Attractors for a class of semi-linear degenerate parabolic equations

Abstract

We consider degenerate parabolic equations of the form

$$\left. \begin{array}{ll}\,\,\, \partial_t u = \Delta_\lambda u + f(u) \\u|_{\partial\Omega} = 0, u|_{t=0} = u_0\end{array}\right.$$

in a bounded domain ${\Omega\subset\mathbb{R}^N}$, where Δ_λ is a subelliptic operator of the type

$$\quad \Delta_\lambda:= \sum_{i=1}^{N} \partial_{x_i}(\lambda_{i}^{2} \partial_{x_i}),\qquad \lambda = (\lambda_1,\ldots, \lambda_N).$$

We prove global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity.