Regularity and energy transfer for a nonlinear beam equation

TL;DR

This paper investigates how a discontinuous forcing term affects the regularity and energy distribution in a nonlinear beam equation, revealing key effects during the transition from attached to detached states.

Contribution

It introduces a spectral decomposition approach to analyze the impact of nonlinearity on solution regularity and energy transfer in a nonlinear beam model.

Findings

Loss of regularity at transition points

Migration of energy across scales due to nonlinearity

Spectral analysis reveals key effects of discontinuous forcing

Abstract

In this paper we study some key effects of a discontinuous forcing term in a fourth order wave equation on a bounded domain, modeling the adhesion of an elastic beam with a substrate through an elastic-breakable interaction. By using a spectral decomposition method we show that the main effects induced by the nonlinearity at the transition from attached to detached states can be traced in a loss of regularity of the solution and in a migration of the total energy through the scales.