A road map to the blow-up for a Kirchhoff equation with external force

TL;DR

This paper constructs a specific forced Kirchhoff equation with a carefully chosen external force that leads to finite-time blow-up of solutions, despite the force being regular and bounded.

Contribution

It introduces a novel example of a forced Kirchhoff equation exhibiting finite-time blow-up under regular, bounded forcing, using heteroclinic connections between simple modes.

Findings

Existence of heteroclinic connections between simple modes.

Construction of a forcing term with maximal regularity that induces blow-up.

Finite-time blow-up occurs despite bounded, regular forcing.

Abstract

It is well-known that the classical hyperbolic Kirchhoff equation admits infinitely many simple modes, namely time-periodic solutions with only one Fourier component in the space variables. In this paper we assume that, for a suitable choice of the nonlinearity, there exists a heteroclinic connection between two simple modes with different frequencies. Under this assumption, we cook up a forced Kirchhoff equation that admits a solution that blows-up in finite time, despite the regularity and boundedness of the forcing term. The forcing term can be chosen with the maximal regularity that prevents the application of the classical global existence results in analytic and quasi-analytic classes.