On the backward problems in time for time-fractional subdiffusion equations

Abstract

The backward problem for subdiffusion equation with the fractional Riemann-Liouville time-derivative of order ρ ∈ (0,1) and an arbitrary positive self-adjoint operator A is considered. This problem is ill-posed in the sense of Hadamard due to the lack of stability of the solution. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists and it is unique. Using this result, we study the inverse problem of initial value identification for subdiffusion equation. The results obtained differ significantly from the corresponding results for the classical diffusion equation (i.e. ρ = 1 ) and even for the subdiffusion equation with the Caputo derivative. A list of examples of operator A is discussed, including linear systems of fractional differential equations, differential models with involution, fractional Sturm-Liouville operators, and many others.

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

“…The uniqueness of the solution can be proved by the standard technique based on completeness of the set of eigenfunctions {v k } in H (see, e.g., [5]).…”

“…The backward problems in case (2) were studied in detail, for example, in [2][3][4]. The work [5] is devoted to the study of the backward problem in case (3). Therefore, in what follows we only consider the case α = 0.…”

“…In other words, a small change of u(T) in the norm of space H leads to large changes in the initial data. As can be seen from the main results of papers [2][3][4][5] (note, in these works 0 < ρ < 1), the situation changes if we take the sufficiently smooth function u(T). Since the problem is ill-posed, many authors have considered various regularization options for finding the initial condition (see, for one-dimensional elliptical part, Liu and Yamamoto [2], for the nonlinear case, Tuan, Huynh, Ngoc, and Zhou [7]).…”

“…As shown in these papers, in contrast to the retrospective inverse problem, the problem ( 5) is coercively solvable in some spaces of differentiable functions. It should also be noted that various nonlocal boundary value problems for parabolic equations reduce to the boundary value problem (5) (see, e.g., [15], Chapter 1).…”

Backward and non-local problems for the Rayleigh-Stokes equation

“…Note that in work [18] the baclward problem for the subdiffusion equation ∂ α t u + Au = f was studied. It is shown that an estimate similar to (5.3) is valid only with the norm ||ω(T )|| 2 .…”