Initial-boundary value problems for coupled systems of time-fractional diffusion equations

Abstract

This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional diffusion equations. Defining the mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior of the solution, which mostly inherit those of a single equation. Owing to the coupling effect, we also obtain the uniqueness for an inverse problem on determining all the fractional orders by the single point observation of a single component of the solution.

Mild Solutions of Fractional Integrodifferential Diffusion Equations with Nonlocal Initial Conditions via the Resolvent Family

“…It is worth noting that since A in [18] is ω-sectorial (ω < 0) and the order of the equations is different from that in this paper, the proof that S α,β (t) is bounded cannot be directly applied in this paper. By the Laplace transform, the definition of a mild solution for (11) is given, and we prove the existence and uniqueness of the mild solutions through several fixed point theorems. In the future, we will further investigate the representation of solutions and the existence and regularity of solutions to fractional diffusion equations with multiple derivatives.…”

“…Lemma 6. Assume that u ∈ C(J, X), D α t u ∈ C((0, T], X), u(t) ∈ D(A) for t ∈ (0, T], and u satisfies (11). Then, u satisfies the formal integral equation…”

“…Proof. By the definitions of the Caputo derivative and the Riemann-Liouville integral [1], we can rewrite (11) as the equivalent integral equation…”

“…As is known, they can provide excellent descriptive models to resolve various problems in reality, and they are applied in various fields, such as control engineering [3], viscoelastic materials [4], fluid mechanics [5], electrochemistry [6], the analysis of epidemic [7] and complex networks [8], statistical mechanics [9], numerical schemes [10], etc. The relevant problems for the diffusion equations have been studied by many scholars, see [11,12]. Significantly, when α is used to represent the order of a fractional diffusion equation, when α ∈ (0, 1), α = 1, and α ∈ (1, 2), the equation describes subdiffusion, regular diffusion, and superdiffusion, respectively.…”

“…Afterwards, we define the mild solutions for (11) using the Laplace transform. In Section 3, the existence and uniqueness of the mild solutions to (11) are established by several fixed point theorems under some assumptions. In Section 4, an example is provided to verify the reasonability of the results.…”