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A piezoelectric beam model with nonlinear interior dampings and supercritical sources

Menglan Liao, Baowei Feng

TL;DR

The paper studies a fully coupled, three-dimensional piezoelectric beam model with complete magnetic effects, nonlinear interior damping, and strong interior sources. Using nonlinear semigroups and monotone operator theory for local well-posedness, and potential-well methods for global existence, it provides a comprehensive view of long-time behavior: exponential, polynomial, or logarithmic energy decay depending on damping exponents, and sharp blow-up results when sources dominate damping. The results include energy decay without introducing lower-order terms and do not impose restrictive relations among model coefficients, broadening the understanding of stabilization and instability in fully dynamic piezoelectric systems. Collectively, the work advances the mathematical theory of strongly coupled piezoelectric PDEs by linking source-damping balance to global existence, decay, and finite-time blow-up phenomena.

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.

A piezoelectric beam model with nonlinear interior dampings and supercritical sources

TL;DR

The paper studies a fully coupled, three-dimensional piezoelectric beam model with complete magnetic effects, nonlinear interior damping, and strong interior sources. Using nonlinear semigroups and monotone operator theory for local well-posedness, and potential-well methods for global existence, it provides a comprehensive view of long-time behavior: exponential, polynomial, or logarithmic energy decay depending on damping exponents, and sharp blow-up results when sources dominate damping. The results include energy decay without introducing lower-order terms and do not impose restrictive relations among model coefficients, broadening the understanding of stabilization and instability in fully dynamic piezoelectric systems. Collectively, the work advances the mathematical theory of strongly coupled piezoelectric PDEs by linking source-damping balance to global existence, decay, and finite-time blow-up phenomena.

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.
Paper Structure (12 sections, 17 theorems, 264 equations)

This paper contains 12 sections, 17 theorems, 264 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Local solutions
  4. Globally Lipschitz sources
  5. Locally Lipschitz sources
  6. Completion of the proof of the local existence
  7. Global existence of potential well solutions
  8. Energy decay rates
  9. Blow-up in finite time
  10. Blow-up in finite time for negative initial energy
  11. Blow-up in finite time for bounded positive initial energy
  12. Blow-up in finite time for arbitrarily positive initial energy

Key Result

Theorem 2.3

Let Assumption ass hold, then there exists a local weak solution $(v,p)$ to problem 1.1 defined on $[0,T]$ for some $T>0$ depending on the initial quadratic energy $E(0)$, where the quadratic energy $E(t)$ is given by In addition, for all $t\in [0,T]$, $(v,p)$ satisfies the following energy identity

Theorems & Definitions (27)

  • Definition 2.2: Weak solution
  • Theorem 2.3: Local existence and energy identity
  • Remark 2.4
  • Theorem 2.5: Global existence
  • Theorem 2.6: Decay decay rates
  • Remark 2.7
  • Theorem 2.8: Blow-up in finite time for negative initial energy
  • Remark 2.9
  • Theorem 2.10: Blow-up in finite time for bounded positive initial energy
  • Corollary 2.11
  • ...and 17 more