Unbounded Sturm attractors for quasilinear equations

TL;DR

This paper investigates the long-term behavior of solutions to quasilinear parabolic equations that can grow unboundedly, revealing the structure of their unbounded attractors and the dynamics at infinity, with applications to black hole event horizons.

Contribution

It introduces a framework for analyzing unbounded attractors in quasilinear equations and characterizes the dynamics at infinity, including heteroclinic connections, under hyperbolicity conditions.

Findings

Unbounded global attractors can be described via a Poincaré projection.

The dynamics at infinity can be gradient and composed of equilibria and heteroclinics.

Explicit conditions for heteroclinic connections between equilibria are provided.

Abstract

We analyze the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e., blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincar\'e projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.