Attractors for a class of semi-linear degenerate parabolic equations with critical exponent

Abstract

We consider the degenerate parabolic equation ${ \partial_t u = \triangle_{\lambda}u + f(u)}$ with Dirichlet boundary condition defined on a bounded domain ${\Omega \subset \mathbb{R}^N}$, where ${\triangle_{\lambda}}$, the so-called ${\triangle_{\lambda}}$ -Laplacian, is a subelliptic operator of the type

$$\triangle_{\lambda} :=\sum_{i=1}^N\partial_{x_i}(\lambda^2_i\partial_{x_i}),\quad \lambda= (\lambda_1(x),\ldots,\lambda_N(x)).$$

In this paper, we will establish the global existence of solutions and its corresponding attractor with critical nonlinearity

$$|f(u)-f(v)| \leq c|u-v|\left(1+|u|^{\frac{4}{Q-2}}+|v|^{\frac{4}{Q-2}} \right),\quad\forall~~u,v\in \mathbb{R}.$$