This book presents recent advances on the global existence and decay estimates of solutions to two important classes of damped wave equations, with particular emphasis on the double dispersion equation and related models. Building on the author's work with collaborators, the volume brings together results previously scattered across research papers, alongside new material published here for the first time. It offers a unified perspective on the long-standing problem of global existence and the asymptotic behavior of dissipative hyperbolic equations, including convergence toward equilibrium and optimal decay properties.

The book is organized into eight chapters. The opening chapter reviews essential tools from Fourier analysis, Sobolev spaces, and fundamental inequalities. Subsequent chapters investigate the asymptotic profiles, decay rates, and pointwise behavior of solutions to regularity-gained double dispersion equations, followed by a systematic study of regularity-loss wave equations. Techniques such as time-weighted energy methods and refined asymptotic analysis are developed to address weak dissipation and nonlinear effects, establishing global existence and precise decay estimates under suitable initial conditions and spatial dimensions.

Combining rigorous theory with clear structure, this book highlights both foundational methods and recent progress in nonlinear partial differential equations. It is intended for graduate students, researchers, and specialists interested in nonlinear wave phenomena, dissipative systems, and modern analytical techniques in PDEs.