Overview

This book provides a comprehensive analysis of the long-term behavior of complex nonlinear systems described by partial differential equations. It focuses on three fundamental phenomena: finite-time blow-up, where solutions become infinite; asymptotic stability, where solutions settle into a steady state; and general decay, where energy dissipates over time. The work examines advanced models including viscoelastic and thermoelastic systems with memory effects, time delays, and singular coefficients. Key achievements include establishing sharp conditions for blow-up with time estimates, proving general decay rates beyond standard exponential or polynomial decay, and demonstrating the well-posedness of these complex systems. The book serves as an advanced reference for researchers in applied mathematics and mathematical physics, offering new analytical frameworks and highlighting future directions like fractional calculus and stochastic systems.

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