Long-time dynamics of Kirchhoff wave models with strong nonlinear damping

TL;DR

This paper investigates the long-term behavior of Kirchhoff wave models with strong nonlinear damping, establishing existence, uniqueness, and finite-dimensional attractors under various conditions, including degeneracy and supercritical sources.

Contribution

It provides new results on global attractors and their properties for nonlinear Kirchhoff wave equations with strong damping, covering cases with degeneracy and supercritical sources.

Findings

Existence and uniqueness of weak solutions.

Finite-dimensional global attractors in various topologies.

Conditions for exponential attractors and determining functionals.

Abstract

We study well-posedness and long-time dynamics of a class of quasilinear wave equations with a strong damping. We accept the Kirchhoff hypotheses and assume that the stiffness and damping coefficients are $C^1$ functions of the $L_2$-norm of the gradient of the displacement. We first prove the existence and uniqueness of weak solutions and study their properties for a rather wide class of nonlinearities which covers the case of possible degeneration (or even negativity) of the stiffness coefficient and the case of a supercritical source term. Our main results deal with global attractors. In the case of strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global attractor whose fractal dimension is finite. In the non-supercritical case the partially strong topology becomes strong and a finite dimensional attractor exists in the strong topology of the energy space. Moreover, in this case we also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals. Our arguments involve a recently developed method based on "compensated" compactness and quasi-stability estimates.