Use of Central Multiplicity Theory in the generalized Kirchhoff equation with applications to physics

Abstract: This diploma thesis explores the solutions of a nonlinear hyperbolic Kirchhoff-type wave equation on unbounded sets across the entire space RNmathbb{R}^NRN. The work investigates both local and total existence of solutions, focusing on the stability of the linearized system through the use of Central Multiplicity Theory and Pego’s transformation. Additionally, the thesis addresses the characteristics of compact and invariant sets, as well as the attractors of the system. Practical examples are provided using Matlab to illustrate the theoretical findings.

Introduction: Nonlinear hyperbolic equations, such as the Kirchhoff-type wave equation, play a significant role in mathematical physics and applied mathematics due to their complex dynamics and wide range of applications. This thesis delves into the solution behavior of these equations on unbounded domains, aiming to establish existence results and analyze the stability of solutions. By utilizing advanced mathematical tools such as Central Multiplicity Theory and Pego’s transformation, the research offers a comprehensive analysis of the equation’s solution space, stability properties, and attractors.

Objectives:

• Study the local and total existence of solutions to the nonlinear hyperbolic Kirchhoff-type wave equation on unbounded sets RNmathbb{R}^NRN.
• Analyze the stability of the linearized system using Central Multiplicity Theory and Pego’s transformation.
• Investigate the nature of compact and invariant sets, and identify the attractors associated with the equation.
• Provide Matlab simulations and examples to demonstrate the theoretical results and enhance understanding of the wave equation’s dynamics.

Methodology: The methodology is divided into several stages:

• Mathematical Formulation: Formulate the nonlinear hyperbolic Kirchhoff-type wave equation and define the unbounded sets over which the equation is studied.
• Existence and Stability Analysis: Employ Central Multiplicity Theory and Pego’s transformation to study the local and total existence of solutions. Analyze the stability of the linearized system and provide stability results.
• Invariant Sets and Attractors: Examine the structure of compact and invariant sets, determining the attractors associated with the problem.
• Numerical Simulations: Utilize Matlab to create simulations and examples that illustrate the theoretical findings, providing practical insights into the behavior of the equation.

Expected Contributions:

• A detailed analysis of the existence and stability of solutions for nonlinear hyperbolic Kirchhoff-type wave equations on unbounded sets.
• Stability results for the linearized system using Central Multiplicity Theory and Pego’s transformation, offering new perspectives on the behavior of wave equations.
• Insights into the nature of compact and invariant sets and the attractors related to the problem.
• Practical demonstrations through Matlab simulations that support the theoretical analysis and provide a visual understanding of the equation’s dynamics.

Prerequisite knowledge: Mathematical Analysis, Knowledge of Matlab, Nonlinear Differential Equations, Nonlinear Dynamical Systems, Partial Differential Equations, Linear Algebra.

Conclusion: This thesis aims to provide a thorough examination of the nonlinear hyperbolic Kirchhoff-type wave equation, focusing on solution existence, stability, and attractor behavior. Through the application of Central Multiplicity Theory, Pego’s transformation, and Matlab simulations, the work contributes to the deeper understanding of wave equation dynamics in mathematical and physical contexts. The results obtained can serve as a foundation for further studies in nonlinear wave equations and dynamical systems.