Fractional Diffusion and Wave Equations

Inhaltsverzeichnis

1. Frontmatter

2. Chapter 1. Preliminaries

   Yong Zhou

   Abstract

   In this chapter, we introduce some notations and basic facts on fractional calculus, the Mittag–Leffler functions, the Wright-type function, integral transforms, and semigroups, which are needed throughout this monograph.

3. Chapter 2. Well-Posedness of Fractional Diffusion Equations

   Yong Zhou

   Abstract

   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).

4. Chapter 3. Inverse Problems of Fractional Diffusion Equations

   Yong Zhou

   This chapter deals with the inverse problems of time fractional diffusion equations of order \(\alpha \in (0,1)\). In Sect. 3.1, we study a backward problem for an inhomogeneous fractional diffusion equation in a bounded domain. By applying the properties of the Mittag-Leffler functions and the method of eigenvalue expansion, we establish some results about the existence, uniqueness, and regularity of the mild solutions and the classical solutions of the proposed problem in a weighted Hölder continuous function space. In Sect. 3.2, we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain D of \( \mathbb {R}^{k}\), \(k\ge 1\), which includes the fractional power \(\mathscr {L}^\beta \), \(0<\beta \leq 1\), of a symmetric uniformly elliptic operator \(\mathscr {L}\) defined on \(L^2(D)\). A representation of solutions is given by using the Laplace transform and the spectrum of \(\mathscr {L}^\beta \). We present some existence and regularity results for our problem in both the linear and nonlinear cases. The materials in Sect. 3.1 are adopted from Zhou, He, Ahmad, and Tuan [245]. The contents in Sect. 3.2 are due to Tuan, Ngoc, Zhou, and O’Regan [210].

5. Chapter 4. Well-Posedness and Regularity of Fractional Wave Equations

   Yong Zhou

   In this chapter, we first study the well-posedness and regularity of mild solutions for a class of time fractional damped wave equations. A concept of mild solutions is introduced to prove the existence for the linear problem, as well as the regularity of the solutions. We also establish a well-posedness result for nonlinear problem. As an application, we discuss a case of time fractional telegraph equations. Section 4.2 studies the semilinear time fractional wave equation on a whole Euclidean space, also known as the superdiffusive equations. Based on the initial data taken in the fractional Sobolev spaces and some known Sobolev embeddings, we prove the local/global well-posedness results of \(L^2\)-solutions for the linear and semilinear problems. In Sect. 4.3, we concern with an exponential nonlinearity for a fractional wave equation in the whole space, and we establish the local existence of solutions in a dense subspace of the Orlicz classification. Moreover, we obtain the global existence of solutions for small initial data in lower dimension \(1\leq d\leq 3\). Our proofs base on the analyticity of the Mittag-Leffler functions, the framework of prior estimates, and the type of exponential nonlinearity. The material in Sect. 4.1 is due to Zhou and He (Monatsh Math 194(2):425–458, 2021) . The results in Sect. 4.2 are taken from Zhou, He, Alsaedi, and Ahmad Zhou et al. (Elec Res Arch 30(8):2981–3003, 2022). The results in Sect. 4.3 are adopted from He and Zhou (Bull Sci Math 189:103357, 2023).

6. Chapter 5. Inverse Problems of Fractional Wave Equations

   Yong Zhou

   In this chapter, we firstly concern with a backward problem (or called initial inverse problem) for a nonlinear time fractional wave equation in a bounded domain. By applying the properties of Mittag–Leffler functions and the method of eigenfunction expansion, we establish some results about the existence and uniqueness of mild solutions of the proposed problem based on the compact technique. Due to the ill-posedness of backward problem in the sense of Hadamard, a general filter regularization method is utilized to approximate the solution, and further we prove the convergence rate for the regularized solutions. In Sect. 5.2, we consider the backward problem for an inhomogeneous time fractional wave equation in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme by using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution under two parameter choice rules. In Sect. 5.3, we consider the terminal value problem of determining the initial value, in a general class of time fractional wave equation with the Caputo derivative, from a given final value. We are concerned with the existence and regularity upon the terminal value data of the mild solution. Under some assumptions of the nonlinear source function, we address and show the well-posedness for the terminal value problem. Some regularity results for the mild solution and its derivatives of first and fractional orders are also derived. The effectiveness of our methods is shown by applying the results to two interesting models: time fractional Ginzburg–Landau equation and time fractional Burgers equation, where time and spatial regularity estimates are obtained. The contents of Sect. 5.1 are taken from He and Zhou (Proc R. Soc Edinburgh Sect A 152(6):1589–1612, 2022). The results in Sect. 5.2 are adopted from Huynh et al. (Appl Anal 100(4):860–878, 2021). Section 5.3 is from Tuan et al. (Nonlinearity 34(3):1448, 2021).

7. Backmatter