SOURCE PARAMETER IDENTIFICATION PROBLEM FOR SOME CLASS OF SEMILINEAR SUBDIFFUSION EQUATIONS

MINISTRY OF EDUCATION AND TRAININGHANOI NATIONAL UNIVERSITY OF EDUCATION ---  ---BUI THI HAI YEN SOURCE PARAMETER IDENTIFICATION PROBLEM FOR SOME CLASS OF SEMILINEAR SUBDIFFUSION EQU

MINISTRY OF EDUCATION AND TRAINING

HANOI NATIONAL UNIVERSITY OF EDUCATION

- 

-BUI THI HAI YEN

SOURCE PARAMETER IDENTIFICATION PROBLEM FOR SOME CLASS OF

SEMILINEAR SUBDIFFUSION EQUATIONS

Speciality : Differential and Integral Equations

Code : 9.46.01.03

SUMMARY OF DOCTORAL DISSERTATION IN MATHEMATICS

Hà Nội - 2025

The dissertation was written on the basis of the author’s research

works carried atHanoi National University of Education

Supervisor: Assoc.Prof Tran Dinh Ke

Assoc.Prof Nguyen Thi Van Anh

Referee 1: Prof Dr Sci Nguyen Minh Tri

Institute of Mathematics, Vietnam Academy of Science and Technology

Referee 2: Assoc.Prof Nguyen Minh Tuan

VNU University of Education, Vietnam National University Hanoi

Referee 3: Assoc.Prof Le Van Hien

Hanoi National University of Education

This dissertation is presented to the examining committee at HanoiNational University of Education, 136 Xuan Thuy Road, Hanoi, Vietnam

At the time of , 2025

Full-text of the dissertation is publicly available and can be accessedat:

- The National Library of Vietnam

- The Library of Hanoi National University of Education

1 Motivation of the problem

The class of classical diffusion equations

has been used to describe heat conduction in homogeneous media and

to model various problems in Physics, Chemistry, and Biology However,

in reality, diffusion processes often occur in media with memory effects

As a result, the classical model becomes inadequate, and new diffusionmodel is proposed, which is governed by a fractional partial differentialequation

where ∂α

tu denotes the Caputo fractional derivative of order α ∈ (0, 1).This model has been shown to provide a more accurate description ofdiffusion processes in porous materials, physical phenomena with mem-ory effects, and heterogeneous media In recent years, various problemsassociated with this class of equations have been extensively studied Itshould be noted that the Caputo fractional derivative is defined by theformula

k(t − τ )[u(τ ) − u(0)]dτ

is called the Caputo-type nonlocal derivative, or memory derivative J.Pr¨uss systematically developed the theory of convolution evolution equa-tions and established an operator-theoretic framework for differentialequations involving generalized convolution derivatives

In recent decades, the class of equations (3) has become a timely andactively studied topic, attracting significant attention from both domes-tic and international mathematicians Notable contributors include ĐangĐuc Trong, Tran Đinh Ke, Nguyen Huy Tuan, M Yamamoto, V Ver-gara, R Zacher, Y Liu, and F Yang, among others The solution prop-erties of the forward problem have been thoroughly studied for variouscases, including homogeneous equations, semilinear equations, and equa-tions with delay These properties encompass existence and uniqueness,stability, regularity, and decay behavior of the solution In contrast to theforward problem, the inverse problem is often ill-posed when the givendata lack sufficient regularity, leading to significant analytical difficul-ties Currently, several approaches have been developed to study inverseproblems for subdiffusion equations, including numerical and regular-ization methods; methods based on maximum principles and Carlemanestimates; and methods employing operator-theoretic frameworks.The problem of identifying source parameters in differential equa-tions has attracted significant attention from mathematicians due toits wide range of practical applications Numerous theoretical studieshave been published for classical partial differential equations, includ-ing works by Prilepko, Orlovsky, and Vasin, as well as by Choulli andYamamoto, and Tikhonov and Eidelman For partial differential equa-tions of fractional order, the source parameter identification problem has

also been investigated in both linear and semilinear settings Assumethat the external force f takes one of the following forms f = g(x)h(t),

f = g(x)h(t) + f1(u) or f = g(x)h(u) In the modeling process, one

of the components of f (e.g g(x)) may remain unknown The problemthen consists in determining this unknown component of the externalforce using an additional measurement, for example, u(T ) = ξ (the finalvalue is known) or u(T ) = 1

T

RT

0 u(s)ds = ξ (the time-averaged value

is known) In the linear case, uniqueness and stability of the solutionhave been extensively investigated by researchers such as Yamamoto,

Z Zhang, and T Wei In the semilinear case, uniqueness and stabilityhave also been studied, though the challenges are more significant due

to the nonlinear nature of the equation Most of the aforementioned sults rely on numerical and regularization methods, as well as approachesbased on the maximum principle and Carleman estimates for Caputo-type fractional equations in sufficiently regular spaces In addition, theoperator-theoretic approach developed by J Pr¨uss for a class of gener-alized Caputo fractional equations has recently attracted the interest ofthe research group led by Assoc Prof Dr Tran Dinh Ke This grouphas obtained significant results concerning the qualitative properties ofsolutions to equation (3), including the forward problem, the final valueproblem, and the source parameter identification problem However, tothe best of our knowledge, qualitative aspects of source parameter identi-fication problems for certain classes of semilinear subdiffusion equationssuch as abstract subdiffusion equations, equations with weak nonlinearperturbations (taking values in Hilbert scales with negative exponents),and equations involving nonlinear-intensity external forces have not yetbeen fully addressed Furthermore, we observe that relaxing some of thetechnical assumptions imposed in previous works is both important andnecessary These considerations form the main motivation for our interest

re-in studyre-ing this topic

2 Object and scope of the thesis

• Research object: The aim is to study the qualitative aspects ofthe source parameter identification problem for certain classes of sub-diffusion equations of the form (3), including the well-posedness of theproblem (existence, uniqueness, and stability of solutions with respect

to local data) and the regularity of solutions

4 Structure of the thesis

In addition to the introduction, conclusion, list of published works,and list of references, the thesis consists of four chapters:

Chapter 1 PRELIMINARIES

1.1 Function spaces

1.2 The power of an operator on a Hilbert space

Let H be a separable infinite-dimensional Hilbert space, A be anunbounded linear satisfying the following hypotheses

(HA) The operator A : D(A) → H is densely defined, self-adjoint, andpositively definite with compact resolvent

Then, there exists an orthonormal basis of H consisting of eigenfunctions{ek} of A Let ω ∈ H, define ωk= (ω, ek) where (·, ·) denotes the innerproduct in H Then, we have

1.3 Fractional Sobolev spaces and Hilbert scales

Take an orthonormal basis of L2(O) consisting of eigenfunctions {ek}

of the negative Laplace operator −∆ with homogeneous Dirichlet ary conditions Then we can write

O

ω(x)ek(x)dx



ek,

where λk > 0 is the eigenvalue associated with the eigenfunction ek of

−∆, for all k ∈ N For ϱ ≥ 0, we define:

5

where the unknown functions s, r are the scalar functions, γ is a rameter Denote by s(·, γ) and r(·, γ) the solutions of (1.1) and (1.2),respectively.

pa-Sonine kernels

The pair (k, l) satisfying condition (P C) is called a Sonine kernel pair

We need the following assumption in the rest of this work

(Hk) The hypothesis (P C) holds with the function l being nonincreasing

• k(t) = g1−α(t)e−ηt, η > 0, 0 < α < 1: We have l(t) = gα(t)e−ηt+η

t

R

gα(s)e−ηsds and ∂t,k is the tempered fractional derivative

One-parameter families on Hilbert spaces

We define the one-parameter families {S(t)}t≥0and {R(t)}t>0as follows:

One-parameter families on Hilbert scales

In the case H = L2(O), A = (−∆)σ, −∆ is the Laplace operator withhomogeneous Dirichlet boundary conditions We construct the relsoventoperators based on the relaxation functions s and r as the following

Chapter 2 SOURCE PARAMETER IDENTIFICATION PROBLEM FOR A CLASS OF ABSTRACT SEMILINEAR

SUBDIFFUSION EQUATIONS

In this chapter, we study the source identification problem for a class

of semilinear subdiffusion equations in an abstract Hilbert space setting,involving abstract operators, with local initial conditions and nonlocalfinal conditions The external force is assumed to be separable in vari-ables, and the perturbation is assumed to take values in a regular Hilbertspace We demonstrate the existence of mild and strong solutions to theproblem Additionally, we investigate the continuous dependence on thegiven data and the regularity of the solutions for this class of equations.The contents of this chapter is written based on the paper [1] in thesection of author’s works related to the thesis that has been published

2.1 Problem setting and Solution representation

2.1.1 Problem setting

Let H be a separable infinite-dimensional Hilbert space A be an bounded linear operator We consider the following problem: find (p, u)such that:

un-∂t,ku(t) + Au(t) = pg(t) + h(t, u(t)), 0 < t < T, (2.1)

we have the inverse problem of parameter identifying with local finalcondition In this model, the unknown function u takes values in H andthe parameter p is in H or in the dual space V−1, which will be specified

in the later section The functions g : R+→ R+

, h : R+× H → H andthe kernel k are given

2.1.2 The assumptions and Solution representation

In this chapter, let h : [0, T ] × H → H, ξ ∈ H We suppose thefollowing assumption imposed on the scalar function g

(Hg) g : R+→ R+ is continuous and

g(t) > g0, for all t ≥ 0,for some g0> 0

When we consider the mild solutions of problem (2.1)-(2.3), we canuse the admissible set of the functions φ given below

1 W is a linear operator and the domain D(W ) = D(A)

2 The following estimate holds

∥W ψ∥−1 ≤ ν(T )∥ψ∥1, (2.6)where

Z t 0

R(t − s)pg(s)ds +

Z t 0

2.2 Solvability and Regularity of mild solutions

2.2.1 The existence and uniqueness of mild solutions

Let (ξ, φ) ∈ H × AdT We give the following hypotheses imposed on

(Hφ) φ ∈ AdT, φ(0) = 0 and φ satisfies

∥φ(u) − φ(v)∥1 ≤ ηφ(r)∥u − v∥∞, ∀∥u∥∞, ∥v∥∞< r,

where ∥ · ∥∞ stands for the sup norm in C([0, T ]; H) and ηφ(·) is

a nonnegative function satisfying that

on [0, T ] satisfying ∥u∥∞≤ ρ provided that κ < 1, where

here ν(T ) is given in (2.7), Λr= k(T )+λ1

1 and G = supt∈[0,T ]g(t).Remark 2.2.2 The condition κ < 1 is fulfilled provided that η∗

φ and η∗hare sufficiently small

2.2.2 The stability of mild solutions

To establish the well-posedness of the problem, we demonstrate theLipschitz continuous dependence of the mild solution on the given data.Theorem 2.2.5 Suppose that all hypotheses of Theorem2.2.1 hold Let(ξ1, φ), (ξ2, φ) ∈ H ×AdT such that ∥ξ1∥, ∥ξ2∥ are small enough to ensurethe existence and uniqueness of the problem If (p1, u1) and (p2, u2) arethe mild solutions with respect to data (ξ1, φ) and (ξ2, φ), then thereexists a constant C(T, k, g, λ1, φ, h) not depending on pi, ui such that thefollowing estimate holds

∥p1− p2∥−1 + ∥u1− u2∥∞≤ C(T, k, g, λ1, φ, h)∥ξ1− ξ2∥ (2.11)

(H∗h) h satisfies (Hh) and there is a positive function θh(·) such that

∥h(t, u) − h(s, u)∥ ≤ θh(r)|t − s|γ∥u∥, ∀t, s ∈ [0, T ], ∀u ∈ H, ∥u∥ ≤ r;

for some γ ∈ (0, 1)

Theorem 2.2.7 Suppose that all asumptions of Theorem 2.2.1 are isfied Moreover, suppose that (H∗h) holds, g ∈ Cγ

sat-([0, T ]; R) for some

γ ∈ (0,12] and λ1 > η∗h Then the identification problem (2.1)-(2.3) has

a unique mild solution (p, u) with u is H¨older continuous of order γ intime in the weak sense

2.3 Solvability and Regularity of strong solutions

In this section, we consider φ ∈ Ad∗T, where

Ad∗T = {φ : C([0, T ]; H) → D(A), φ is continuous}

2.3.1 The existence and uniqueness of strong solutions

For the function φ considered above, we establish that the H¨oldercontinuous mild solution coincides with the strong solution of the prob-lem

Theorem 2.3.2 Suppose that all asumptions of Theorem 2.2.7 are isfied Then, the identification problem (2.1)-(2.3) has a unique strongsolution on [0, T ]

2.3.2 The stability of strong solutions

Theorem 2.3.5 Assume that all asumptions of Theorem 2.3.2 holdand there exists a nonnegative function ζφ(·) such that

∥φ(u) − φ(v)∥D(A)≤ ζφ(r)∥u − v∥∞,

for all u, v ∈ C([0, T ]; H), ∥u∥∞; ∥v∥∞< r,

and

ζφ∗ := lim sup

r→0 +

ζφ(r) < ∞

Let (ξ1, φ), (ξ2, φ) ∈ H × Ad∗T such that ∥ξ1∥, ∥ξ2∥ are small enough

to ensure the existence and uniqueness of mild solutions of (2.1)-(2.3)

If (p1, u1) and (p2, u2) are the strong solutions with respect to the givendata (ξ1, φ) and (ξ2, φ), then there exists a constant C∗(T, k, g, λ1, φ, h)independent on pi, ui, ξi, i = 1, 2 such that the following estimate holds

∥p1− p2∥ + ∥u1− u2∥∞≤ C∗(T, k, g, λ1, φ, h)∥ξ1− ξ2∥, (2.12)

provided that

ϑ∗= 1 − ν(T )GΛr ζφ∗+ ηh∗Λr
− η∗

hΛr> 0

2.3.3 The H¨older regularity of mild solutions

Theorem 2.3.7 Suppose that all asumptions of Theorem 2.3.2 are isfied Then the unique strong solution (p, u) of the identification problem(2.1)-(2.3) satisfying u ∈ Cγρ,ρ′ for every γ ∈ (0, 1)

sat-Chapter 3 SOURCE PARAMETER IDENTIFICATION PROBLEM FOR A CLASS OF SUBDIFFUSION EQUATIONS WITH

WEAK NONLINEAR PERTURBATIONS

In this chapter, we investigate the source identification problem for aclass of subdiffusion equations with nonlocal initial and final conditions,where the external force is separable in variables and the nonlinear per-turbation may take values in a Hilbert scale space with negative expo-nent By establishing appropriate estimates for the solution operatorsand employing embedding theorems, we derive results concerning theexistence, uniqueness, stability, and H¨older regularity of mild solutions

Remark: The approach to this problem is, in essence, analogous to that

in Chapter 1 However, in the current setting, the problem is formulatedwithin a Hilbert scale framework, and the noisy data takes values in theHilbert scale with negative indices As a result, the necessary estimatesmust be established within the Hilbert scale, leading to the emergence

of technical conditions that require more delicate and intricate sis Furthermore, when examining concrete examples, the noise estima-tion necessitates the application of embedding lemmas between Sobolevspaces and the Hilbert scale Consequently, it is essential to suitably ad-just the conditions related to the indices of the function spaces to ensurethe consistency and feasibility of the analytical methodology

analy-3.1.2 The assumptions and Solution representation

We denote the sup norm in C([0, T ]; Hκ) by

∥φ(u) − φ(v)∥Hκ ≤ ηφ∥u − v∥κ,∞, ∀u, v ∈ C([0, T ]; Hκ),

for some ηφ> 0, where ∥·∥κ,∞stands for the sup norm in C([0, T ]; Hκ)

(Hψ) ψ : C([0, T ]; Hκ

) → Hκ satisfies ψ(0) = 0,

∥ψ(u) − ψ(v)∥Hκ≤ ηψ∥u − v∥κ,∞, ∀u, v ∈ C([0, T ]; Hκ),

for some ηψ > 0, where ∥ · ∥κ,∞ stands for the sup norm inC([0, T ]; Hκ)

(Hh) h : [0, T ] × Hκ→ Hθ with the numbers

Remark 3.1.1 The novelty in the condition (Hh) is the fact that h can

be taken values in scale Hilbert Hθ with θ ≤ 0 (e.g., polynomial-type orgradient-type perturbations) It is much more improved in comparisonwith thoses used in the chapter 2 and the previously works

r(T − s, λσn)g(s)ds

!−1

unen (3.6)

Lemma 3.1.2 Let W : Hκ→ Hκ−2σ be defined as (3.6) Then,

(1) W is a linear operator and the domain D(W ) = D((−∆)σ);(2) The following estimate holds

∥W ψ∥Hκ−2σ ≤ νT∥ψ∥Hκ, (3.7)

where

g (1 − s(T, λσ)). (3.8)