Cauchy problem for fractional partial differential equations

Abstract

This thesis is devoted to the analytical study of time–space Fractional Partial Differential Equations (FPDEs) involving generalized fractional derivatives and nonlocal spatial operators. The main objective is to inves tigate the existence, uniqueness, and qualitative behavior of solutions for a class of fractional Cauchy problems governed by the φ-Caputo fractional derivative in time and the fractional Laplacian in space. The analysis is based on the construction of explicit fundamental solutions, expressed in terms of the Fox H function, which provides a unified and powerful representation framework for fractional diffusion and superdiffusion models. These representations allow us to derive sharp decay estimates in Lebesgue and Lorentz spaces, highlighting the dispersive and smoothing effects induced by fractional dynamics. Building on the linear theory, local and global existence results for nonlinear fractional diffusion equations are established using fixed-point arguments in suitable functional spaces. Moreover, sufficient conditions for finite-time blow-up are derived for nonlinear superdiffusion equations, showing that the interplay between nonlinear effects and fractional orders in time and space plays a decisive role in the long-time behavior of solutions. The results presented in this thesis contribute to the theoretical development of fractional partial differential equations with generalized fractional operators and provide a rigorous analytical framework that extends several classical models.