This book presents recent investigations on nonlinear evolution partial differential equations with time delay, with particular focus on hyperbolic PDEs and their modern applications. Hyperbolic PDEs describe wave propagation and signals traveling at finite speeds and arise in many areas of science and engineering, including acoustics, elasticity, fluid dynamics, electromagnetism, traffic flow, and quantum mechanics.The book also examines dissipative hyperbolic systems, where energy decreases over time due to damping or internal resistance, as well as models involving time delay, in which the future state of a system depends on its past states. Such features appear naturally in systems with memory, feedback mechanisms, and nonlinear interactions, but they also present significant mathematical challenges concerning the existence, stability, and long-term behavior of solutions.The book consists of eight chapters. Chapter 1 introduces basic definitions and foundational results related to delay terms and damping mechanisms. The subsequent chapters study several important models, including the nonlinear Euler-Bernoulli beam with neutral delay, delayed transmission problems with infinite memory, plate equations with variable delay, wave equations with dynamical boundary conditions, wave equations with variable-exponent nonlinearities, thermoelastic laminated beams with time-varying delay, and delayed Timoshenko beam systems.Written with scholarly rigor, this book is intended for mathematicians, physicists, engineers, and researchers in applied sciences. It can serve both as a graduate-level text and as a reference for researchers working on partial differential equations and related fields.
