A piezoelectric beam model with nonlinear dampings and supercritical sources

TL;DR

This paper analyzes a complex piezoelectric beam model with nonlinear damping and strong sources, establishing existence, decay, and blow-up results using advanced mathematical techniques, and improves previous models by weakening certain conditions.

Contribution

It introduces a novel mathematical framework for the model, removing previous restrictions and providing comprehensive stability and blow-up analyses.

Findings

Existence of local and global weak solutions.

Energy decay rates without lower-order terms.

Blow-up results for strong sources and initial energies.

Abstract

This paper aims to investigate a three-dimensional fully magnetic effected piezoelectric beam model with strong sources and nonlinear interior dampings. By employing nonlinear semigroups and the theory of monotone operators, the existence of local weak solutions is established. By the potential well method, we obtain the global existence of potential well solutions. Decay rates of the total energy are obtained in terms of the behavior of the damping terms. The main advantage in this work is that the stabilization estimate does not generate lower-order terms, and in addition we remove some strong conditions in previous results to obtain a weaker energy decay. Finally, when the initial total energy is negative, positive but small, respectively, the blow-up results for weak solutions if the source terms are stronger than damping terms are obtained according to the differential inequality technique. Moreover, if interior dampings are linear, a blow-up result with arbitrarily high initial energy is established by the concavity method and an upper bound for the blow-up time is also derived. All results are independent of any relation among the model coefficients.