2. Well-Posedness of Fractional Diffusion Equations

This chapter deals with the time fractional diffusion equations. In Sect. 2.1, we study a Cauchy problem for a space-time fractional diffusion equation with exponential nonlinearity. Based on the standard \(L^p\)-\(L^q\) estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in the Orlicz space \(\exp L^p({\mathbb R}^d)\) and a time weighted \(L^r({\mathbb R}^d)\) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space. Section 2.2 is devoted to the study of a semilinear diffusion problem with distributed order fractional derivative on \(\mathbb R^N\), which can be used to characterize the ultraslow diffusion processes with time-dependent logarithmic law attenuation. We use the resolvents approach to present the local well-posedness of mild solutions belonging to \(L^r(\mathbb R^N)~(r>2)\), in which the \(L^p\)-\(L^q\) estimates and continuity of the operator are first established. Then, under the assumption on the initial value belonging to \(L^p(\mathbb R^N)\), the global well-posedness of mild solutions is derived. Moreover, a decay estimate in \(L^r\)-norm is included. Section 2.3 discusses an analysis of approximate controllability from the exterior of distributed order fractional diffusion problem with the fractional Laplace operator subject to the nonzero exterior condition. We first establish some well-posedness results, such as the existence, uniqueness, and regularity of the solutions allowing the weighted function \(\mu \) that may be noncontinuous. Especially, we show that the solutions can be represented by the series for the integral of a real-valued function. After giving the unique continuation property of the adjoint system, approximate controllability of the system is also included. The material in Sect. 2.1 is taken from He et al. (Nonlinear Anal Model Control 29(2):286–304, 2024). Section 2.2 is taken from Peng et al. (Monatsh Math 198:445–463, 2022). The results in Sect. 2.3 are taken from Peng and Zhou (Appl Math Optim 86(2):22, 2022).