This bookpresents recent advances on the global existence and decay estimates ofsolutions to two important classes of damped wave equations, with particularemphasis on the double dispersion equation and related models. Building on theauthor’s work with collaborators, the volume brings together results previouslyscattered across research papers, alongside new material published here for thefirst time. It offers a unified perspective on the long-standing problem ofglobal existence and the asymptotic behavior of dissipative hyperbolicequations, including convergence toward equilibrium and optimal decayproperties.

The book isorganized into eight chapters. The opening chapter reviews essential tools fromFourier analysis, Sobolev spaces, and fundamental inequalities. Subsequentchapters investigate the asymptotic profiles, decay rates, and pointwisebehavior of solutions to regularity-gained double dispersion equations, followedby a systematic study of regularity-loss wave equations. Techniques such astime-weighted energy methods and refined asymptotic analysis are developed toaddress weak dissipation and nonlinear effects, establishing global existenceand precise decay estimates under suitable initial conditions and spatialdimensions.

Combiningrigorous theory with clear structure, this book highlights both foundationalmethods and recent progress in nonlinear partial differential equations. It isintended for graduate students, researchers, and specialists interested innonlinear wave phenomena, dissipative systems, and modern analytical techniquesin PDEs.