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This book provides a unified analysis and scheme for the existence and uniqueness of strong and mild solutions to certain fractional kinetic equations. This class of equations is characterized by the presence of a nonlinear time-dependent source, generally of arbitrary growth in the unknown function, a time derivative in the sense of Caputo and the presence of a large class of diffusion operators. The global regularity problem is then treated separately and the analysis is extended to some systems of fractional kinetic equations, including prey-predator models of Volterra-Lotka type and chemical reactions models, all of them possibly containing some fractional kinetics.
Besides classical examples involving the Laplace operator, subject to standard (namely, Dirichlet, Neumann, Robin, dynamic/Wentzell and Steklov) boundary conditions, the framework also includes non-standard diffusion operators of "fractional" type, subject to appropriate boundary conditions.
This book is aimed at graduate students and researchers in mathematics, physics, mathematical engineering and mathematical biology, whose research involves partial differential equations.
List of contents
1. Introduction.-1.1 Historical remarks .-1.2 On overview of main results and applications .-1.3 Results on nonlocal reaction-diffusion systems 2. The functional framework.-2.1 The fractional-in-time linear Cauchy problem .-2.2 Ultracontractivity and resolvent families .-2.3 Examples of sectorial operators .-3 The semilinear parabolic problem.-3.1 Maximal mild solution theory .-3.2 Maximal strong solution theory .-3.3 Differentiability properties in the case 0 < a < .-3.4 Global a priori estimates .- 3.5 Limiting behavior as a ®1.- 3.6 Nonnegativity of mild solutions .-3.7 An application: the fractional Fisher-KPP equation .-4 Systems of fractional kinetic equations .-4.1 Nonlinear fractional reaction-diffusion .-4.2 The fractional Volterra-Lotka model .-4.3 A fractional nuclear reactor model .-5 Final remarks and open problems .-A Some supporting technical tools .-B Integration by parts formula for the regional fractional Laplacian .-C A zoo of fractional kinetic equations.-C.1Fractional equation with nonlocality in space.-C.2 Fractional equation with nonlocality in time.-C.3 Space-time fractional nonlocal equation.-References.-Index.
About the author
Ciprian Gal is Associate Professor at Florida International University, Miami, Florida (USA). His research focuses on the analysis of nonlinear partial differential equations including nonlocal PDEs.Mahamadi Warma is Professor at George Mason University in Fairfax, Virginia (USA). His reseach focuses on linear and nonlinear partial differential equations, fractional PDEs and their controllability-observability properties.
