Morse – Smale Inequalities and Chafee – Infante Attractors

In this paper, we are concerned with the shape of the attractor $\mathcal{A}^\lambda$ of the scalar Chafee – Infante equation. We construct a Morse – Smale vector field in the disk $\mathbb{D}^k$ topologically equivalent to infinite-dimensional dynamics of the Chafee – Infante equation. As a consequence, we obtain geometric properties of $\mathcal{A}^\lambda$ using the Morse – Smale inequalities.