Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

Abstract

In this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$ u ∈ L ∞ ( ( 0 , T ) , H 0 1 ( Ω ) ) to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$ H 0 1 ( Ω ) . We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$ C ( [ 0 , T ] , H 0 1 ( Ω ) ∗ ) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$ d d t ( k ∗ v ) ( t ) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.

Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Time-Fractional Equations with Non-Smooth Solutions

“…In this section, we studied ISPs (2), ( 3) and ( 2), (4) using a variational approach. We used the following lemmas (see [2] (Lemma 2.1), [2] (Corollary 3.1), and [2] (Theorem 6.1)) to study the uniqueness of the solution to the ISPs.…”

“…If we assumed that solution u satisfied additionally 2 (Ω) , then we could show the uniqueness of a solution to ISP (2) and (3), with time-dependent coefficients in operator L.…”

“…Then, there exists at most one couple u, f ∈ L ∞ (0, T), 2 (Ω) , such that Problem (2), together with Condition (3), is satisfied.…”

“…where the symbol ' * ' stands for the classical convolution product defined by ( f * g)(t) = t 0 f (t − s)g(s) ds. This convolution structure is crucial when establishing the result in [2] (Lemma 2.1), which is used throughout this paper.…”

A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions

“…In case of direct problems of time-fractional diffusion equation (initial value problems and initial boundary value problems), the literature is fruitful with many studies about them. For some theoretical results concerned with uniqueness and existence, we refer to [26,29,18,35,36]. Numerical implementations for that direct problems are tackled in the literature also by using different numerical methods.…”