Sturm Attractors for Quasilinear Parabolic Equations with Singular Coefficients

Abstract

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.

Sturm attractors for quasilinear parabolic equations

“…The only difference lying in the functional setting: convergence in L 2 should be replaced by L 2 w with weigth w := sin(θ), and the topology in C 2α+β has to be replaced with an appropriate metric at C 2α+β w as in [16].…”

Space of initial data for self-similar Schwarzschild solutions of the Einstein equations

“…The other describes some symmetries of certain metric at r 1 . Both are rigorously proved in [13], [14] and [15].…”

“…where α, β ∈ (0, 1) are respectively the Hölder exponent and a fractional power exponent. See [14]. Note that from Lemma 1 in [25], if v 0 > 0, then v(t) > 0 for t > 0 and the space of positive functions is invariant guaranteeing strict parabolicity of (1.6).…”

“…Note that v ≥ ǫ > 0 in the phase-space X. If R depends on ∇v, there exists a Lyapunov function for axially symmetric solutions, as in [14]. For more general radial foliations, we could possibly obtain a fully nonlinear parabolic equation, instead of (1.2).…”

“…For more general radial foliations, we could possibly obtain a fully nonlinear parabolic equation, instead of (1.2). In this case a Lyapunov function for axisymmetric solutions can be available by incorporating the weigth from [14] to [16].…”