# A source identification problem in a bi-parabolic equation: convergence
rates and some optimal results

Subhankar Mondal and M. Thamban Nair  (SM): TIFR Centre for Applicable
Mathematics, Bangalore-560065, India; (MTN): Department of Mathematics, BITS
Pilani, Goa campus, Goa-403726<EMAIL_ADDRESS><EMAIL_ADDRESS>pilani.ac.in

###### Abstract.

This paper is concerned with identification of a spatial source function from
final time observation in a bi-parabolic equation, where the full source
function is assumed to be a product of time dependent and a space dependent
function. Due to the ill-posedness of the problem, recently some authors have
employed different regularization method and analysed the convergence rates.
But, to the best of our knowledge, the quasi-reversibility method is not
explored yet, and thus we study that in this paper. As an important
implication, the Hölder rates for the apriori and aposteriori error estimates
obtained in this paper improve upon the rates obtained in earlier works. Also,
in some cases we show that the rates obtained are of optimal order. Further,
this work seems to be the first one that has broaden the applicability of the
problem by allowing the time dependent component of the source function to
change sign. To the best of our knowledge, the earlier known work assumed the
fixed sign of the time dependent component by assuming some bounded below
condition.

Keywords: bi-parabolic equation, source identification, quasi-reversibility
method, apriori and aposteriori parameter choice, optimal rate

MSC 2010: 35R25, 47A52, 65M30, 41A25

##  1\. Introduction

In the theory of heat transfers, the bi-parabolic equations have many
applications [28]. It is reported, for instance in [11, 19], that the
classical parabolic equations fail to describe the heat conduction processes
accurately. To overcome this shortcoming, several models were proposed which
include the bi-parabolic model proposed in [12]. In addition to this, bi-
parabolic models are used to describe the special phenomenon of process
dynamics filter image [20] and also to model various natural phenomena which
includes ice formation, fluid on the lungs, design of distinctive curves on
the surface [13]. Thus, from application point of view the bi-parabolic
systems are interesting and relevant topic.

In this paper, we consider an inverse source identification problem associated
with a bi-parabolic system. More precisely, let
$\Omega\subset{\mathbb{R}}^{d}(d\geq 1),$ be a bounded domain with Lipschitz
boundary $\partial\Omega$ and $\tau>0$ be fixed. We consider the following
system

(1.1) $\begin{cases}u_{tt}-2\Delta
u_{t}+\Delta^{2}u=\psi(t)f(x),&\quad\text{in}\quad\Omega\times(0,\tau),\\\
u=0=\Delta u,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
u(\cdot,0)=0=u_{t}(\cdot,0),&\quad\text{in}\quad\Omega,\\\
u(\cdot,\tau)=h,&\quad\text{in}\quad\Omega,\end{cases}$

where $\psi\in C([0,\tau])$ and $f,h\in L^{2}(\Omega).$ The inverse problem is
to identify the spatially dependent exact source $f$ from the knowledge of
exact final time observation $h.$ But in practice since we will always have
access to a measurement data of the final value, noise in the data is
inevitable. In this paper, we consider the deterministic noise model. That is,
for $\delta>0,$ let $h^{\delta}$ be the measured final time noisy observation
satisfying

(1.2) $\|h-h^{\delta}\|_{L^{2}(\Omega)}\leq\delta.$

This type of problem belongs to the class of inverse source identification
problem which has a vast literature in the area of inverse problems (cf. [18])
because of their tremendous real world applications. Indeed, the source
identification problem is used in detecting the source of pollution in water
surfaces or atmospheric media [2], and also for geophysical prospecting [23],
to name a few.

The inverse source problem that is considered in this paper is known to be
ill-posed [30], in the sense that a small perturbation in the final time
observation may lead to a large deviation in the corresponding source function
and thus, some regularization method (cf. [9, 25]) has to be employed in order
to obtain some stable approximations for the source function. For $\psi\equiv
1$ in $[0,\tau]$, an iterative regularization method has been proposed in [38]
to obtain stable approximations for the identification of the spatially
dependent source function. For the general source term of the separable form
that is considered in this paper, it seems the work on the regularization
aspect is in the initial stage. To the best of our knowledge, the very recent
paper of Quoc Nam et al. [30] is the first paper where the authors considered
the same problem of source identification that is considered here, and have
employed the Tikhonov regularization method for the regularization purpose.
Subsequently, in a follow-up work, Tuan [35] has used a Fourier truncation
method for the regularization purpose again for a similar source
identification problem and analysed the case when the observed data is in any
$L^{p}$ spaces. In [29] the authors employed the quasi-boundary value method
for the regularization purpose for the source identification problem that is
considered in this paper. To the best of our knowledge, another important type
of regularization method, namely the quasi reversibility method (QRM) has not
been explored yet in the setting of source identification associated with the
bi-parabolic equation (1.1). In this paper we employ the QRM for the
regularization purpose and obtain the error estimates.

The QRM was popularised by Lattès and Lions [21]. The idea behind this method
is to perturb the governing differential equation by a suitable operator so
that solving the resulting problem becomes a well-posed problem. More
specifically, in [21] the authors considered a final value problem of the form

$\begin{cases}\partial_{t}v+\mathcal{L}(t)v=0\\\ v(\tau)=v_{\tau}\end{cases}$

and considered the problem of identifying the initial value $v(0),$ where the
operator $\mathcal{L}(t)$ is a spatial differential operator, for example, one
may have $\mathcal{L}(t)$ to be the negative spatial Laplacian $-\Delta$ or of
the form $-\nabla\cdot a(x,t)\nabla$, where $a(x,t)$ is sufficiently smooth.
In order to solve the backward problem the authors proposed the following
perturbed equation

$\begin{cases}\partial_{t}v_{\epsilon}+\mathcal{L}(t)v_{\epsilon}+\epsilon\mathcal{L}^{*}(t)\mathcal{L}(t)v_{\epsilon}=0\\\
v_{\epsilon}(\tau)=v^{\epsilon}_{\tau}\end{cases}$

for $\epsilon>0$. The problem of obtaining the initial value $v_{\epsilon}(0)$
from the final value $v^{\epsilon}_{\tau}$ is a well-posed problem and
$v_{\epsilon}(0)$ gives an approximation of $v(0)$ in some suitable topology.
Since the work in [21], several researchers have analysed the QRM in a wide
range of problems. The application of QRM can be found, for example, in [3]
for the elliptic setting, [7, 24, 27, 32, 36] for the parabolic setting and
[22, 26] for the hyperbolic setting. Also, with the increasing work in the
backward time fractional diffusion equation, the QRM has found its application
in several recent works related to time/space fractional setting, see for
instance [8, 37]. For more details on QRM and its several implications and
modifications, the interested reader may refer to [1, 4, 5, 17, 31].

Motivated by the recent work of Duc et al. [8], where a QRM is analysed for a
source identification problem associated with a time-space fractional
parabolic equation, we analyse a QRM for the inverse problem considered in
this paper. More precisely, for $b\geq 2$ and $\alpha>0,$ we consider the
following bi-parabolic system

$\begin{cases}v_{tt}-2\Delta
v_{t}+\Delta^{2}v=(I+\alpha(-\Delta)^{b})\psi(t)\tilde{f}(x),&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v=0=\Delta v,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(\cdot,0)=0=v_{t}(\cdot,0),&\quad\text{in}\quad\Omega,\\\
v(\cdot,\tau)=\tilde{h},&\quad\text{in}\quad\Omega,\end{cases}$

where $I:L^{2}(0,\tau;L^{2}(\Omega))\to L^{2}(0,\tau;L^{2}(\Omega))$ is the
identity map, and consider the problem of identifying $\tilde{f}$ from the
final time measurement $\tilde{h}$ using the mild solution (see Definition
4.2). We show that this is an well-posed problem, and the obtained solutions
are indeed an approximations for the exact source $f.$

We now point out the novelty and main contributions of this paper.

  * •

In all the earlier known works [30, 35, 29] that considered the identification
of $f$ from $h$ associated with the bi-parabolic system (1.1), the analysis
has been done by assuming a bounded below assumption on $\psi,$ the time
dependent component of the source, which ensures a fixed sign property for
$\psi$. In contrast to this, our analysis works for a more general case, which
includes the fixed sign case and also allows $\psi$ to change sign (see
Assumption 3.1). In that sense, this work seems to be the first paper that
allows $\psi$ to change sign in the context of the considered inverse problem.

  * •

This work seems to be the first work where the QRM is analysed for the
regularization purpose in the context of the bi-parabolic system (1.1). As
mentioned earlier, the Tikhonov regularization method, quasi-boundary value
method and the Fourier truncation method have already been analysed by some
authors for the same problem.

  * •

We obtain Hölder type error estimates for both the apriori and aposteriori
parameter choice strategies using a conditional stability estimates. Under
some additional assumptions on $b$, the order of the error estimates in
apriori case is better than the previously known rates (see Remarks 4.5 and
4.8).

  * •

We show that in some cases the obtained error estimates are of optimal order
(in the sense of worst case error) for both the apriori and aposteriori cases.
This fills the gap in the arguments for the order optimality claim made in [8,
15] (see Remark 5.5).

The remainder of the paper is organized as follows. In Section 2 we recall all
the necessary results and state some facts that will be used throughout in
this paper. Section 3 perhaps is the base of the paper. In this section, we
specify certain assumptions on $\psi$, see their consequences and as a by
product we obtain a conditional stability estimate. In Section 4 we consider
the regularization by quasi-reversibility method, obtain error estimates for
both apriori and aposteriori parameter choice strategies. Finally, in Section
5 we prove that some of the error estimate that we obtain are of optimal
order.

##  2\. preliminaries

In this section we recall some basic results from the spectral theory of
elliptic PDE, and discuss about the solutions associated to (1.1), and also
present some basic results that will be used later.

First, let us fix some notations that will be used throughout.

  * •

Let $\mathbb{X}$ denote the space of real-valued functions defined on
$\Omega$, for e.g., $L^{2}(\Omega).$ Then, for $z\in
L^{2}(0,\tau;\mathbb{X}),$ we may use the notation $z(t)$ to denote
$z(\cdot,t)$ for almost all $t\in[0,\tau].$

  * •

$C([0,\tau];L^{2}(\Omega))$ denotes the standard space of all
$L^{2}(\Omega)$-valued continuous function in $[0,\tau].$

  * •

If $\chi\in C([0,\tau])$, then $\|\chi\|_{\infty}$ denotes the standard $\sup$
norm.

  * •

For $g_{1},g_{2}\in L^{2}(\Omega)$, $\langle g_{1},g_{2}\rangle$ denotes the
standard inner-product in $L^{2}(\Omega).$

  * •

For normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$, and a linear operator
$\mathbb{A}:\mathbb{X}\to\mathbb{Y},$ $\text{\bf dom}(\mathbb{A})$, $\text{\bf
ran}(\mathbb{A})$ and $N(\mathbb{A})$ shall denote the domain, range and the
kernel of $\mathbb{A},$ respectively.

For $\lambda\in{\mathbb{R}},$ it is known that the spectral problem

(2.1) $\begin{cases}-\Delta v=\lambda v\quad&\text{in}\quad\Omega,\\\ \quad
v=0\quad&\text{on}\quad\partial\Omega,\end{cases}$

admits a sequence of eigenvalues satisfying
$0<\lambda_{1}\leq\lambda_{2}\leq\ldots\to+\infty$, and the corresponding
eigenfunctions $\varphi_{n}$ are such that $\varphi_{n}\in
H^{1}_{0}(\Omega)\cap H^{2}(\Omega)$ and $\\{\varphi_{n}:n\in{\mathbb{N}}\\}$
is an orthonormal basis of $L^{2}(\Omega)$ (see for e.g., [10]).

For $p\geq 0,$ we define

$\mathbb{H}_{p}:=\\{\phi\in
L^{2}(\Omega):\sum_{n=1}^{\infty}\lambda_{n}^{2p}|\langle\phi,\varphi_{n}\rangle|^{2}<\infty\\},$

where $\langle\cdot,\cdot\rangle$ denotes the inner product
$\langle\cdot,\cdot\rangle_{L^{2}(\Omega)}$ on the space $L^{2}(\Omega)$. It
can be seen that $\mathbb{H}_{p}$ is a Hilbert space with respect to the inner
product

$\langle\phi,\psi\rangle_{{\mathbb{H}}_{p}}=\sum_{n=1}^{\infty}\lambda_{n}^{2p}\langle\phi,\varphi_{n}\rangle\langle\varphi_{n},\psi\rangle,\quad\phi,\psi\in{\mathbb{H}}_{p},$

so that the corresponding norm is given by

$\|\phi\|_{\mathbb{H}_{p}}=\left(\sum_{n=1}^{\infty}\lambda_{n}^{2p}|\langle\phi,\varphi_{n}\rangle|^{2}\right)^{\frac{1}{2}},\quad\phi\in{\mathbb{H}}_{p}.$

Clearly, when $p=0,$ then $\mathbb{H}_{p}=L^{2}(\Omega).$

We now look into the existence and uniqueness of the solution of the following
initial boundary value problem which will help to obtain a representation of
the source term $f$ in terms of $\psi$ and $h$. Let $\psi\in C([0,\tau])$,
$f\in L^{2}(\Omega)$. We consider the following system

(2.2) $\begin{cases}v_{tt}-2\Delta
v_{t}+\Delta^{2}v=\psi(t)f(x),&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v=0=\Delta v,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(\cdot,0)=0=v_{t}(\cdot,0),&\quad\text{in}\quad\Omega.\end{cases}$

Following [38, 30] (see also [35]) the problem (2.2) has a solution $u,$ given
by

(2.3) $u(t)=\sum_{n=1}^{\infty}\mu_{n}(t)\langle
f,\varphi_{n}\rangle\varphi_{n},$

with

(2.4) $\mu_{n}(t):=\int_{0}^{t}e^{-\lambda_{n}(t-s)}(t-s)\psi(s)\,ds.$

Thus, taking into account the final value condition $u(\tau)=h$ in (1.1), we
have

(2.5) $\langle
h,\varphi_{n}\rangle=\int_{0}^{\tau}e^{-\lambda_{n}(\tau-s)}(\tau-s)\psi(s)\,ds\,\langle
f,\varphi_{n}\rangle=\mu_{n}\langle f,\varphi_{n}\rangle,$

where

(2.6)
$\mu_{n}:=\mu_{n}(\tau)=\int_{0}^{\tau}e^{-\lambda_{n}(\tau-s)}(\tau-s)\psi(s)\,ds.$

In particular, the inverse source identification problem associated with (1.1)
is same as the problem of solving the operator equation

(2.7) ${\mathbb{T}}f=h,$

where ${\mathbb{T}}:L^{2}(\Omega)\to L^{2}(\Omega)$ is defined by

(2.8)
${\mathbb{T}}\varphi=\sum_{n=1}^{\infty}\mu_{n}\langle\varphi,\varphi_{n}\rangle\varphi_{n},\quad\varphi\in
L^{2}(\Omega).$

Since $\lambda_{n}\to\infty$ and $\psi\in C[0,\tau]$, it can be seen that
$|\mu_{n}|\to 0$ as $n\to\infty$. In fact,

(2.9)
$|\mu_{n}|\leq\|\psi\|_{\infty}\int_{0}^{\tau}e^{-\lambda_{n}(\tau-s)}(\tau-s)\,ds=\|\psi\|_{\infty}\frac{1-(1+\tau\lambda_{n})e^{-\lambda_{n}\tau}}{\lambda_{n}^{2}}\leq\|\psi\|_{\infty}\,\frac{1}{\lambda_{n}^{2}}.$

Hence, from the representation (2.8) of $\mathbb{T}$, it follows that
${\mathbb{T}}$ is a compact self-adjoint operator with eigenvalues
$\mu_{n},\,n\in{\mathbb{N}}.$ In particular, we can infer that the inverse
problem under consideration, which is same as the problem of solving the
operator equation (2.7) is an ill-posed problem. We shall show, under certain
assumptions on $\psi$, that $\mu_{n}\not=0$ for all $n\in{\mathbb{N}}$ so that
under those conditions, the equation (2.7) has a unique solution provided

$\sum_{n=1}^{\infty}\frac{|\langle
h,\varphi_{n}\rangle|^{2}}{|\mu_{n}|^{2}}<\infty,$

and in that case the solution $f$ is given by

(2.10) $f=\sum_{n=1}^{\infty}\frac{\langle
h,\varphi_{n}\rangle}{\mu_{n}}\varphi_{n}.$

We now state two elementary results in the form of a lemma, whose proofs
follow easily from basic calculus.

######  LEMMA 2.1.

  1. (i)

Let $a>0$ be fixed and $\eta_{a}:[\lambda_{1},\infty)\to{\mathbb{R}}$ be
defined by $\eta_{a}(s)=(1+as)e^{-as}$. Then $\eta_{a}$ is strictly decreasing
and hence attains its maximum at $s=\lambda_{1}.$

  2. (ii)

Let $\alpha,p>0$, and $b>p.$ Let $\eta_{\alpha,b,p}:[0,\infty)\to[0,\infty)$
be defined by $\eta_{\alpha,b,p}(s)=\frac{\alpha s^{b-p}}{1+\alpha s^{b}}.$
Then $\eta_{\alpha,b,p}$ attains global maximum at
$s_{0}=\left(\frac{b-p}{\alpha p}\right)^{\frac{1}{b}}$ and
$\eta_{\alpha,b,p}(s_{0})=\frac{p}{b}\left(\frac{b-p}{p}\right)^{\frac{b-p}{b}}\,\alpha^{\frac{p}{b}}.$

##  3\. assumption on $\psi$, its consequences and a stability result

The results of this section are the base of this paper. We begin with the
following assumption.

######  Assumption 3.1.

There exists $\tau_{0}\in[0,\tau)$ and $\kappa_{1}>0$ such that
$|\psi|\geq\kappa_{1}$ on $[\tau_{0},\tau],$ and one of the following holds:

  1. (i)

$\psi$ does not change sign on $[0,\tau].$

  2. (ii)

If $\psi$ changes sign on $[0,\tau]$ then $\psi$ is differentiable and there
exists $\kappa_{2}>0$ such that $|\psi^{\prime}|\leq\kappa_{2}$ on $[0,\tau],$
and $|\psi|\leq M:=\frac{(\tau-\tau_{0})^{2}\kappa_{1}}{4\tau\tau_{0}}$ on
$I_{\text{sc}}:=\\{t\in[0,\tau]:\psi(t)\psi(\tau)\leq 0\\}.$

######  Remark 3.2.

The above assumption on $\psi$ is less restrictive than the assumptions made
in [30, 35, 29] (where the analysis is done by assuming $|\psi|$ is bounded
below by a positive constant, which by virtue of continuity implies $\psi$ has
a fixed sign, i.e., either $\psi>0$ or $<0$ in $[0,\tau]$) in the following
sense: firstly, the bounded below assumption for the fixed sign case does not
necessarily has to be on the whole interval $[0,\tau]$, and secondly, the sign
changing assumption obviously implies $\psi$ can take much general form than
only the fixed sign case. $\lozenge$

We now give an example of a function that satisfies Assumption 3.1 (ii).

######  Example 3.3.

Let $\tau=\pi.$ Consider the function

$\psi(t)=\begin{cases}\frac{1}{48\sqrt{2}}\cos t,\quad&\text{if}\quad
t\in[0,\frac{\pi}{2}],\\\ \cos t,\quad&\text{if}\quad
t\in[\frac{\pi}{2},\pi].\end{cases}$

Clearly $|\psi(t)|\geq\frac{1}{\sqrt{2}}$ in $[\frac{3\pi}{4},\pi],$ and hence
$\tau_{0}:=\frac{3\pi}{4}$ and $\kappa_{1}:=\frac{1}{\sqrt{2}}.$ Since
$\psi(\pi)<0$, $I_{sc}:=[0,\frac{\pi}{2}].$ Now, observe that
$\max\limits_{I_{sc}}|\psi|=\frac{1}{48\sqrt{2}}\leq
M:=\frac{(\tau-\tau_{0})^{2}\kappa_{1}}{4\tau\tau_{0}}=\frac{1}{48\sqrt{2}}.$
Thus, $\psi$ satisfies all the requirements of Assumption 3.1 (ii). $\lozenge$

######  Remark 3.4.

Although the Assumption 3.1 is motivated from [8, Assumption H], there is a
difference to that. The difference is w.r.t. the bound $M$ of $|\psi|$ on
$I_{\text{sc}}.$ Note that $M=\frac{(\tau-\tau_{0})}{4\tau}M_{0},$ where
$M_{0}:=\frac{(\tau-\tau_{0})\kappa_{1}}{\tau_{0}}$ is the bound in [8].
$\lozenge$

######  Theorem 3.5.

Let Assumption 3.1 holds and $\mu_{n}$ be as in (2.6). Then there exists a
constant $C>0$ such that

$|\mu_{n}|\geq\frac{C}{\lambda_{n}^{2}}\quad\forall\,n\in{\mathbb{N}}.$

In particular, $\mu_{n}\not=0$ for all $n\in{\mathbb{N}}$.

###### Proof.

Case 1. Suppose the item (i) of Assumption 3.1 holds. Then either $\psi(s)>0$
or $\psi(s)<0$ for all $s\in[0,\tau].$ Since
$e^{\lambda_{n}(s-\tau)}(\tau-s)\geq 0$ for $s\in[0,\tau]$, we obtain

$|\mu_{n}|=\Big{|}\int_{0}^{\tau}e^{\lambda_{n}(s-\tau)}(\tau-s)\psi(s)\,ds\Big{|}=\int_{0}^{\tau}e^{\lambda_{n}(s-\tau)}(\tau-s)|\psi(s)|\,ds.$

Note that

$\displaystyle\lambda_{n}^{2}\int_{0}^{\tau}e^{\lambda_{n}(s-\tau)}(\tau-s)|\psi(s)|ds$
$\displaystyle\geq$
$\displaystyle\kappa_{1}\lambda_{n}^{2}\int_{\tau_{0}}^{\tau}e^{\lambda_{n}(s-\tau)}(\tau-s)\,ds$
$\displaystyle=$
$\displaystyle\kappa_{1}[1-(1+\lambda_{n}(\tau-\tau_{0}))e^{-\lambda_{n}(\tau-\tau_{0})}]$
$\displaystyle\geq$
$\displaystyle\kappa_{1}[1-(1+\lambda_{1}(\tau-\tau_{0}))e^{-\lambda_{1}(\tau-\tau_{0})}].$

The last inequality follows from the fact that the function
$t\mapsto(1+t)e^{-t}$ is strictly decreasing on $(0,\infty)$, thanks to Lemma
2.1(i). Since
$(1+\lambda_{1}(\tau-\tau_{0}))e^{-\lambda_{1}(\tau-\tau_{0})}<1$, we obtain
the required inequality with
$C=\kappa_{1}[1-(1+\lambda_{1}(\tau-\tau_{0}))e^{-\lambda_{1}(\tau-\tau_{0})}]$.

Case 2. Suppose that item (ii) of Assumption 3.1 holds, and let
$\tau_{1}=\max\\{t\in[0,\tau]:\psi(t)=0\\}.$ It is easy to observe that
$0<\tau_{1}<\tau_{0}.$ Let
$I_{\text{su}}:=\\{t\in[0,\tau]:\psi(t)\psi(\tau)\geq 0\\}.$ Then
$[\tau_{1},\tau]\subset I_{\text{su}}$ and $I_{\text{sc}}\subset[0,\tau_{1}].$
Let $\tilde{C}$ be such that
$\tilde{C}>\frac{|\psi(0)|+2\tau\kappa_{2}}{\tau|\psi(\tau)|}.$

Sub-case 2 (i). $\lambda_{n}\leq\tilde{C}.$

Recall that by Assumption 3.1 (ii) we have $|\psi|\leq
M:=\frac{(\tau-\tau_{0})^{2}\kappa_{1}}{4\tau\tau_{0}}$ on
$I_{\text{sc}}:=\\{t\in[0,\tau]:\psi(t)\psi(\tau)\leq 0\\}.$ Hence,

$\displaystyle\lambda_{n}^{2}|\mu_{n}|$ $\displaystyle\geq$
$\displaystyle\lambda_{n}^{2}\Big{[}\int_{I_{\text{su}}}e^{-\lambda_{n}(\tau-s)}(\tau-s)|\psi(s)|\,ds-\int_{I_{\text{sc}}}e^{-\lambda_{n}(\tau-s)}(\tau-s)|\psi(s)|\,ds\Big{]}$
$\displaystyle\geq$
$\displaystyle\lambda_{n}^{2}\Big{[}\int_{\tau_{1}}^{\tau}e^{-\lambda_{n}(\tau-s)}(\tau-s)|\psi(s)|\,ds-M\int_{0}^{\tau_{1}}e^{-\lambda_{n}(\tau-s)}(\tau-s)\,ds\Big{]}$
$\displaystyle=$
$\displaystyle\lambda_{n}^{2}\Big{[}\int_{\tau_{1}}^{\tau_{0}}e^{-\lambda_{n}(\tau-s)}(\tau-s)|\psi(s)|\,ds+\int_{\tau_{0}}^{\tau}e^{-\lambda_{n}(\tau-s)}(\tau-s)|\psi(s)|\,ds-M\int_{0}^{\tau_{1}}e^{-\lambda_{n}(\tau-s)}(\tau-s)\,ds\Big{]}$
$\displaystyle\geq$
$\displaystyle\lambda_{n}^{2}\Big{[}\int_{\tau_{1}}^{\tau_{0}}e^{-\lambda_{n}(\tau-s)}(\tau-s)|\psi(s)|\,ds+\kappa_{1}\int_{\tau_{0}}^{\tau}e^{-\lambda_{n}(\tau-s)}(\tau-s)\,ds-M\int_{0}^{\tau_{1}}e^{-\lambda_{n}(\tau-s)}(\tau-s)\,ds\Big{]}.$

We now obtain some bounds for the integrals

$I_{1}:=\int_{\tau_{1}}^{\tau_{0}}e^{-\lambda_{n}(\tau-s)}(\tau-s)|\psi(s)|\,ds,\quad
I_{2}:=\int_{\tau_{0}}^{\tau}e^{-\lambda_{n}(\tau-s)}(\tau-s)\,ds,\quad
I_{3}:=\int_{0}^{\tau_{1}}e^{-\lambda_{n}(\tau-s)}(\tau-s)\,ds.$

It is easy to see that

$I_{1}\geq(\tau-\tau_{0})e^{-\lambda_{n}(\tau-\tau_{1})}\int_{\tau_{1}}^{\tau_{0}}|\psi(s)|\,ds.$

Let $\varepsilon=\frac{\tau-\tau_{0}}{2}$. Then

$I_{2}\geq\int_{\tau_{0}}^{\tau-\varepsilon}e^{-\lambda_{n}(\tau-s)}{(\tau-s)}\,ds\geq\frac{\varepsilon{(\tau-\tau_{0}-\varepsilon)}}{e^{\lambda_{n}(\tau-\tau_{0})}}\geq\frac{\varepsilon{(\tau-\tau_{0}-\varepsilon)}}{e^{\lambda_{n}(\tau-\tau_{1})}}{=\frac{(\tau-\tau_{0})^{2}}{4e^{\lambda_{n}(\tau-\tau_{1})}}}.$

Similar to $I_{1}$ it is easy to check that
$I_{3}\leq\tau\tau_{0}e^{-\lambda_{n}(\tau-\tau_{1})}.$ Thus, using these
estimates we obtain

$\displaystyle\lambda_{n}^{2}|\mu_{n}|$ $\displaystyle\geq$
$\displaystyle\lambda_{n}^{2}[\,(\tau-\tau_{0})e^{-\lambda_{n}(\tau-\tau_{1})}\int_{\tau_{1}}^{\tau_{0}}|\psi(s)|\,ds+{\frac{(\tau-\tau_{0})^{2}\kappa_{1}}{4}}e^{-\lambda_{n}(\tau-\tau_{1})}-M\tau{\tau_{0}}e^{-\lambda_{n}(\tau-\tau_{1})}\,]$
$\displaystyle\geq$
$\displaystyle\lambda_{n}^{2}e^{-\tilde{C}(\tau-\tau_{1})}[\,(\tau-\tau_{0})\int_{\tau_{1}}^{\tau_{0}}|\psi(s)|\,ds+{\frac{(\tau-\tau_{0})^{2}\kappa_{1}}{4}}-M\tau{\tau_{0}}\,]$
$\displaystyle\geq$
$\displaystyle\lambda_{1}^{2}e^{-\tilde{C}(\tau-\tau_{1})}[\,(\tau-\tau_{0})\int_{\tau_{1}}^{\tau_{0}}|\psi(s)|\,ds+{\frac{(\tau-\tau_{0})^{2}\kappa_{1}}{4}}-M\tau{\tau_{0}}\,]$
$\displaystyle=$
$\displaystyle\lambda_{1}^{2}e^{-\tilde{C}(\tau-\tau_{1})}(\tau-\tau_{0})\int_{\tau_{1}}^{\tau_{0}}|\psi(s)|\,ds.$

Thus, the required inequality is satisfied with
$C=\lambda_{1}^{2}e^{-\tilde{C}(\tau-\tau_{1})}(\tau-\tau_{0})\int_{\tau_{1}}^{\tau_{0}}|\psi(s)|\,ds.$

Sub-case 2 (ii). $\lambda_{n}\geq\tilde{C}.$

Recall that by Assumption 3.1 (ii), $\psi$ is differentiable in $[0,\tau]$ and
there exists $\kappa_{2}>0$ such that $|\psi^{\prime}(s)|\leq\kappa_{2}$ on
$[0,\tau].$ Therefore, we have

$\displaystyle\lambda_{n}^{2}|\mu_{n}|$ $\displaystyle=$
$\displaystyle\lambda_{n}^{2}\Big{|}\big{[}\frac{\psi(\tau)}{\lambda_{n}^{2}}-\psi(0)(\frac{\tau
e^{-\lambda_{n}\tau}}{\lambda_{n}}+\frac{e^{-\lambda_{n}\tau}}{\lambda_{n}^{2}})\big{]}-\int_{0}^{\tau}\psi_{s}(s)\big{[}\frac{(\tau-s)e^{-\lambda_{n}(\tau-s)}}{\lambda_{n}}+\frac{e^{-\lambda_{n}(\tau-s)}}{\lambda_{n}^{2}}\big{]}\,ds\Big{|}$
$\displaystyle\geq$
$\displaystyle|\psi(\tau)|-\lambda_{n}^{2}|\psi(0)|\big{[}\frac{\tau
e^{-\lambda_{n}\tau}}{\lambda_{n}}+\frac{e^{-\lambda_{n}\tau}}{\lambda_{n}^{2}}\big{]}-\lambda_{n}^{2}\int_{0}^{\tau}|\psi_{s}(s)|\big{[}\frac{(\tau-s)e^{-\lambda_{n}(\tau-s)}}{\lambda_{n}}+\frac{e^{-\lambda_{n}(\tau-s)}}{\lambda_{n}^{2}}\big{]}\,ds$
$\displaystyle\geq$
$\displaystyle|\psi(\tau)|-\lambda_{n}^{2}|\psi(0)|\big{[}\frac{\tau
e^{-\lambda_{n}\tau}}{\lambda_{n}}+\frac{e^{-\lambda_{n}\tau}}{\lambda_{n}^{2}}\big{]}-\kappa_{2}\lambda_{n}^{2}\int_{0}^{\tau}\big{[}\frac{(\tau-s)e^{-\lambda_{n}(\tau-s)}}{\lambda_{n}}+\frac{e^{-\lambda_{n}(\tau-s)}}{\lambda_{n}^{2}}\big{]}\,ds$
$\displaystyle=$
$\displaystyle|\psi(\tau)|-\lambda_{n}^{2}|\psi(0)|\big{[}\frac{\lambda_{n}\tau
e^{-\lambda_{n}\tau}+e^{-\lambda_{n}\tau}}{\lambda_{n}^{2}}\big{]}-\lambda_{n}\kappa_{2}\big{[}\frac{1}{\lambda_{n}^{2}}-\frac{\lambda_{n}\tau
e^{-\lambda_{n}\tau}+e^{-\lambda_{n}\tau}}{\lambda_{n}^{2}}\big{]}-\kappa_{2}\big{[}\frac{1}{\lambda_{n}}-\frac{e^{-\lambda_{n}\tau}}{\lambda_{n}}\big{]}$
$\displaystyle\geq$
$\displaystyle|\psi(\tau)|-\lambda_{n}^{2}|\psi(0)|\big{[}\frac{\lambda_{n}\tau
e^{-\lambda_{n}\tau}+e^{-\lambda_{n}\tau}}{\lambda_{n}^{2}}\big{]}-\frac{2\kappa_{2}}{\lambda_{n}}$
$\displaystyle\geq$
$\displaystyle|\psi(\tau)|-\frac{|\psi(0)|}{\tau\lambda_{n}}-\frac{2\kappa_{2}}{\lambda_{n}}$
$\displaystyle\geq$
$\displaystyle|\psi(\tau)|-\frac{|\psi(0)|}{\tau\tilde{C}}-\frac{2\kappa_{2}}{\tilde{C}}$
$\displaystyle=$
$\displaystyle|\psi(\tau)|-\frac{1}{\tilde{C}}\big{[}\frac{|\psi(0)|+2\tau{\kappa_{2}}}{\tau}\big{]}.$

From this we obtain the required inequality with
$C=|\psi(\tau)|-\frac{1}{\tilde{C}}\big{[}\frac{|\psi(0)|+2\tau{\kappa_{2}}}{\tau}\big{]}.$
∎

Let $\varrho>0$ be fixed and $p>0$. We now obtain a conditional stability
estimate for the source set

(3.1) $S_{\varrho,\,p}=\\{g\in
L^{2}(\Omega):\sum_{n=1}^{\infty}\lambda_{n}^{2p}|\langle
g,\varphi_{n}\rangle|^{2}\leq\varrho^{2}\\}=\\{g\in\mathbb{H}_{p}:\|g\|_{\mathbb{H}_{p}}\leq\varrho\\}.$

######  Theorem 3.6.

(Conditional stability estimate) Let the exact source $f\in S_{\varrho,\,p}$,
$h\in L^{2}(\Omega)$ and $C$ be the constant as in Theorem 3.5. Then, we have

$\|f\|_{L^{2}(\Omega)}\leq
C^{-\frac{p}{p+2}}\varrho^{\frac{2}{p+2}}\|h\|_{L^{2}(\Omega)}^{\frac{p}{p+2}}.$

###### Proof.

From the Fourier expansion of $f$, we have

$\displaystyle\|f\|^{2}_{L^{2}(\Omega)}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}|\langle
f,\varphi_{n}\rangle|^{2}=\sum_{n=1}^{\infty}\lambda_{n}^{\frac{4p}{p+2}}|\langle
f,\varphi_{n}\rangle|^{\frac{4}{p+2}}\lambda_{n}^{-\frac{4p}{p+2}}|\langle
f,\varphi_{n}\rangle|^{\frac{2p}{p+2}}$ $\displaystyle\leq$
$\displaystyle\left(\sum_{n=1}^{\infty}\lambda_{n}^{2p}|\langle
f,\varphi_{n}\rangle|^{2}\right)^{\frac{2}{p+2}}\left(\sum_{n=1}^{\infty}\lambda_{n}^{-4}|\langle
f,\varphi_{n}\rangle|^{2}\right)^{\frac{p}{p+2}}$ $\displaystyle\leq$
$\displaystyle\varrho^{\frac{4}{p+2}}\left(\sum_{n=1}^{\infty}\lambda_{n}^{-4}|\langle
f,\varphi_{n}\rangle|^{2}\right)^{\frac{p}{p+2}}.$

Hence, using (2.5) and Theorem 3.5, we have

$\|f\|^{2}_{L^{2}(\Omega)}\leq\varrho^{\frac{4}{p+2}}\left(\sum_{n=1}^{\infty}\lambda_{n}^{-4}\Big{|}\frac{\langle
h,\varphi_{n}\rangle}{\mu_{n}}\Big{|}^{2}\right)^{\frac{p}{p+2}}\\\
\leq\varrho^{\frac{4}{p+2}}C^{-\frac{2p}{p+2}}\|h\|_{L^{2}(\Omega)}^{\frac{2p}{p+2}}.$

Hence, $\|f\|_{L^{2}(\Omega)}\leq
C^{-\frac{p}{p+2}}\varrho^{\frac{2}{p+2}}\|h\|_{L^{2}(\Omega)}^{\frac{p}{p+2}}.$
∎

##  4\. the quasi-reversibility method and error estimates

We have already observed that the inverse source problem under consideration
is ill-posed (also see, [30, sec 3.1]). We also illustrate this fact using an
example.

######  Example 4.1.

(Ill-posed) Let $\tilde{h}=0$ and
$\tilde{h}_{k}=\frac{1}{\lambda_{k}}\varphi_{k}$ for all $k\in{\mathbb{N}}.$
Let $\tilde{f}$ and $\tilde{f}_{k}$ denote the source functions corresponding
to the final values $\tilde{h}$ and $\tilde{h}_{k},$ respectively. Then by
(2.10), we have $\tilde{f}=0$ and
$\tilde{f}_{k}=\frac{1}{\lambda_{k}\mu_{k}}\,\varphi_{k}.$ Thus, we have
$\|\tilde{h}-\tilde{h}_{k}\|_{L^{2}(\Omega)}\to 0$ as $k\to\infty.$ But, from
(2.9), we have
$\|\tilde{f}_{k}-\tilde{f}\|_{L^{2}(\Omega)}\geq\frac{1}{\|\psi\|_{\infty}}\,\lambda_{k}$
and hence $\|\tilde{f}_{k}-\tilde{f}\|_{L^{2}(\Omega)}\to\infty$ as
$k\to\infty.$ $\lozenge$

Thus, some some regularization method has to be employed for obtaining stable
approximate solutions. As mentioned earlier, we will be using the quasi-
reversibility method (QRM) as follows.

Let $b\geq 2$ and $\alpha>0.$ We consider the following system

(4.1) $\begin{cases}v_{tt}-2\Delta
v_{t}+\Delta^{2}v=(I+\alpha(-\Delta)^{b})\psi(t)\tilde{f}(x),&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v=0=\Delta v,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(\cdot,0)=0=v_{t}(\cdot,0),&\quad\text{in}\quad\Omega,\\\
v(\cdot,\tau)=\tilde{h},&\quad\text{in}\quad\Omega.\end{cases}$

Suppose $\psi\in L^{2}(0,\tau),\,\,\tilde{h}\in L^{2}(\Omega)$ and
$\tilde{f}\in\mathbb{H}_{b}$, the domain of $(-\Delta)^{b}$. Suppose $v$ is a
solution of (4.1). Then following [30, 35] we observe the following:

Taking the $L^{2}$ inner product on both sides of the governing equation of
(4.1) with $\varphi_{n}$, for all $n\in{\mathbb{N}},$ we obtain

$\psi(t)\langle(I+\alpha(-\Delta)^{b})\tilde{f},\varphi_{n}\rangle=\frac{d^{2}}{dt^{2}}\langle
v(t),\varphi_{n}\rangle-2\frac{d}{dt}\langle\Delta
v(t),\varphi_{n}\rangle+\langle\Delta^{2}v(t),\varphi_{n}\rangle.$

Now for $\tilde{f}\in\mathbb{H}_{b},$ we have
$\displaystyle(I+(-\Delta)^{b})\tilde{f}=\sum_{n=1}^{\infty}(1+\alpha\lambda_{n}^{b})\langle\tilde{f},\varphi_{n}\rangle\varphi_{n}.$
Therefore,

$\psi(t)\langle\tilde{f},\varphi_{n}\rangle=\frac{1}{1+\alpha\lambda_{n}^{b}}\left(\frac{d^{2}}{dt^{2}}\langle
v(t),\varphi_{n}\rangle-2\frac{d}{dt}\langle\Delta
v(t),\varphi_{n}\rangle+\langle\Delta^{2}v(t),\varphi_{n}\rangle\right)$

Now, doing repeated integration by parts and making use of the boundary
condition in (4.1), we obtain

$\langle\Delta v(t),\varphi_{n}\rangle=\langle
v(t),\Delta\varphi_{n}\rangle=-\lambda_{n}\langle v(t),\varphi_{n}\rangle$

and

$\langle\Delta^{2}v(t),\varphi_{n}\rangle=\langle
v(t),\Delta^{2}\varphi_{n}\rangle=\lambda_{n}^{2}\langle
v(t),\varphi_{n}\rangle.$

Therefore, we have

(4.2)
$\psi(t)\langle\tilde{f},\varphi_{n}\rangle=\frac{1}{1+\alpha\lambda_{n}^{b}}\left(\frac{d^{2}}{dt^{2}}\langle
v(t),\varphi_{n}\rangle+2\lambda_{n}\frac{d}{dt}\langle
v(t),\varphi_{n}\rangle+\lambda_{n}^{2}\langle
v(t),\varphi_{n}\rangle\right).$

Now it is easy to check that

$\langle
v(t),\varphi_{n}\rangle=(1+\alpha\lambda_{n}^{b})\mu_{n}(t)\langle\tilde{f},\varphi_{n}\rangle,$

where $\mu_{n}(t)$ is as in (2.4). Hence, the solution $v$ is given by

(4.3)
$v(t):=\sum_{n=1}^{\infty}(1+\alpha\lambda_{n}^{b})\mu_{n}(t)\langle\tilde{f},\varphi_{n}\rangle\varphi_{n},$

Now, analogous to the observation (2.9), we have

$|\mu_{n}(t)|=\Big{|}\int_{0}^{t}e^{-\lambda_{n}(t-s)}(t-s)\psi(s)\,ds\Big{|}\leq\frac{\|\psi\|_{\infty}}{\lambda_{n}^{2}}.$

Thus, in order to ensure $v(t)\in L^{2}(\Omega)$ we need
$\tilde{f}\in\mathbb{H}_{b-2}$ only, and not necessarily in $\mathbb{H}_{b},$
the domain of $(-\Delta)^{b}$.

In view of the above discussion, we introduce the following definition.

######  Definition 4.2.

For $\psi\in L^{2}(0,\tau)$ and $\tilde{f}\in\mathbb{H}_{b-2},$ a function
$v\in C([0,\tau];L^{2}(\Omega))$ with the representation given by (4.3) is
said to be a mild solution of (4.1). $\lozenge$

From the above discussion, we observe that for $\tilde{h}\in L^{2}(\Omega)$
and $\tilde{f}\in\mathbb{H}_{b-2},\,b\geq 2,$ (4.1) has a unique mild
solution, and we call it as $v.$ Now, taking into account the final value
condition $v(\cdot,\tau)=\tilde{h}$, we obtain

$\langle\tilde{f},\varphi_{n}\rangle=\frac{\langle\tilde{h},\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}.$

Thus, the problem of identification of $\tilde{f}$ from the final value
$\tilde{h}$ associated with (4.1) is same as the problem of solving the
operator equation

(4.4) $\mathbb{T}_{\alpha}\tilde{f}=\tilde{h},$

where $\mathbb{T}_{\alpha}:\text{\bf dom}(\mathbb{T}_{\alpha})\subset
L^{2}(\Omega)\to L^{2}(\Omega)$ is defined by

$\mathbb{T}_{\alpha}\phi:=\sum_{n=1}^{\infty}(1+\alpha\lambda_{n}^{b})\mu_{n}\langle\phi,\varphi_{n}\rangle\varphi_{n}.$

Since $|\mu_{n}|\sim\frac{1}{\lambda_{n}^{2}}$, it is easy to check that
$\text{\bf dom}(\mathbb{T}_{\alpha})=\mathbb{H}_{b-2}.$ Also, it is easy to
check that $\mathbb{T}_{\alpha}$ is a densely defined closed self-adjoint
operator. Now, for $b\geq
2,\,\,(1+\alpha\lambda_{n}^{b})^{b}\geq(1+\alpha\lambda_{n}^{b})^{2}\geq\alpha^{2}\lambda_{n}^{2b}$
and hence $(1+\alpha\lambda_{n}^{b})\geq\alpha^{\frac{2}{b}}\lambda_{n}^{2}$.
In view of this and from Theorem 3.5, we have

$\|\mathbb{T}_{\alpha}\phi\|_{L^{2}(\Omega)}\geq
C\alpha^{\frac{2}{b}}\|\phi\|_{L^{2}(\Omega)}\quad\text{for
all}\quad\phi\in\text{\bf dom}(\mathbb{T}_{\alpha}).$

Thus, $\mathbb{T}_{\alpha}^{-1}:\text{\bf ran}(\mathbb{T}_{\alpha})\to
L^{2}(\Omega)$ is a closed operator which is also bounded, and hence
$\text{\bf ran}(\mathbb{T}_{\alpha})$ is closed (cf. [25]). Now from the fact
$\text{\bf
ran}(\mathbb{T}_{\alpha})=N(\mathbb{T}_{\alpha})^{\perp}=\\{0\\}^{\perp}$, we
have $\text{\bf ran}(\mathbb{T}_{\alpha})=L^{2}(\Omega).$ Therefore, it
follows that $\mathbb{T}_{\alpha}$ has a bounded inverse with

$\|\mathbb{T}_{\alpha}^{-1}\phi\|_{L^{2}(\Omega)}\leq\frac{1}{C\alpha^{\frac{2}{b}}}\|\phi\|_{L^{2}(\Omega)}\quad\text{for
all}\quad\phi\in L^{2}(\Omega).$

Thus, from the above discussion, we observe that solving equation (4.4) is an
well-posed problem. Moreover, for $\alpha>0$, let
$f_{\alpha}:=\mathbb{T}_{\alpha}^{-1}h$ and
$f_{\alpha}^{\delta}:=\mathbb{T}_{\alpha}^{-1}h^{\delta}.$ Then, we have

$\|f_{\alpha}-f_{\alpha}^{\delta}\|_{L^{2}(\Omega)}\leq\frac{1}{C\alpha^{\frac{2}{b}}}\|h-h^{\delta}\|_{L^{2}(\Omega)}.$

Recall that,
$f_{\alpha}=\mathbb{T}_{\alpha}^{-1}h=\sum_{n=1}^{\infty}\frac{\langle
h,\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}\varphi_{n}$ and
$f_{\alpha}^{\delta}=\mathbb{T}_{\alpha}^{-1}h^{\delta}=\sum_{n=1}^{\infty}\frac{\langle
h^{\delta},\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}\varphi_{n}.$
Therefore, we have

(4.5) $\langle f_{\alpha},\varphi_{n}\rangle=\frac{\langle
h,\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}\quad\text{and}\quad\langle
f_{\alpha}^{\delta},\varphi_{n}\rangle=\frac{\langle
h^{\delta},\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}.$

Before proceeding further, let us mention that Young’s inequality will be used
in many places in the upcoming analysis without mentioning explicitly. The
Young’s inequality states that: for $a,b\geq 0$ and
$\theta,\theta^{\prime}\geq 1$ satisfying
$\frac{1}{\theta}+\frac{1}{\theta^{\prime}}=1$, we have
$ab\leq\frac{a^{\theta}}{\theta}+\frac{b^{\theta^{\prime}}}{\theta^{\prime}}.$

###  4.1. Apriori parameter choice

We are now in a position to state and prove one of the main results of this
paper, the order of the error estimate for an apriori parameter choice
strategy. First we derive some general error estimates in terms of
$(\alpha,\delta)$.

######  Theorem 4.3.

Let $\alpha,\delta>0,b\geq 2$ and $h^{\delta}\in L^{2}(\Omega)$ be as in
(1.2). Let $f\in S_{\varrho,\,p}$ for some $\varrho,p>0$ and $C$ be as in
Theorem 3.5. Then we have the following estimates:

  1. (i)

$\|f_{\alpha}-f_{\alpha}^{\delta}\|_{L^{2}(\Omega)}\leq\frac{1}{C}\frac{\delta}{\alpha^{\frac{2}{b}}}.$

  2. (ii)

$\|f-f_{\alpha}\|_{L^{2}(\Omega)}\leq\begin{cases}C_{\rm
apri1}\,\varrho\,\alpha^{\frac{p}{b}},&\quad\text{if}\quad 0<p<b,\\\ C_{\rm
apri2}\,\varrho\,\alpha,&\quad\text{if}\quad p\geq b,\end{cases}$
where $C_{\rm apri1}=\frac{p}{b}\,\big{(}\frac{b-p}{p}\big{)}^{\frac{b-p}{b}}$
and $C_{\rm apri2}=\lambda_{1}^{(b-p)}.$

###### Proof.

(i). First, we observe that for $b\geq 2,$

$(1+\alpha\lambda_{n}^{b})^{b}\geq(1+\alpha\lambda_{n}^{b})^{2}\geq\alpha^{2}\lambda_{n}^{2b}$

and hence $1+\alpha\lambda_{n}^{b}\geq\alpha^{\frac{2}{b}}\lambda_{n}^{2}.$
Therefore, from (4.5) it follows that

$\displaystyle\|f_{\alpha}^{\delta}-f_{\alpha}\|^{2}_{L^{2}(\Omega)}$
$\displaystyle=$ $\displaystyle\sum_{n=1}^{\infty}\Big{|}\frac{\langle
h^{\delta}-h,\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}\Big{|}^{2}\leq\frac{1}{\alpha^{\frac{4}{b}}}\sum_{n=1}^{\infty}\Big{|}\frac{\langle
h^{\delta}-h,\varphi_{n}\rangle}{\lambda_{n}^{2}\mu_{n}}\Big{|}^{2}$
$\displaystyle\underbrace{\leq}_{\rm Theorem\ref{non_zero_lower_bound}}$
$\displaystyle\frac{1}{\alpha^{\frac{4}{b}}C^{2}}\sum_{n=1}^{\infty}|\langle
h^{\delta}-h,\varphi_{n}\rangle|^{2}\leq\frac{1}{C^{2}}\frac{\delta^{2}}{\alpha^{\frac{4}{b}}}.$

Hence, we obtain

(4.6)
$\|f_{\alpha}^{\delta}-f_{\alpha}\|_{L^{2}(\Omega)}\leq\frac{1}{C}\frac{\delta}{\alpha^{\frac{2}{b}}}.$

(ii). From (4.5) we have

$\displaystyle\|f-f_{\alpha}\|^{2}_{L^{2}(\Omega)}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\Big{|}\frac{\langle
h,\varphi_{n}\rangle}{\mu_{n}}-\frac{\langle
h,\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}\Big{|}^{2}=\sum_{n=1}^{\infty}\Big{|}\frac{\alpha\lambda_{n}^{b}\langle
h,\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})\mu_{n}}\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{\alpha^{2}\lambda_{n}^{2b}}{(1+\alpha\lambda_{n}^{b})^{2}}|\langle
f,\varphi_{n}\rangle|^{2}=\sum_{n=1}^{\infty}\frac{\alpha^{2}\lambda_{n}^{2b-2p}}{(1+\alpha\lambda_{n}^{b})^{2}}\lambda_{n}^{2p}|\langle
f,\varphi_{n}\rangle|^{2}.$

Now, we consider two cases, namely, $0<p<b$ and $p\geq b$. Case 1. $0<p<b$ :
In this case, by Lemma 2.1(ii), we have

$\frac{\alpha^{2}\lambda_{n}^{2b-2p}}{(1+\alpha\lambda_{n}^{b})^{2}}\leq\left(\frac{p}{b}\right)^{2}\left(\frac{b-p}{p}\right)^{\frac{2(b-p)}{b}}\alpha^{\frac{2p}{b}}$

and hence

(4.7) $\|f-f_{\alpha}\|_{L^{2}(\Omega)}\leq C_{\rm
apri1}\,\varrho\,\alpha^{\frac{p}{b}},$

where $C_{\rm apri1}=\frac{p}{b}\,\left(\frac{b-p}{p}\right)^{\frac{b-p}{b}}.$

Case 2. $p\geq b$ : In this case,

$\frac{\alpha^{2}\lambda_{n}^{2b-2p}}{(1+\alpha\lambda_{n}^{b})^{2}}\leq\alpha^{2}\lambda_{1}^{2b-2p},$

and hence

(4.8) $\|f-f_{\alpha}\|_{L^{2}(\Omega)}\leq C_{\rm apri2}\,\varrho\,\alpha,$

where $C_{\rm apri2}=\lambda_{1}^{(b-p)}.$ ∎

Now we are in a position to state the main result of this subsection.

######  Theorem 4.4.

Let $\alpha,\delta>0,b\geq 2$ and $h^{\delta}\in L^{2}(\Omega)$ be as in
(1.2). Let $f\in S_{\varrho,\,p}$ for some $\varrho,p>0.$ Then the following
holds.

  1. (i)

If $0<p<b$ then for
$\alpha\sim\left(\frac{\delta}{\varrho}\right)^{\frac{b}{p+2}}$ there exists a
constant $C_{1}:=C_{1}(C,p,b)>0$ such that

$\|f-f_{\alpha}^{\delta}\|_{L^{2}(\Omega)}\leq
C_{1}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}.$

  2. (ii)

If $p\geq b$ then for
$\alpha\sim\left(\frac{\delta}{\varrho}\right)^{\frac{b}{b+2}}$ there exists a
constant $C_{2}:=C_{2}(C,\lambda_{1},p,b)>0$ such that

$\|f-f_{\alpha}^{\delta}\|_{L^{2}(\Omega)}\leq
C_{2}\,\varrho^{\frac{2}{b+2}}\,\delta^{\frac{b}{b+2}}.$

###### Proof.

By Theorem 4.3, we have

$\|f-f_{\alpha}^{\delta}\|_{L^{2}(\Omega)}\leq\begin{cases}C_{\rm
apri1}\,\varrho\,\alpha^{\frac{p}{b}}+\frac{1}{C}\,\frac{\delta}{\alpha^{\frac{2}{b}}},&\quad\text{if}\quad
0<p<b,\\\ C_{\rm
apri2}\,\varrho\,\alpha+\frac{1}{C}\frac{\delta}{\alpha^{\frac{2}{b}}},&\quad\text{if}\quad
p\geq b.\end{cases}$

Hence the proof follows by appropriately choosing the constants $C_{1}$ and
$C_{2}.$ ∎

######  Remark 4.5.

If $b=2,$ then by Theorem 4.4, the best possible order of the error estimate
in the apriori case is $\mathcal{O}(\delta^{\frac{1}{2}})$ which is better
than the order $\mathcal{O}(\delta^{\frac{1}{3}})$ obtained in [30, Theorem
4.1] using Tikhonov regularization method and the order
$\mathcal{O}(\delta^{\frac{1}{4}})$ obtained in [29, Theorem 3.2] using
modified quasi-boundary value method for the apriori case.

Also, it is easy to observe that if $b\geq 4$ then
$\frac{b}{b+2}\geq\frac{2}{3}$ and hence by Theorem 4.4 it follows that the
best possible order can exceed the benchmark order of
$\mathcal{O}(\delta^{\frac{2}{3}})$ for Tikhonov regularization. $\lozenge$

###  4.2. Aposteriori parameter choice

We now proceed on to aposteriori case, where we will do the analysis using
Morozov-type discrepancy principle, see for e.g., [9, 25]. For $\alpha>0,$ let
$\mathbb{B}_{\alpha}:L^{2}(\Omega)\to L^{2}(\Omega)$ be defined by

$\mathbb{B}_{\alpha}\tilde{h}:=\sum_{n=1}^{\infty}\frac{\langle\tilde{h},\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})}\varphi_{n}.$

Then we have the following result.

######  LEMMA 4.6.

(cf. [14]) For $\alpha>0,$ let
$\zeta(\alpha):=\|\mathbb{B}_{\alpha}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}.$
Then

  1. (i)

$\zeta$ is a continuous function.

  2. (ii)

$\lim\limits_{\alpha\to 0^{+}}\zeta(\alpha)=0$ and
$\lim\limits_{\alpha\to+\infty}\zeta(\alpha)=h^{\delta}.$

  3. (iii)

$\zeta$ is strictly increasing.

We are now in a position to state another main result of this paper, namely
the error estimates for an aposteriori parameter choice strategies.

######  Theorem 4.7.

Let the Assumption 3.1 holds, $f\in S_{\varrho,p}$, $\delta>0$ and
$h^{\delta}$ be as in (1.2). Let $\sigma\in(0,1)$ and $b\geq 2.$ Suppose that
$\delta^{\sigma}<\|h^{\delta}\|_{L^{2}(\Omega)}.$ Choose $\xi>1$ such that
$0<\xi\delta^{\sigma}\leq\|h^{\delta}\|_{L^{2}(\Omega)}.$ Then we have the
following estimates.

  1. (i)

If $b\neq 2$ then for each $\delta>0$ there exists a unique
$\alpha_{\delta}>0$ such that
$\|\mathbb{B}_{\alpha_{\delta}}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}=\xi\delta.$
Further, there exists constants
$C_{4}:=C_{4}(C,\|\psi\|_{\infty},\xi,\lambda_{1},p,b),C_{6}:=C_{6}(C,\|\psi\|_{\infty},\xi,p),C_{\rm
apost}:=C_{\rm apost}(C,\xi,p)>0$ such that

$\|f-f_{\alpha_{\delta}}^{\delta}\|_{L^{2}(\Omega)}\leq\begin{cases}C_{6}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}},&\quad\text{if}\quad
0<p<b-2\\\ C_{\rm
apost}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}+C_{4}\,\varrho^{\frac{2}{b}}\,\delta^{\frac{b-2}{b}},&\quad\text{if}\quad
p\geq b-2.\end{cases}$

  2. (ii)

If $b=2$ then for each $\delta>0$ there exists a unique $\alpha_{\delta}>0$
such that
$\|\mathbb{B}_{\alpha_{\delta}}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}=\xi\delta^{\sigma}.$
Further, there exists constant
$C_{5}:=C_{5}(C,\|\psi\|_{\infty},\xi,\lambda_{1},p)>0$ (and $C_{\rm apost}$
as in (i)) such that

$\|f-f_{\alpha_{\delta}}^{\delta}\|_{L^{2}(\Omega)}\leq C_{\rm
apost}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p\sigma}{p+2}}+C_{5}\,\varrho\,\delta^{1-\sigma}.$

###### Proof.

Let $w$ be the mild solution (see Definition (4.2)) of

$\begin{cases}v_{tt}-2\Delta
v_{t}+\Delta^{2}v=(I+\alpha(-\Delta)^{b})\psi(t)f_{\alpha}(x),&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v=0=\Delta v,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(\cdot,0)=0=v_{t}(\cdot,0),&\quad\text{in}\quad\Omega,\\\
v(\cdot,\tau)=h,&\quad\text{in}\quad\Omega.\end{cases}$

Then it is easy to check that $z:=\mathbb{B}_{\alpha}w$ is a solution of

(4.9) $\begin{cases}v_{tt}-2\Delta
v_{t}+\Delta^{2}v=\psi(t)f_{\alpha}(x),&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v=0=\Delta v,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(\cdot,0)=0=v_{t}(\cdot,0),&\quad\text{in}\quad\Omega,\\\
v(\cdot,\tau)=\mathbb{B}_{\alpha}h,&\quad\text{in}\quad\Omega.\end{cases}$

Now,

$\|f-f_{\alpha}\|^{2}_{\mathbb{H}_{p}}=\sum_{n=1}^{\infty}\lambda_{n}^{2p}\big{[}\langle
f,\varphi_{n}\rangle-\frac{\langle
h,\varphi_{n}\rangle}{(1+\alpha_{\delta}\lambda_{n}^{b})\mu_{n}}\big{]}^{2}=\sum_{n=1}^{\infty}\lambda_{n}^{2p}\big{[}\frac{\alpha_{\delta}\lambda_{n}^{b}\langle
f,\varphi_{n}\rangle}{1+\alpha_{\delta}\lambda_{n}^{b}}\big{]}^{2}\leq\|f\|^{2}_{\mathbb{H}_{p}}\leq\varrho^{2}$

and hence

(4.10) $\|f-f_{\alpha}\|_{\mathbb{H}_{p}}\leq\varrho.$

Proof of (i). By Lemma 4.6 it follows that there exists a unique
$\alpha_{\delta}>0$ such that
$\|\mathbb{B}_{\alpha_{\delta}}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}=\xi\delta.$
Also, we have

$\displaystyle\|u(\tau)-z(\tau)\|_{L^{2}(\Omega)}$ $\displaystyle=$
$\displaystyle\|h-\mathbb{B}_{\alpha_{\delta}}h\|_{L^{2}(\Omega)}\leq\|\mathbb{B}_{\alpha_{\delta}}h-\mathbb{B}_{\alpha_{\delta}}h^{\delta}\|_{L^{2}(\Omega)}+\|\mathbb{B}_{\alpha_{\delta}}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}+\|h^{\delta}-h\|_{L^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle\|\sum_{n=1}^{\infty}\frac{\langle
h-h^{\delta},\varphi_{n}\rangle}{(1+\alpha\lambda_{n}^{b})}\varphi_{n}\|_{L^{2}(\Omega)}+\xi\delta+\delta\leq(\xi+2)\delta.$

Note that $u-z$ is a solution of

$\begin{cases}v_{tt}-2\Delta
v_{t}+\Delta^{2}v=\psi(t)\,(f-f_{\alpha_{\delta}})(x),&\quad\text{in}\quad\Omega\times(0,\tau),\\\
v=0=\Delta v,&\quad\text{on}\quad\partial\Omega\times(0,\tau),\\\
v(\cdot,0)=0=v_{t}(\cdot,0),&\quad\text{in}\quad\Omega,\\\
v(\cdot,\tau)=h-\mathbb{B}_{\alpha_{\delta}}h,&\quad\text{in}\quad\Omega.\end{cases}$

and (4.10) shows that $f-f_{\alpha_{\delta}}\in S_{\varrho,p}.$ Therefore,
following the proof of Theorem 3.6, it is easy to check that
$\|f-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq
C^{-\frac{p}{p+2}}\varrho^{\frac{2}{p+2}}[(\xi+2)\delta]^{\frac{p}{p+2}}$ and
hence

(4.11) $\|f-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq C_{\rm
apost}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}},$

where $C_{\rm apost}=\left(\frac{\xi+2}{C}\right)^{\frac{p}{p+2}}.$ We now try
to obtain a bound for $\alpha_{\delta}$ in terms of $\delta$ using the
discrepancy principle. Now,

$\displaystyle\xi\delta$ $\displaystyle=$
$\displaystyle\|\mathbb{B}_{\alpha_{\delta}}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}=\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{b}}{1+\alpha_{\delta}\lambda_{n}^{b}}\langle
h^{\delta},\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{b}}{1+\alpha_{\delta}\lambda_{n}^{b}}\langle
h,\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}+\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{b}}{1+\alpha_{\delta}\lambda_{n}^{b}}\langle
h^{\delta}-h,\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{b}}{1+\alpha_{\delta}\lambda_{n}^{b}}\mu_{n}\,\langle
f,\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}+\delta.$

Therefore,

$(\xi-1)\delta\leq\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{b}}{1+\alpha_{\delta}\lambda_{n}^{b}}\mu_{n}\,\langle
f,\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}.$

Thus,

(4.12)
$(\xi-1)^{2}\delta^{2}\leq\sum_{n=1}^{\infty}\left(\frac{\alpha_{\delta}\lambda_{n}^{b-2}}{1+\alpha_{\delta}\lambda_{n}^{b}}\right)^{2}\big{(}\langle
f,\varphi_{n}\rangle\|\psi\|_{\infty}\big{)}^{2}$

Case 1. $b>2$ and $0<p<b-2.$

In this case,

$\displaystyle\alpha_{\delta}\lambda_{n}^{b}+1$ $\displaystyle\geq$
$\displaystyle\frac{b-2-p}{b}\alpha_{\delta}\lambda_{n}^{b}+\frac{p+2}{b}$
$\displaystyle=$
$\displaystyle\frac{b-2-p}{b}\big{[}(\alpha_{\delta}\lambda_{n}^{b})^{\frac{b-2-p}{b}}\big{]}^{\frac{b}{b-2-p}}+\frac{p+2}{b}[1^{\frac{p+2}{b}}]^{\frac{b}{p+2}}$
$\displaystyle\geq$
$\displaystyle\left(\alpha_{\delta}\lambda_{n}^{b}\right)^{\frac{b-2-p}{b}},$

and hence
$\frac{\alpha_{\delta}\lambda_{n}^{b-2}}{1+\alpha_{\delta}\lambda_{n}^{b}}\leq\alpha_{\delta}^{\frac{p+2}{b}}\lambda_{n}^{p}.$
Therefore,

$\displaystyle[(\xi-1)\delta]^{2}\leq\|\psi\|^{2}_{\infty}\,\alpha_{\delta}^{\frac{2(p+2)}{b}}\sum_{n=1}^{\infty}\lambda_{n}^{2p}|\langle
f,\varphi_{n}\rangle|^{2}\leq\|\psi\|^{2}_{\infty}\,\alpha_{\delta}^{\frac{2(p+2)}{b}}\,\varrho^{2}$
$\displaystyle\implies$
$\displaystyle\frac{1}{\alpha_{\delta}}\leq\left(\frac{\|\psi\|_{\infty}}{\xi-1}\right)^{\frac{b}{p+2}}\left(\frac{\varrho}{\delta}\right)^{\frac{b}{p+2}}.$

Thus, from (4.6), we have

$\|f_{\alpha_{\delta}}^{\delta}-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq\frac{1}{C}\frac{\delta}{{\alpha_{\delta}}^{\frac{2}{b}}}\leq\frac{1}{C}\,\delta\,\left(\frac{\|\psi\|_{\infty}}{\xi-1}\right)^{\frac{2}{p+2}}\left(\frac{\varrho}{\delta}\right)^{\frac{2}{p+2}}=\frac{\|\psi\|^{\frac{2}{p+2}}_{\infty}}{C(\xi-1)^{\frac{2}{p+2}}}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}},$

that is,

(4.13)
$\|f_{\alpha_{\delta}}^{\delta}-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq
C_{3}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}},$

where
$C_{3}=\frac{\|\psi\|^{\frac{2}{p+2}}_{\infty}}{C(\xi-1)^{\frac{2}{p+2}}}.$

Case 2. $b>2$ and $p\geq b-2.$

In this case, from (4.12) we have,

$\displaystyle[(\xi-1)\delta]^{2}\leq\|\psi\|^{2}_{\infty}\sum_{n=1}^{\infty}\Big{[}\frac{\alpha_{\delta}\lambda_{n}^{b-2-p}}{1+\alpha_{\delta}\lambda_{n}^{b}}\lambda_{n}^{p}\langle
f,\varphi_{n}\rangle\Big{]}^{2}\leq\frac{\|\psi\|^{2}_{\infty}}{\lambda_{1}^{2(p-b+2)}}\,\alpha_{\delta}^{2}\,\varrho^{2},$
$\displaystyle\implies$
$\displaystyle\frac{1}{\alpha_{\delta}}\leq\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p-b+2}}\,\frac{\varrho}{\delta}.$

Therefore, from (4.6), we have

$\|f_{\alpha_{\delta}}^{\delta}-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq\frac{1}{C}\frac{\delta}{{\alpha_{\delta}}^{\frac{2}{b}}}\leq\frac{1}{C}\,\delta\,\Big{[}\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p-b+2}}\Big{]}^{\frac{2}{b}}\,\left(\frac{\varrho}{\delta}\right)^{\frac{2}{b}}=\frac{1}{C}\Big{[}\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p-b+2}}\Big{]}^{\frac{2}{b}}\,\varrho^{\frac{2}{b}}\,\delta^{\frac{b-2}{b}}$

that is,

(4.14)
$\|f_{\alpha_{\delta}}^{\delta}-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq
C_{4}\,\varrho^{\frac{2}{b}}\,\delta^{\frac{b-2}{b}},$

where
$C_{4}=\frac{1}{C}\Big{[}\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p-b+2}}\Big{]}^{\frac{2}{b}}.$

Thus, from (4.11), (4.13) and (4.14) we have

$\|f-f_{\alpha_{\delta}}^{\delta}\|_{L^{2}(\Omega)}\leq\begin{cases}C_{\rm
apost}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}+C_{3}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}},&\quad\text{if}\quad
0<p<b-2\\\ C_{\rm
apost}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}+C_{4}\,\varrho^{\frac{2}{b}}\,\delta^{\frac{b-2}{b}},&\quad\text{if}\quad
p\geq b-2.\end{cases}$

Now the proof follows by choosing the constant $C_{6}$ appropriately.

Proof of (ii). Again by Lemma 4.6, it follows that there exists a unique
$\alpha_{\delta}>0$ such that
$\|\mathbb{B}_{\alpha_{\delta}}(h^{\delta})-h^{\delta}\|_{L^{2}(\Omega)}=\xi\delta^{\sigma}$
for $\sigma\in(0,1).$ Now, we have

$\displaystyle\|u(\tau)-z(\tau)\|_{L^{2}(\Omega)}$ $\displaystyle=$
$\displaystyle\|h-\mathbb{B}_{\alpha_{\delta}}h\|_{L^{2}(\Omega)}\leq\|\mathbb{B}_{\alpha_{\delta}}h-\mathbb{B}_{\alpha_{\delta}}h^{\delta}\|_{L^{2}(\Omega)}+\|\mathbb{B}_{\alpha_{\delta}}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}+\|h^{\delta}-h\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\delta+\xi\delta^{\sigma}+\delta\leq(\xi+2)\delta^{\sigma}.$

Therefore from (4.10) and Theorem 3.6, we have
$\|f-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq
C^{-\frac{p}{p+2}}\varrho^{\frac{2}{p+2}}[(\xi+2)\delta^{\sigma}]^{\frac{p}{p+2}}$
and hence

(4.15) $\|f-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq C_{\rm
apost}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p\sigma}{p+2}},$

where $C_{\rm apost}=\Big{(}\frac{\xi+2}{C}\Big{)}^{\frac{p}{p+2}}.$

Now, similar to Case 1, we obtain a bound for $\alpha_{\delta}$ in terms of
$\delta$ using the discrepancy principle. Now,

$\displaystyle\xi\delta^{\sigma}$ $\displaystyle=$
$\displaystyle\|\mathbb{B}_{\alpha_{\delta}}h^{\delta}-h^{\delta}\|_{L^{2}(\Omega)}=\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{2}}{1+\alpha_{\delta}\lambda_{n}^{2}}\langle
h^{\delta},\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{2}}{1+\alpha_{\delta}\lambda_{n}^{2}}\mu_{n}\,\langle
f,\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}+\delta$

so that

$(\xi-1)\delta^{\sigma}\leq\|\sum_{n=1}^{\infty}\frac{\alpha_{\delta}\lambda_{n}^{2}}{1+\alpha_{\delta}\lambda_{n}^{2}}\mu_{n}\,\langle
f,\varphi_{n}\rangle\varphi_{n}\|_{L^{2}(\Omega)}.$

Thus, by (2.9), we have

$\displaystyle(\xi-1)^{2}\delta^{2\sigma}\leq\sum_{n=1}^{\infty}\Big{[}\frac{\alpha_{\delta}}{1+\alpha_{\delta}\lambda_{n}^{2}}\langle
f,\varphi_{n}\rangle\|\psi\|_{\infty}\Big{]}^{2}\leq\frac{\|\psi\|^{2}_{\infty}}{\lambda_{1}^{2p}}\,\varrho^{2}\,\alpha_{\delta}^{2}$
$\displaystyle\implies$
$\displaystyle\frac{1}{\alpha_{\delta}}\leq\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p}}\,\frac{\varrho}{\delta^{\sigma}}.$

Hence, from (4.6), we have

$\|f_{\alpha_{\delta}}^{\delta}-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq\frac{1}{C}\frac{\delta}{\alpha_{\delta}}\leq\frac{1}{C}\,\delta\,\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p}}\,\frac{\varrho}{\delta^{\sigma}}=\frac{1}{C}\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p}}\,\varrho\,\delta^{1-\sigma},$

that is,

(4.16)
$\|f_{\alpha_{\delta}}^{\delta}-f_{\alpha_{\delta}}\|_{L^{2}(\Omega)}\leq
C_{5}\,\varrho\,\delta^{1-\sigma},$

where $C_{5}=\frac{1}{C}\frac{\|\psi\|_{\infty}}{(\xi-1)\lambda_{1}^{p}}.$

Thus, from (4.15) and (4.16), we have

$\|f-f_{\alpha_{\delta}}^{\delta}\|_{L^{2}(\Omega)}\leq C_{\rm
apost}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p\sigma}{p+2}}+C_{5}\,\varrho\,\delta^{1-\sigma}.$

This completes the proof. ∎

######  Remark 4.8.

From Theorem 4.7 (i) it follows that if $2<p<b-2$ or if $p>b-2>2$, then the
order of the estimate is better than $\mathcal{O}(\delta^{\frac{1}{2}}),$ the
best possible rate obtained for the aposteriori case in [30, Theorem 4.2]
using the Tikhonov regularization method.

Also, if $4<p<b-2$ or $p>b-2>4,$ the by Theorem 4.7 (i) it follows that the
order of the estimate is better than $\mathcal{O}(\delta^{\frac{2}{3}}).$
$\lozenge$

##  5\. Optimality

In this section we analyse the order optimality of the error estimates
obtained in the preceding section for the source set $S_{\varrho,p}$. We have
already observed in Section 2 that the considered inverse source
identification problem is equivalent to solving a linear operator equation

${\mathbb{T}}f=h,$

where ${\mathbb{T}}:L^{2}(\Omega)\to L^{2}(\Omega)$ is a compact self-adjoint
bounded linear operator defined by (2.8), that is,

${\mathbb{T}}\varphi=\sum_{n=1}^{\infty}\mu_{n}\langle\varphi,\varphi_{n}\rangle\varphi_{n},\quad\varphi\in
L^{2}(\Omega)$

with $\mu_{n}$ as in (2.6).

We now formulate some useful representation of the source set $S_{\varrho,p}$
in terms of the operator $\mathbb{T}.$ For this, we observe from Theorem 3.5
and (2.9) that

(5.1)
$\frac{\lambda_{n}^{2}}{\|\psi\|_{\infty}}\leq\frac{1}{|\mu_{n}|}\leq\frac{\lambda_{n}^{2}}{C}.$

Thus, in particular we have

(5.2)
$\frac{\lambda_{n}^{2p}}{\|\psi\|^{p}_{\infty}}\leq\frac{1}{|\mu_{n}|^{p}}\leq\frac{\lambda_{n}^{2p}}{C^{p}}.$

Therefore, from (5.2) it follows that

(5.3)
$\sum_{n=1}^{\infty}\lambda_{n}^{2p}|\langle\tilde{f},\varphi_{n}\rangle|^{2}\leq\varrho^{2}\quad\text{if
and only
if}\quad\sum_{n=1}^{\infty}\frac{|\langle\tilde{f},\varphi_{n}\rangle|^{2}}{|\mu_{n}|^{p}}\leq\tilde{\varrho}^{2},$

where, for a given $\varrho>0$, $\tilde{\varrho}:=C^{-\frac{p}{2}}\varrho$,
and for a given $\tilde{\varrho}>0$,
$\varrho:=\|\psi\|_{\infty}^{\frac{p}{2}}\tilde{\varrho}$.

Next, recall that the operator ${\mathbb{T}}:L^{2}(\Omega)\to L^{2}(\Omega)$
defined above is a compact self-adjoint operator. Hence,
${\mathbb{T}}^{*}{\mathbb{T}}={\mathbb{T}}^{2}$ is a positive self-adjoint
operator with representation

${\mathbb{T}}^{*}{\mathbb{T}}\varphi=\sum_{n=1}^{\infty}\mu_{n}^{2}\langle\varphi,\varphi_{n}\rangle\,\varphi_{n},\quad\varphi\in
L^{2}(\Omega),$

and its square-root, conventionally denoted by $|{\mathbb{T}}|$, has the
representation

$|{\mathbb{T}}|\varphi=\sum_{n=1}^{\infty}|\mu_{n}|\langle{\varphi},\varphi_{n}\rangle\,\varphi_{n},\quad\varphi\in
L^{2}(\Omega).$

Consequently, for any $\nu>0$, we have

$|{\mathbb{T}}|^{\nu}{\varphi}=\sum_{n=1}^{\infty}|\mu_{n}|^{\nu}\langle{\varphi},\varphi_{n}\rangle\,\varphi_{n},\quad\varphi\in
L^{2}(\Omega).$

We may also observe that, for $\nu>0$ if

${\mathcal{D}}_{\nu}:=\\{\varphi\in
L^{2}(\Omega):\sum_{n=1}^{\infty}|\mu_{n}|^{-2\nu}|\langle{\varphi},\varphi_{n}\rangle|^{2}<\infty\\},$

then ${\mathcal{D}}_{\nu}$ is the range of $|{\mathbb{T}}|^{\nu}$ so that
${\mathcal{D}}_{\nu}$ is the domain of the unbounded operator
$|{\mathbb{T}}|^{-\nu}$, which has the representation

$|{\mathbb{T}}|^{-\nu}\varphi=\sum_{n=1}^{\infty}|\mu_{n}|^{-\nu}\langle{\varphi},\varphi_{n}\rangle\,\varphi_{n},\quad\varphi\in{\mathcal{D}}_{\nu}.$

Therefore, from (5.3), we have

$\sum_{n=1}^{\infty}\lambda_{n}^{2p}|\langle
f,\varphi_{n}\rangle|^{2}\leq\varrho^{2}\quad\text{if and only
if}\quad\||\mathbb{T}|^{-\frac{p}{2}}f\|^{2}_{L^{2}(\Omega)}\leq\tilde{\varrho}^{2}$

whenever $f\in{\mathcal{D}}_{p/2}$. Thus, for $r>0$, we have

(5.4) $M_{r,p}:=\\{g\in
L^{2}(\Omega):g=|\mathbb{T}|^{\frac{p}{2}}\tilde{g},\,\|\tilde{g}\|_{L^{2}(\Omega)}\leq
r\\}=\\{g\in{\mathcal{D}}_{p/2}:\||{\mathbb{T}}|^{-p/2}g\|_{L^{2}(\Omega)}\leq
r\\}.$

Now, it is easy to see that $S_{\varrho,p}\subseteq M_{r_{1},p}$ and
$M_{r_{2},p}\subseteq S_{\varrho,p}$, where $r_{1}=C^{-\frac{p}{2}}\varrho$
and $r_{2}=\|\psi\|_{\infty}^{-\frac{p}{2}}\,\varrho.$

We may recall that (cf. [9, 25]), corresponding to $\delta>0$, the worst case
error for a reconstruction method described by an operator
$\mathcal{R}:L^{2}(\Omega)\to L^{2}(\Omega)$ approximating the inverse of
$\mathbb{T}$ with respect to a bounded set $\mathcal{M}\subset L^{2}(\Omega)$
is defined to be the quantity

$\triangle(\mathcal{R},\delta,\mathcal{M}):=\sup\\{\|\mathcal{R}\tilde{h}-\tilde{f}\|:\tilde{f}\in\mathcal{M},\,\tilde{h}\in
L^{2}(\Omega),\,\|\mathbb{T}\tilde{f}-\tilde{h}\|\leq\delta\\}.$

It is known that (cf. [9, 25]) the worst case error is bounded below by the
corresponding modulus of continuity

$\omega(\delta,\mathcal{M}):=\sup\\{\|\tilde{f}_{1}-\tilde{f}_{2}\|_{L^{2}(\Omega)}:\tilde{f}_{1},\tilde{f}_{2}\in\mathcal{M}\,,\,\|\mathbb{T}\tilde{f}_{1}-\mathbb{T}\tilde{f}_{2}\|\leq\delta\\},$

up to a constant multiple, in the sense that

(5.5)
$\triangle(\mathcal{R},\delta,\mathcal{M})\geq\frac{1}{2}\omega(2\delta,\mathcal{M})\quad\text{for
all}\quad\delta>0.$

We also observe some easily verifiable facts:

(5.6)
$\omega(\delta_{1},\mathcal{M})\leq\omega(\delta_{2},\mathcal{M})\quad\text{for}\quad
0<\delta_{1}\leq\delta_{2},$

and

(5.7)
$\omega(\delta,\mathcal{M}_{1})\leq\omega(\delta,\mathcal{M}_{2})\quad\text{for}\quad\mathcal{M}_{1}\subseteq\mathcal{M}_{2},\quad\delta>0.$

For more details on results related to modulus of continuity for linear
operators, we refer to [16, 25].

We will denote the spectrum of a bounded linear operator $\mathbb{K}$ by
${\rm\bf spec}(\mathbb{K}).$ The following result is motivated from Trong and
Hai [34]. In its proof, we shall make use of the function $\rho(\cdot)$
defined by

$\rho(t):=t^{\frac{p+2}{p}},\quad t>0,$

and the relation $\rho(t)=t\varphi^{-1}(t),\,t>0,$ where
$\varphi(t):=t^{\frac{p}{2}}$ for $t>0$. Clearly, $\varphi$ and $\rho$
satisfies the assumption in [33, Assumption 1.1], that is,
$\displaystyle\lim_{t\to 0}\varphi(t)=0,$ $\varphi$ is strictly increasing and
$\rho$ is convex. Note that
$|\mathbb{T}|^{2}\varphi(|\mathbb{T}|^{2})=|\mathbb{T}|^{p+2}$ and
$M_{r,p}=\\{g\in
L^{2}(\Omega):g=[\varphi(|\mathbb{T}|^{2})]^{\frac{1}{2}}\tilde{g},\,\|\tilde{g}\|_{L^{2}(\Omega)}\leq
r\\}$.

######  Theorem 5.1.

Let $\delta>0,r>0$ and $M_{r,p}$ be as in (5.4). Then the following results
hold.

  1. (i)

If $\frac{\delta^{2}}{r^{2}}\in{\rm{\bf
spec}}\big{(}|\mathbb{T}|^{p+2}\big{)},$ then
$\omega(\delta,M_{r,p})=r^{\frac{2}{p+2}}\delta^{\frac{p}{p+2}}.$

  2. (ii)

If $\frac{\delta^{2}}{r^{2}}\notin{\rm{\bf
spec}}\big{(}|\mathbb{T}|^{p+2}\big{)}$, then for $\delta\leq
r\sup\limits_{n}|\mu_{n}|^{\frac{p+2}{2}}$ either

$\inf_{n}\frac{|\mu_{n}|^{\frac{p}{2}}}{|\mu_{n+1}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}\leq\omega(\delta,M_{r,p})\leq\sup_{n}\frac{|\mu_{n+1}|^{\frac{p}{2}}}{|\mu_{n}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}},$

or

$\inf_{n}\frac{|\mu_{n+1}|^{\frac{p}{2}}}{|\mu_{n}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}\leq\omega(\delta,M_{r,p})\leq\sup_{n}\frac{|\mu_{n}|^{\frac{p}{2}}}{|\mu_{n+1}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}.$

###### Proof.

Case 1. $\frac{\delta^{2}}{r^{2}}\in{\rm{\bf
spec}}\big{(}|\mathbb{T}|^{p+2}\big{)}.$

In this case, by the result of Tautenhahn [33, Theorem 2.1], it follows that

$\omega(\delta,M_{r,p})=r\sqrt{\rho^{-1}\left(\frac{\delta^{2}}{r^{2}}\right)}=r\left(\frac{\delta}{r}\right)^{\frac{p}{p+2}}=r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}.$

Case 2. $\frac{\delta^{2}}{r^{2}}\notin{\rm{\bf
spec}}\big{(}|\mathbb{T}|^{p+2}\big{)}.$

Recall that the set of eigenvalues of the compact self-adjoint operator
$|\mathbb{T}|^{p+2}$ is precisely the set
$\Big{\\{}|\mu_{n}|^{p+2}:n\in{\mathbb{N}}\Big{\\}},$ and that $|\mu_{n}|\to
0$ as $n\to\infty.$ Thus, there exists $n_{0}\in{\mathbb{N}}$ such that
$|\mu_{n_{0}}|=\sup\limits_{n}\Big{\\{}|\mu_{n}|:n\in{\mathbb{N}}\Big{\\}}.$
Therefore, for $\frac{\delta^{2}}{r^{2}}\notin{\rm{\bf
spec}}\big{(}|\mathbb{T}|^{p+2}\big{)}$ and for $0<\delta\leq
r\sup\limits_{n}|\mu_{n}|^{\frac{p+2}{2}}$, there are only two possibilities
given below as sub-cases.

Sub-case 2(i). There exists $n\in{\mathbb{N}}$ such that

$|\mu_{n}|^{p+2}\leq\frac{\delta^{2}}{r^{2}}\leq|\mu_{n+1}|^{p+2},$

that is,

(5.8) $r\,|\mu_{n}|^{\frac{p+2}{2}}\leq\delta\leq
r\,|\mu_{n+1}|^{\frac{p+2}{2}}.$

Therefore, by (5.6) and (5.8), we have

(5.9)
$\omega(\delta,M_{r,p})\geq\omega(r\,|\mu_{n}|^{\frac{p+2}{2}},M_{r,p})\underbrace{=}_{{\bf
Case1}}r^{\frac{2}{p+2}}\left(r\,|\mu_{n}|^{\frac{p+2}{2}}\right)^{\frac{p}{p+2}}.$

From (5.8) we have

$\frac{r}{\delta}\,|\mu_{n+1}|^{\frac{p+2}{2}}\geq 1$

and hence by (5.9)

$\omega(\delta,M_{r,p})\geq
r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}\,\frac{|\mu_{n}|^{\frac{p}{2}}}{|\mu_{n+1}|^{\frac{p}{2}}}.$

Again from (5.6) and (5.8), we have

(5.10)
$\omega(\delta,M_{r,p})\leq\omega(r\,|\mu_{n+1}|^{\frac{p+2}{2}},M_{r,p})\underbrace{=}_{{\bf
Case1}}r^{\frac{2}{p+2}}\,\left(r\,|\mu_{n+1}|^{\frac{p+2}{2}}\right)^{\frac{p}{p+2}}.$

Now from (5.8), we have

$\frac{r}{\delta}\,|\mu_{n}|^{\frac{p+2}{2}}\leq 1$

and hence by (5.10), we have

$\omega(\delta,M_{r,p})\leq
r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}\,\frac{|\mu_{n+1}|^{\frac{p}{2}}}{|\mu_{n}|^{\frac{p}{2}}}.$

Thus, we have

$\inf_{n}\frac{|\mu_{n}|^{\frac{p}{2}}}{|\mu_{n+1}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}\leq\omega(\delta,M_{r,p})\leq\sup_{n}\frac{|\mu_{n+1}|^{\frac{p}{2}}}{|\mu_{n}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}.$

Sub-case 2(ii). There exists $n\in{\mathbb{N}}$ such that

$|\mu_{n+1}|^{p+2}\leq\frac{\delta^{2}}{r^{2}}\leq|\mu_{n}|^{p+2}.$

Then using similar arguments as done for Sub-case 2(i), we obtain

$\inf_{n}\frac{|\mu_{n+1}|^{\frac{p}{2}}}{|\mu_{n}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}\leq\omega(\delta,M_{r,p})\leq\sup_{n}\frac{|\mu_{n}|^{\frac{p}{2}}}{|\mu_{n+1}|^{\frac{p}{2}}}\,r^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}.$

∎

######  Remark 5.2.

From (2.9) it follows that
$\sup\limits_{n}|\mu_{n}|\leq\frac{\|\psi\|_{\infty}}{\lambda_{1}^{2}}.$
Therefore, Theorem 5.1 (ii) holds for all $0<\delta\leq\delta_{0},$ where
$\delta_{0}:=r\,\sup\limits_{n}|\mu_{n}|^{\frac{p+2}{2}}<\infty.$ $\lozenge$

######  Remark 5.3.

We now show that the $infimums$ and the $supremums$ in Theorem 5.1(ii) are
indeed non-zero and finite, respectively. In order to see this, first we
recall a result from Courant and Hilbert [6], which says that there exist
constants $e_{1},e_{2}>0$ such that

$e_{1}n^{\frac{2}{d}}\leq\lambda_{n}\leq e_{2}n^{\frac{2}{d}}\quad\text{for
all}\quad n\in{\mathbb{N}}.$

From this, we obtain

$\frac{e_{1}}{e_{2}}\Big{(}\frac{1}{2}\Big{)}^{\frac{2}{d}}\leq\frac{e_{1}}{e_{2}}\Big{(}\frac{n}{n+1}\Big{)}^{\frac{2}{d}}\leq\frac{\lambda_{n}}{\lambda_{n+1}}\leq\frac{e_{2}}{e_{1}}\Big{(}\frac{n}{n+1}\Big{)}^{\frac{2}{d}}\leq\frac{e_{2}}{e_{1}}.$

This together with the inequalities in (5.1) imply the required assertion.
$\lozenge$

######  Remark 5.4.

(Order optimality) Let $\triangle(QRM,\delta,S_{\varrho,p})$ denote the worst
case error for the quasi-reversibilty method. Recall that
$M_{r_{2},p}\subseteq S_{\varrho,p},$ where
$r_{2}=\|\psi\|_{\infty}^{-\frac{p}{2}}\varrho.$ Let $0<\delta\leq\delta_{0},$
where $\delta_{0}:=\frac{r_{2}}{2}\,\sup\limits_{n}|\mu_{n}|^{\frac{p+2}{2}}$.

Then by Theorem 4.4 (i) for $0<p<b$ in the apriori case, and for any
reconstruction method $\mathcal{R}$, we have

$\displaystyle\triangle(QRM,\delta,S_{\varrho,p})$ $\displaystyle\leq$
$\displaystyle
C_{1}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}\,=\frac{C_{1}}{C_{\omega}}\,C_{\omega}\,\varrho^{\frac{2}{p+2}}\,\delta^{\frac{p}{p+2}}$
$\displaystyle\underbrace{\leq}_{{\rm Theorem}\ref{minmax_bounds_mod_cont}}$
$\displaystyle\frac{C_{1}}{C_{\omega}}\,\omega(2\delta,M_{r_{2},p})\underbrace{\leq}_{\eqref{set_monotone_prop_mod_cont}}\frac{C_{1}}{C_{\omega}}\,\omega(2\delta,S_{\varrho,p})\underbrace{\leq}_{\eqref{uni_lower_bound_for_worst_case_error}}\frac{2C_{1}}{C_{\omega}}\,\triangle(\mathcal{R},\delta,S_{\varrho,p}),$

where $C_{\omega}$ is accordingly chosen from the set

$\Big{\\{}2^{\frac{p}{p+2}}\|\psi\|_{\infty}^{-\frac{p}{p+2}},\quad\inf_{n}\frac{|\mu_{n}|^{\frac{p}{2}}}{|\mu_{n+1}|^{\frac{p}{2}}}\,2^{\frac{p}{p+2}}\|\psi\|_{\infty}^{-\frac{p}{p+2}},\quad\inf_{n}\frac{|\mu_{n+1}|^{\frac{p}{2}}}{|\mu_{n}|^{\frac{p}{2}}}\,2^{\frac{p}{p+2}}\|\psi\|_{\infty}^{-\frac{p}{p+2}}\Big{\\}}.$

Similarly, by Theorem 4.7 (i) for $b>2$ and $0<p<b-2$ in the aposteriori case,
and for any reconstruction method $\mathcal{R},$ we have

$\triangle(QRM,\delta,S_{\varrho,p})\leq\frac{2C_{6}}{C_{\omega}}\,\triangle(\mathcal{R},\delta,S_{\varrho,p}).$

Therefore, in the apriori case for $0<p<b$ and in the aposteriori case for
$b>2$ and $0<p<b-2$ the error estimates obtained in Theorem 4.4 and Theorem
4.7, respectively, are of optimal order for the source set $S_{\varrho,p}$.
$\lozenge$

######  Remark 5.5.

In [8, Theorem 2] and [15, Remark 2] the authors have claimed the order
optimality of their estimates in respective context by appealing to the result
of Tautenhahn [33, Theorem 2.1], which relies on the fact that
$\frac{\delta^{2}}{\varrho^{2}}$ is in the spectrum of the specified operators
under consideration. Since this condition need not be satisfied for a given
$\delta>0$, however small it may be, the arguments for the order optimality in
[8, Theorem 2] and [15, Remark 2] is obviously incomplete. Our arguments for
Theorem 5.1 (ii) and Corollary 5.4 can be modified appropriately in those
settings to fill the gaps in their arguments. $\lozenge$

##  6\. conclusion

We have considered an inverse source identification problem from final time
observation associated with a bi-parabolic system, which is known to be an
ill-posed problem. In order to obtain stable approximations of the sought
source, we have employed the quasi-reversibilty (QRM) method, which has not
been explored yet. We obtained Hölder type error estimates for both the
apriori and aposteriori parameter choice strategies. Our rates seems to exceed
the rates obtained in earlier works by using Tikhonov regularization and
quasi-boundary value method. Moreover, we have also shown the order optimality
of the obtained rates for some cases. The arguments used for proving order
optimality in this paper, can be used to fill the gap in the arguments of
order optimality claimed in some earlier works in different context by
different authors.

Moreover, this paper seems to be the first one that has broaden the
applicability of the considered problem by enlarging the admissible class of
source functions. This is done so by allowing the time dependent component of
the source function to change sign. Note that in the context of the considered
inverse problem, all previously known works have assumed some fixed sign
property of the time dependent component by assuming some bounded below
conditions.

Acknowledgments. The first author, Subhankar Mondal, is supported by the
postdoctoral fellowship of TIFR Centre for Applicable Mathematics, Bangalore,
and the second author, M. Thamban Nair, gratefully acknowledges the support
received from BITS Pilani, K.K. Birla Goa Campus, where he is a Visiting
Professor.

## References

  * [1] K.A. Ames, G.W. Clark, J.F. Epperson and S.F. Oppenheimer, A comparison of regularizations for an ill-posed problem. Math. Comp. 67 (1998), no. 224, 1451–1471.
  * [2] M. Andrle and A. El Badia, Identification of multiple moving pollution sources in surface waters or atmospheric media with boundary observations. Inverse Problems 28 (2012), no. 7, 075009, 22 pp.
  * [3] L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s equation. Inverse Problems 22 (2006), no. 2, 413–430.
  * [4] G.W. Clark and S.F. Oppenheimer, Quasireversibility methods for non-well-posed problems. Electron. J. Differential Equations 1994, No. 08, approx. 9 pp.
  * [5] C. Clason and M.V. Klibanov, The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. 30 (2007/08), no. 1, 1–23.
  * [6] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I. Interscience Publishers, Inc., New York, N.Y., 1953. xv+561 pp.
  * [7] J.R. Dorroh and X. Ru, The application of the method of quasi-reversibility to the sideways heat equation. J. Math. Anal. Appl. 236 (1999), no. 2, 503–519.
  * [8] N.V. Duc, N.V. Thang and N.T. Thành, The quasi-reversibility method for an inverse source problem for time-space fractional parabolic equations. J. Differential Equations 344 (2023), 102–130.
  * [9] H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems. Kluwer, Dordrecht, 1996.
  * [10] L.C. Evans, Partial differential equations. Second edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp. ISBN: 978-0-8218-4974-3 35-01.
  * [11] G. Fichera, Is the Fourier theory of heat propagation paradoxical? Rend. Circ. Mat. Palermo (2) 41 (1992), no. 1, 5–28.
  * [12] V.I. Fushchich, A.S. Galitsyn, A.S. Polubinskii, A new mathematical model of heat conduction processes. Ukrainian Math. J. 42 (1990), no. 2, 210–216.
  * [13] B.J. Greer, A.L. Bertozzi and G. Sapiro, Fourth order partial differential equations on general geometries. J. Comput. Phys. 216 (2006), no. 1, 216–246.
  * [14] D.N. Háo, N.V. Duc, N.V. Thang and N.T. Thánh, Regularization of backward time-fractional parabolic equations by Sobolev-type equations. J. Inverse Ill-Posed Probl. 28 (2020), no. 5, 659–676.
  * [15] D.N. Háo, J. Liu, N.V. Duc and N.V. Thang, Stability results for backward time-fractional parabolic equations. Inverse Problems 35 (2019), no. 12, 125006, 25 pp.
  * [16] B. Hofmann, P. Mathé and M. Schieck, Modulus of continuity for conditionally stable ill-posed problems in Hilbert space. J. Inverse Ill-Posed Probl. 16 (2008), no. 6, 567–585.
  * [17] Y. Huang, Modified quasi-reversibility method for final value problems in Banach spaces. J. Math. Anal. Appl. 340 (2008), no. 2, 757–769.
  * [18] V. Isakov, Inverse problems for partial differential equations. Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006. xiv+344 pp.
  * [19] D.D. Joseph and L. Preziosi, Heat waves. Rev. Mod. Phys. 61 (1989), no. 1, 41-73.
  * [20] V. Kalantarov and S. Zelik, Finite-dimensional attractors for the quasi-linear strongly-damped wave equation. J. Differential Equations 247 (2009), no. 4, 1120–1155.
  * [21] R. Lattès and J.-L. Lions, Méthode de quasi-réversibilité et applications. (French) Travaux et Recherches Mathématiques, No. 15 Dunod, Paris 1967 xii+368 pp.
  * [22] T.T. Le, L.H. Nguyen, T.P. Nguyen and W. Powell, The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations. J. Sci. Comput. 87 (2021), no. 3, Paper No. 90, 23 pp.
  * [23] N. Magnoli and G.A. Viano, The source identification problem in electromagnetic theory. J. Math. Phys. 38 (1997), no. 5, 2366–2388.
  * [24] G. Mophou and M. Warma, Quasi-reversibility methods of optimal control for ill-posed final value diffusion equations. J. Math. Anal. Appl. 517 (2023), no. 2, Paper No. 126618, 35 pp.
  * [25] M.T. Nair, Linear Operator Equations: Approximation and Regularization. World Scientific, Hackensack, 2009.
  * [26] L.H. Nguyen, An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method. Inverse Problems 35 (2019), no. 3, 035007, 28 pp.
  * [27] H.T. Nguyen, V.A. Khoa and V.A. Vo, Analysis of a quasi-reversibility method for a terminal value quasi-linear parabolic problem with measurements. SIAM J. Math. Anal. 51 (2019), no. 1, 60–85.
  * [28] L.E. Payne and J.C. Song, On a proposed model for heat conduction. IMA J. Appl. Math. 71 (2006), no. 4, 590–599.
  * [29] N.D. Phuong, N.H. Luc and L.D. Long, Modified quasi boundary value method for inverse source problem of the bi-parabolic equation. Advances in the Theory of Nonlinear Analysis and its Applications 4 (2020), no. 3, 132-142.
  * [30] D.H. Quoc Nam, L.D. Long, D. O’Regan, T.B. Ngoc and N.H. Tuan, Identification of the right-hand side in a bi-parabolic equation with final data. Appl. Anal. 101 (2022), no. 4, 1157–1175.
  * [31] R.E. Showalter, The final value problem for evolution equations. J. Math. Anal. Appl. 47 (1974), 563–572.
  * [32] R.E. Showalter, Quasi-reversibility of first and second order parabolic evolution equations. Improperly posed boundary value problems (Conf., Univ. New Mexico, Albuquerque, N.M., 1974), pp. 76–84. Res. Notes in Math., No. 1, Pitman, London, 1975.
  * [33] U. Tautenhahn, Optimality for ill-posed problems under general source conditions. Numer. Funct. Anal. Optim. 19 (1998), no. 3-4, 377–398.
  * [34] D.D. Trong and D.N.D. Hai, Backward problem for time-space fractional diffusion equations in Hilbert scales. Comput. Math. Appl. 93 (2021), 253–264.
  * [35] N.H. Tuan, On some inverse problem for bi-parabolic equation with observed data in $L^{p}$ spaces. Opuscula Math. 42 (2022), no. 2, 305–335.
  * [36] N.H. Tuan, E. Nane and D.D. Trong, Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data. Illinois J. Math. 65 (2021), no. 4, 793–845.
  * [37] F. Yang, Y.P. Ren and X.X. Li, The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source. Math. Methods Appl. Sci. 41 (2018), no. 5, 1774–1795.
  * [38] F. Zouyed and S. Djemoui, An iterative regularization method for identifying the source term in a second order differential equation. Math. Probl. Eng. 2015, Art. ID 713403, 9 pp.