PARTIAL DIFFERENTIAL EQUATIONS IN SOBOLEV & ANALYTIC SPACES

Partial Differential Equations (PDEs) are fundamental in fields such as physics and engineering, underpinning our understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics. They also arise in areas like differential geometry and the calculus of variations.

This book focuses on recent investigations of PDEs in Sobolev and analytic spaces. It consists of twelve chapters, starting with foundational definitions and results on linear, metric, normed, and Banach spaces, which are essential for introducing weak solutions to PDEs. Subsequent chapters cover topics such as Lebesgue integration, Lp spaces, distributions, Fourier transforms, Sobolev and Bourgain spaces, and various types of KdV equations. Advanced topics include higher order dispersive equations, local and global well-posedness, and specific classes of Kadomtsev-Petviashvili equations.

This book is intended for specialists like mathematicians, physicists, engineers, and biologists. It can serve as a graduate-level textbook and a reference for multiple disciplines.

Contents:
PreliminariesLebesgue IntegrationThe Lp SpacesDistributions: The Fourier TransformSobolev Spaces: Analytic SpacesOriginal Method for the KdV Equation in Hs(ℝ)Fifth-Order Shallow Water EquationHigher-Order Nonlinear Dispersive EquationKadomtsev–Petviashvili in Analytic SpacesGeneralized Kadomtsev–Petviashvili I EquationCoupled System of KDV Equations in Gevrey SpacesSystem of Generalized KdV Equations
Readership: This book is addressed to a wide audience of specialists such as mathematicians, physicists, engineers and biologists. It can be used as a textbook at the graduate level and as a reference book for several disciplines. It is suitable for researchers in PDEs, mathematics, physics, biology, chemistry and informatics.

Aissa Boukarou is Assistant professor at University of Science and Technology Houari Boumediene. His research interests lie in Partial differential equations, Harmonic analysis, Probability Theory, Stochastic partial differential equations, Numerical Analysis, Image processing.Khaled Zennir is associate professor at Qassim University, KSA. He received his PhD in mathematics in 2013. He obtained his Habilitation in mathematics from Constantine University, Algeria in 2015. His research interests lie in nonlinear partial differential equations: global existence, blow up and long-time behavior. Svetlin G Georgiev is a mathematician who has worked in various areas of mathematics. He currently focuses on harmonic analysis, functional analysis, partial differential equations, ordinary differential equations, Clifford and quaternion analysis, integral equations and dynamic calculus on time scales.