On the solution for a nonlinear wave equation with variable exponent nonlinearity and a varying delay.

Area of Science:

• Differential Equations
• Mathematical Analysis
• Nonlinear Dynamics

Background:

• Nonlinear wave equations are fundamental in physics and engineering.
• Understanding solutions and stability is crucial for modeling complex phenomena.
• Damping and delay terms introduce significant analytical challenges.

Purpose of the Study:

• To investigate a nonlinear wave equation with a variable exponent damping term and a time-varying delay.
• To establish the existence of global solutions under specific conditions.
• To analyze the energy decay rate of the system.

Main Methods:

• The Faedo-Galerkin method, a compactness technique, was employed to prove the existence of global solutions.
• The multiplier technique combined with a Komornik-type integral inequality was used for energy decay analysis.

Main Results:

• The existence of global solutions for the specified nonlinear wave equation is demonstrated.
• A quantitative assessment of the system's energy decay rate has been established.

Conclusions:

• The study successfully addresses the existence and stability of solutions for a complex wave equation.
• The findings provide valuable insights into the behavior of systems with variable damping and delays.
• This research contributes to the theoretical understanding of nonlinear partial differential equations.