The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovský and Fiedler (1986), generalized by Fiedler and Rocha (1996) and later for quasilinear equations by the author (2017). In particular, we construct heteroclinic connections between hyperbolic equilibria, stating necessary and sufficient conditions for heteroclinics to occur. Such conditions can be computed through a permutation of the equilibria. Lastly, an example is computed yielding the well known Chafee-Infante attractor.Keywords: parabolic equations, singular coefficients, infinite dimensional dynamical systems, global attractor, Sturm attractor. 0 < ǫ ≤ a(x, u, p) ≤ δ 1. the global attractor A of (1.2) consists of finitely many equilibria E and their heteroclinic orbits H.2. there exists a heteroclinic u(t) ∈ H between u − , u + ∈ E such thatif, and only if, u − and u + are adjacent and i(u − ) > i(u + ).The first claim follows due to the existence of a Lyapunov functional constructed by Matano [29] and Zelenyak [44]. A modification of such functional for the case of singular coefficients is done in Section 2.1.

The goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations when solution can also grow-up, and hence the attractor is unbounded and induces a flow at infinity. In particular, we construct heteroclinic connections between bounded and/or unbounded hyperbolic equilibria when the diffusion is asymptotically linear.Keywords: quasilinear parabolic equations, infinite dimensional dynamical systems, grow-up, unbounded global attractor, Sturm attractor. Main resultsConsider the scalar quasilinear parabolic differential equation. We suppose that 2α + β > 1 so that solutions are at least C 1 ([0, π]). The appropriate functional setting is described in Section 2.1.We are interested in the asymptotic behaviour of solutions of (1.1) when grow-up can occur, namely, when solutions grows unboundedly as t → ∞. Sufficient conditions for grow-up to occur is when b > 0, as we will prove later in Lemma 2.1. This class of asymptotics is also known as slowly non-dissipative. In this setting, there does not exist a global attractor in the usual sense, namely a maximal compact invariant set that attracts all bounded sets. Emitting the compactness condition, there is an unbounded global attractor A ⊆ X α , defined as the minimal invariant non-empty set in X α attracting all bounded sets, firstly introduced by Chepyzhov and Goritskii [7]. The goal of this paper is to decompose A into smaller invariant sets, describe them and show how they are related. This geometric description of the attractor A in the semilinear dissipative case was carried out by Brunovský and Fiedler [5] for f (u), by Fiedler and Rocha [10] for f (x, u, u x ), for periodic boundary conditions by Fiedler, Rocha and Wolfrum [11], and for quasilinear equations by Lappicy [15]. Such attractors are known as Sturm attractors. When solutions can grow-up, the semilinear case was previously studied by Hell [13] in order to give an understanding of the structure at infinity, Ben-Gal [4] for f (u), Pimentel and Rocha [22] for f (x, u, u x ). The case of periodic boundary condition was treated separately in Pimentel [21]. Such attractors are known as unbounded Sturm attractors.Despite non-dissipativity, there still exist a Lyapunov function constructed by Matano [19]. Hence, the following dichotomy hold: either solutions converge to a bounded equilibrium as t → ∞, or it is a grow-up solution.

Lyapunov functions are used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano's method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.