Inverse Moving Source Problems for (Time-Fractional)
Diﬀusion(-Wave) Equations†
Yikan LIU∗

Within the last decade, evolution equations with fractional derivatives have gatheredconsiderable attention among researchers from both theoretical and applied disciplines
due to their feasibility in modeling physical processes such as anomalous diﬀusion inheterogeneous media. Meanwhile, the corresponding inverse problems on determining
various ingredients in the equations have also been investigated intensively in view of theirmathematical and practical signiﬁcance. In this talk, we consider the initial-boundaryvalue problem for a (time-fractional) diﬀusion(-wave) equation
(∂tα + L)u = Fin Ω × (0, T ),{ u=0if 0 < α ≤ 1,in Ω × {0},(1)u=∂u=0if1<α≤2tu = 0on ∂Ω × (0, T ),where ∂tα (0 < α ≤ 2) denotes the Caputo derivative in time, Ω ⊂ Rd (d = 1, 2, . . .) is abounded domain with a smooth boundary ∂Ω, and L is a second order elliptic operator inΩ. Here the source term F describes the external force of the system, which models thecontaminant source in the context of anomalous diﬀusion. If F takes the form of separated
variables, there have been a lot of existing literature on determining either the temporalor the spatial component (see [3] and the references therein).
Unlike the conventional formulation, this talk is concerned with a class of inverse
moving source problem where F depicts some objects moving along orbits. More precisely,
we study the following two problems.
Problem 1 (Determination of orbits). Let u be the solution to (1) with F (x, t) = g(x −γ(t)), and pick N interior points xj ∈ Ω (j = 1, . . . , N ). Provided that the source profileg(x) is suitably given, determine the source orbit γ(t) (0 ≤ t ≤ T ) by the multiple point
observations of u at {xj }N
j=1 × [0, T ].
∗Research Institute for Electronic Science, Hokkaido University. N12W7, Kita-Ward, Sapporo 060-0812,
Japan. E-mail: ykliu@es.hokudai.ac.jp†
This is a joint work with Guanghui Hu (Beijing Computational Science Research Center) and Masahiro
Yamamoto (The University of Tokyo).

Problem 2 (Determination of proﬁles). Let u be the solution to (1) with{g(x − p t),0 < α ≤ 1,F (x, t) =g(x − p t) + h(x − q t), 1 < α ≤ 2,(2)and ω ⊂ Ω be a suitably chosen nonempty subdomain of Ω. Provided that q, q ∈ Rd are
given constant vectors such that p ̸= q, determine the source profile g, h by the partial
interior observation of u in ω × (0, T ).
It reveals that Problem 1 is nonlinear and Problem 2 is linear. In Problem 1, the orbit
γ : [0, T ] −→ Rd is assumed to be a smooth vector-valued function with d components, andthus the number N of observation points should basically be no less than d. In Problem
2, we assume the parallel movement of the source(s), and it is required to determine two
proﬁles g and h simultaneously in the case of 1 < α ≤ 2.
For Problem 1, we basically restrict the unknown γ in the admissible set
U := {γ ∈ (C ∞ [0, T ])d | γ(0) = 0, ∥γ∥C[0,T ] ≤ ε, ∥γ ′ ∥C[0,T ] ≤ K}
with positive constants ε, K. Especially, the restriction ∥γ∥C[0,T ] ≤ ε means that γ is alocalized moving source. In this situation, we pick the minimum necessary d observation
points {xj }dj=1 and make the following key assumption:(∇g(y 1 ) ∇g(y 2 ) · · ·)−1∇g(y d )≤ C,∀ y j ∈ Bε (xj ) , j = 1, . . . , d.

(3)()
In other words, we assume that the matrix ∇g(y 1 ) · · · ∇g(y d ) is invertible for all∏(y 1 , . . . , y d ) ∈ dj=1 Bε (xj ) . Then we can state the Lipschitz stability and uniquenessresults for Problem 1.
Theorem 1 (see [1]). Fix γ1 , γ2 ∈ U and denote by u1 and u2 the solutions to (1) withγ = γ1 and γ = γ2 , respectively. If the set of observation points {xj }dj=1 satisfies (3),then there exists a constant C > 0 depending on g, {xj }dj=1 and U such that
∥γ1 − γ2 ∥C[0,T ] ≤ Cd∑∥∂tα (u1 − u2 )(xj , · )∥C[0,T ] .

j=1

Especially, u1 = u2 at {xj }dj=1 × [0, T ] implies γ1 = γ2 on [0, T ].
The key to proving the above theorem is the reduction of the original problem to avector-valued Volterra integral equation of the second kind with respect to the diﬀerence
γ1 − γ2 . To this end, we employ the frequently used Mittag-Leﬄer functions and a newly
established fractional Duhamel’s principle to obtain the representation of solutions to (1).
On the other hand, provided that the key assumption (3) holds globally in Ω, we canremove the localization condition ∥γ∥C[0,T ] ≤ ε and obtain a uniqueness result at the costof very dense observation points.
For Problem 2, we state the uniqueness result as follows.

Theorem 2 (see [2]). Let u be the solution to (1) with F given by (2). For α ∈ (0, 2]\{1},assume that ω ⊂ Ω is a nonempty subdomain such that ∂ω ⊃ ∂Ω.
(a) Let 0 < α ≤ 1 and g ∈ H01 (Ω). Then u = 0 in ω × (0, T ) implies g ≡ 0 in Ω.
(b) Let 1 < α ≤ 2 and g, h ∈ H 2 (Ω)∩H01 (Ω). For α = 2, further assume T > 2diam(Ω).
Then u = 0 in ω × (0, T ) implies g = h ≡ 0 in Ω.
In the above theorem, the assumption ∂ω ⊃ ∂Ω means that ω covers the whole boundary ∂Ω, which is only absent in the case of α = 1. For α = 2, the observation duration T
is additionally assumed to be suﬃciently long due to the ﬁnite propagation speed of wave.
The key to proving Theorem 2 is a reductions to the inverse problem on the determinationof the initial condition by introducing auxiliary functions{J 1−α (∂t + p · ∇)u,0 < α ≤ 1,v :=J 2−α (∂t + p · ∇)(∂t + q · ∇)u, 1 < α ≤ 2,
where J β denotes the Riemann-Liouville integral operator. Then we can turn to a uniquecontinuation argument to conclude the vanishing of the initial condition for v to complete
the proof.

References[1] G. Hu, Y. Liu and M. Yamamoto, Inverse moving source problems for fractional diﬀusion(-wave)
equations: Determination of orbits, Inverse Problems and Related Topics, Springer Proceedings inMathematics & Statistics 310, Springer, Singapore, 2020, accepted.
[2] Y. Liu, G. Hu and M. Yamamoto, Inverse moving source problems for fractional diﬀusion(-wave)
equations: Determination of proﬁles, preprint.
[3] Y. Liu, Z. Li and M. Yamamoto, Inverse problems of determining sources of the fractional partial diﬀerential equations, Handbook of Fractional Calculus with Applications. Volume 2: Fractional Diﬀerential
Equations, De Gruyter, Berlin, 2019, 411–430.