Kirchhoff equations with strong damping

TL;DR

This paper investigates Kirchhoff equations with strong damping, establishing local and global existence results depending on the damping exponent, with weaker regularity assumptions than classical models.

Contribution

It provides new existence results for Kirchhoff equations with strong damping under weaker regularity assumptions and different regimes based on the damping exponent.

Findings

Global existence when damping exponent > 1/2

Local existence with small data when exponent < 1/2

Weaker regularity assumptions than classical results

Abstract

We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the "elastic" operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space we prove local existence, and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces.