Methods for studying blow-up phenomena in differential equations

Abstract

This thesis investigates the dynamics of a coupled nonlinear wave system with

internal fractional damping, a model relevant to viscoelastic materials and engineering ap- plications. The study focuses on two key aspects: establishing the local existence and

uniqueness of solutions and analyzing conditions leading to finite-time blow-up. Employing

functional analysis and C0-semigroup theory, we reformulate the system as an abstract evo- lution equation in a Hilbert space, proving local well-posedness under specific conditions on

the nonlinear exponents. A carefully designed Lyapunov functional is used to derive criteria

for solution blow-up, highlighting the interplay between nonlinearity and fractional damp- ing. The results contribute to the theoretical understanding of fractional partial differential

equations and offer insights for practical applications, such as vibration control and material stability. This work advances the mathematical framework for nonlocal systems and lays the groundwork for future research into global existence and numerical simulations.