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.