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This book investigates the stability and instability of solutions to nonlinear viscoelastic Kirchhoff-type equations under various damping and delay mechanisms including variable-exponents. It provides a systematic analysis of three key phenomena: the rate of energy decay, finite-time blow-up of solutions, and the long-term growth behavior of global solutions.Designed as a practical guide, the book walks readers through advanced analytical techniques including the energy method, Lyapunov functions, and Komornik's lemma making it accessible for students and researchers entering the field.