Abstract Time-Fractional Cauchy Problems Associated to m-Dissipative Operators

This book presents a self-contained and unified treatment of time-fractional Cauchy problems associated with m dissipative operators. Fractional diffusion equations arise in the modeling of anomalous diffusion, and diffusion processes with memory, and have become an important area of research in analysis, partial differential equations, and applied mathematics. The book develops a systematic framework for studying fractional evolution equations by combining operator theory with Laplace transform techniques. Both constant-order and variable-order time-fractional derivatives are treated. A central theme is the use of m dissipative operators to establish existence, uniqueness, and qualitative properties of solutions in a unified manner. The presentation covers fractional derivatives of orders in (0, 1) as well as between (1, 2), highlighting the similarities and differences between these cases. Special attention is given to self-adjoint m-dissipative operators, which provide a particularly useful setting for the analysis of fractional Cauchy problems. The book also discusses explicit solution representations for diagonalizable operators, semilinear equations, perturbed equations, and problems involving damping terms. In addition, variable-order fractional Cauchy problems in L ²-spaces are investigated, and existence and uniqueness results are established for both distribution and strong solutions. The final chapter is devoted to inverse problems arising from time-fractional partial differential equations. Topics include the determination of lower-order coefficients, inverse source problems, and backward problems, together with corresponding uniqueness and well-posedness results. This book provides a coherent introduction to modern operator-theoretic approaches to fractional evolution equations and serves as a valuable reference for graduate students and researchers in partial differential equations, functional analysis, and related areas of applied mathematics.