag, Berlin–Heidelberg, 1981 | DOI | MR | Zbl

[28] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1992 | MR | Zbl

[29] A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progr. Nonlinear Differential Equations Appl., 16, Birkhäuser, Basel, 1995 | MR | Zbl

[30] G. Simonett, “Center manifolds for quasilinear reaction-diffusion systems”, Differential Integral Equations, 8:4 (1995), 753–796 | MR | Zbl

[31] J. Escher, G. Simonett, “A centre manifold analysis for the Mullins–Sekerka model”, J. Differential Equations, 143:2 (1998), 267–292 | DOI | MR | Zbl

[32] L. A. Bagirov, V. A. Kondratev, “Ob asimptoticheskikh svoistvakh reshenii differentsialnykh uravnenii”, Tr. sem. im. I. G. Petrovskogo, 22 (2002), 37–70 ; L. A. Bagirov, V. A. Kondratiev, “On asymptotic properties of solutions of diffusion equations”, J. Math. Sci. (N.Y.), 114:4 (2003), 1407–1428 | MR | DOI