Regular attractor for a non-linear elliptic system in a cylindrical domain

M. I. Vishik   ; S. V. Zelik

Sbornik. Mathematics, Tome 190 (1999) no. 6, pp. 803-834

Résumé

The system of second-order elliptic equations \begin{equation} a(\partial_t^2u+\Delta_xu)-\gamma\partial_tu-f(u)=g(t), \quad u\big|_{\partial\omega}=0, \enskip u\big|_{t=0}=u_0, \enskip (t,x)\in\Omega _+, \tag{1} \end{equation} is considered in the half-cylinder $\Omega_+=\mathbb R_+\times\omega$, $\omega\subset\mathbb R^n$. Here $u=(u^1,\dots,u^k)$ is an unknown vector-valued function, $a$ and $\gamma$ are fixed positive-definite self-adjoint $(k\times k)$-matrices, $f$ and $g(t)=g(t,x)$ are fixed functions. It is proved under certain natural conditions on the matrices $a$ and $\gamma$, the non-linear function $f$, and the right-hand side $g$ that the boundary-value problem (1) has a unique solution in the space $W^{2,p}_{\mathrm{loc}}(\Omega_+,\mathbb R^k)$, $p>(n+1)/2$, that is bounded as $t\to\infty$. Moreover, it is established that the problem (1) is equivalent in the class of such solutions to an evolution problem in the space of “initial data” $u_0\in V_0\equiv\operatorname{Tr}_{t=0}W^{2,p}_{\mathrm{loc}}(\Omega_+,\mathbb R^k)$. In the potential case $(f=\nabla _x P$, $g(t,x)\equiv g(x))$ it is shown that the semigroup $S_t\colon V_0\to V_0$ generated by (1) possesses an attractor in the space $V_0$ which is generically the union of finite-dimensional unstable manifolds $\mathscr M^+(z_i)$ corresponding to the equilibria $z_i$ of $S_t$ $(S_tz_i=z_i)$. In addition, an explicit formula for the dimensions of these manifolds is obtained.

Bibliographie

[1] Sobolev S. L., Nekotorye prilozheniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988 | MR

[2] Tribel Kh., Teoriya interpolyatsii, funktsionalnye prostranstva, differentsialnye operatory, Mir, M., 1980 | MR

[3] Babin A. V., “Attraktor obobschennoi polugruppy, porozhdennoi ellipticheskim uravneniem v tsilindricheskoi oblasti”, Izv. RAN. Ser. matem., 58:2 (1994), 3–18 | MR | Zbl

[4] Vishik M. I., Zelik S. V., “Traektornyi attraktor nelineinoi ellipticheskoi sistemy v tsilindricheskoi oblasti”, Matem. sb., 187:12 (1996), 21–56 | MR | Zbl

[5] Zelik S. V., “Ogranichennost reshenii nelineinoi ellipticheskoi sistemy v tsilindricheskoi oblasti”, Matem. zametki, 61:3 (1997), 447–450 | MR | Zbl

[6] Babin A. V., “Inertial manifolds for travelling-wave solutions of reaction-diffusion systems”, Comm. Pure Appl. Math., 48:2 (1995), 167–198 | DOI | MR | Zbl

[7] Brunovsky P., Mora X., Polacik P., Sola-Morales J., “Asymptotic behavior of solutions of semilinear elliptic equations on an unbounded strip”, Acta Math. Univ. Comenian, 60:2 (1991), 163–183 | MR | Zbl

[8] Calsina A., Mora X., Sola-Morales J., “The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic singular limit”, J. Differential Equations, 102:2 (1993), 244–304 | DOI | MR | Zbl

[9] Kirchgässner K., “Wave-solutions of reversible systems and applications”, J. Differential Equations, 45 (1982), 113–127 | DOI | MR | Zbl

[10] Mielke A., “Essential manifolds for an elliptic problem in an infinite strip”, J. Differential Equations, 110:2 (1994), 322–355 | DOI | MR | Zbl

[11] Scheel A., “Existence of fast travelling waves for some parabolic equations: A dynamical systems approach”, J. Dynam. Differential Equations, 8:4 (1996), 469–548 | DOI | MR

[12] Babin A. V., Vishik M. I., Attraktory evolyutsionnykh uravnenii, Nauka, M., 1989 | MR | Zbl

[13] Hale J. K., Asymptotic behaviour of dissipative systems, Math. Surveys Monographs, 25, 1988 | MR | Zbl

[14] Agmon S., Nirenberg L., “Lower bounds and uniqueness theorems for solutions of differential equations in a Hilbert space”, Comm. Pure Appl. Math., 20 (1967), 207–229 | DOI | MR | Zbl

[15] Klement F., Kheimans Kh., Angenent A. i dr., Odnoparametricheskie polugruppy, Mir, M., 1992 | MR

[16] Iosida K., Funktsionalnyi analiz, Mir, M., 1967 | MR

[17] Khenri D., Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Mir, M., 1985 | MR

[18] Davies E., One-parameter semigroups, Academic Press, London, 1980 | MR | Zbl

[19] Gokhberg I., Krein M., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[20] Khirsh M., Differentsialnaya topologiya, Mir, M., 1979 | MR | Zbl