#  $L^{q}$ Carleman estimates with boundary observations and applications to
inverse problems

###### Abstract.

We consider coupled linear parabolic systems and we establish estimates in
$L^{q}$-norm for the sources in terms of observations on the corresponding
solutions on a part of the boundary. The main tool is a family of Carleman
estimates in $L^{q}$-norm with boundary observations.

###### Key words and phrases:

Parabolic systems, Carleman estimates, Inverse problems, Source estimates.

###### 2010 Mathematics Subject Classification:

35R30, 35K10, 35K57, 93B07

This work was co-funded by the European Social Fund, through Operational
Programme Human Capital 2014-2020, project number POCU/993/6/13/153322,
project title ”Educational and training support for PhD students and young
researchers in preparation for insertion into the labor market”.

Elena-Alexandra Melnig

Faculty of Mathematics, ”Al. I. Cuza” University of Iaşi, Romania

Octav Mayer Institute of Mathematics, Romanian Academy, Iaşi Branch

E-mail address<EMAIL_ADDRESS>
## Introduction

In this paper we consider linear parabolic systems coupled in zero order terms
and we obtain estimates in $L^{q}$ spaces, $q\geq 2$ for the sources in terms
of measurements of the solutions on a part of the boundary. This research is
based on previous results in $L^{2}$ and uses the regularity properties of the
heat flow. Such an inverse problem is motivated by the fact that the boundary
of a domain is more accessible for measurements on the solution.

This type of problems was considered by Imanuvilov and Yamamoto in [4] where
they obtained source estimates in $L^{2}$ norm with observations on the
solution on a subdomain or on the boundary. Their method employed the coupling
of two $L^{2}$-Carleman estimates, a technique no longer applicable in the
$L^{q}$ framework, as explained in [9]. Furthermore, they require more
detailed information from the measurements, as their estimates involve the
solution together with its time and space derivatives, something we overcome
in the present paper.

The problem we address here comes as a continuation of the work done in [9]
where we have established such $L^{q}$ stability estimates for the sources of
a linear system in terms of the solution measured on a subdomain. Also, in
[10] we established estimates of this type for the sources in a semilinear
system with observations on the solution in a subdomain.

An important aspect in the above mentioned papers as well as in the present
study is the positivity of the solutions and of the sources, as the systems we
consider are intended to model reaction-diffusion processes. In this context
we need two fundamental tools: strong invariance principles and respectively
Carleman estimates. The first tool relies on the strong maximum principle for
parabolic equations and on strong invariance results for weakly coupled
parabolic systems. For such maximum principles or invariance results in the
framework of classical solutions we refer to [11] for equations and to [12]
for systems. For the case of variational solutions we mention [8].

The other main tool in our approach is a family of Carleman estimates in
$L^{q}$ norm, $q\geq 2$, with boundary observations and general weights of
exponential type obtained for parabolic systems with general boundary
conditions. The Carleman inequalities in $L^{2}$ were first used in the
framework of controllability of the heat equation with controls distributed in
a subdomain bu O.Yu Imanuvilov, [3]. Since then they have found applications
in obtaining observability inequalities in controllability problems, unique
continuation properties and inverse problems. The Carleman estimates in
$L^{q}$ are derived through a bootstrap argument, starting from the
$L^{2}$-Carleman estimates proved in [7] and using the regularizing effect of
the parabolic equation. This type of argument can be found in V. Barbu [1],
J.-M. Coron, S.Guerrero, L.Rosier [2], K. Le Balc’h [6], in the context of
controllability.

##  1\. Preliminaries and main results

Let $T>0$ and $\Omega\subset\mathbb{R}^{N}$ be an annular domain, that is a
domain which is diffeomorphic to $B_{2}(0)\setminus B_{1}(0)$, with boundary
$\partial\Omega$ of class $C^{2}$. Consider $\Gamma_{1}\subset\partial\Omega$
be the exterior boundary and
$\Gamma_{0}=\partial\Omega\setminus\overline{\Gamma}_{1}$ be the inner
boundary, and let $Q=(0,T)\times\Omega$,
$\Sigma_{1}:=(0,T)\times\Gamma_{1},\Sigma_{0}:=(0,T)\times\Gamma_{0}$.

In the following, whenever we will refer to vector valued functions from a
corresponding Sobolev space, we will write $L^{q}(Q)$ instead of
$[L^{q}(Q)]^{n}$, $W^{2,1}_{q}(Q)\cap L^{\infty}(Q)$ instead of
$[W^{2,1}_{q}(Q)\cap L^{\infty}(Q)]^{n}$.

We consider linear parabolic systems coupled in zero order terms with the
elliptic part of the operator in divergence form and general homogeneous
boundary conditions (Dirichlet, Neumann or Robin) on each connected component
of the boundary and for each component of the system:

(S)
$\left\\{\begin{array}[]{ll}D_{t}y_{i}-\sum\limits_{j,k=1}^{N}D_{j}(a_{i}^{jk}D_{k}y_{i})+\sum\limits_{k=1}^{N}b_{i}^{k}D_{k}y_{i}+\sum\limits_{l=1}^{N}c_{i}^{l}y_{l}=g_{i}&(0,T)\times\Omega,\\\
\beta_{i}(x)\frac{\partial y_{i}}{\partial
n_{A}}+\eta_{i}(x)y_{i}=0&(0,T)\times\partial\Omega,\\\ \end{array}\quad
i=\overline{1,n}\right.$

where

  * (H1)

$a_{i}^{jk}\in W^{1,\infty}(\Omega)$, $b^{k}_{i},c_{i}^{l}\in
L^{\infty}(\Omega)$ and $a_{i}^{jk}$ satisfy the ellipticity condition:

$\exists\mu>0\text{ {s.t.}
}\sum_{j,k=1}^{N}a_{i}^{jk}(x)\xi_{j}\xi_{k}\geq\mu|\xi|^{2},\quad\forall\xi\in\mathbb{R}^{N},\quad(t,x)\in
Q,i=\overline{1,n};$

  * (H2)

the sources are positive, $g_{i}\in L^{q}(Q)$, $g_{i}\geq 0,i=\overline{1,n};$

  * (H3)

the coupling coefficients are non-positive $c_{i}^{l}\leq 0,i\neq l$;

  * (H4)

$\beta_{i},\eta_{i}\in L^{\infty}(\partial\Omega),\,\beta_{i},\eta_{i}\geq
0\text{ and }\beta_{i}>0\text{ or }\beta_{i}\equiv 0\text{ and }\eta_{i}\equiv
1$ on each connected component of $\partial\Omega$.

The boundary observation on the solution is
$\zeta=(\zeta_{i}(y_{i}))_{i=\overline{1,n}}$:

(O) $\zeta_{i}(y_{i})=\gamma_{i}(x)\frac{\partial y_{i}}{\partial
n_{A}}+\delta_{i}(x)y_{i},\quad(0,T)\times\Gamma_{1},\quad\Gamma_{1}\subset\partial\Omega,\quad
i=\overline{1,n},$

with given $\gamma_{i},\delta_{i}\in L^{\infty}(\Gamma_{1})$.

We impose an independence condition between the boundary conditions and the
observations expressed as:

  * (H5)

$\begin{vmatrix}\gamma_{i}&\delta_{i}\\\ \beta_{i}&\eta_{i}\end{vmatrix}\neq
0\text{ on }\Gamma_{1},\quad i=\overline{1,n}.$

For $2\leq q\leq\infty$ and $k>0$ we consider the following classes of
sources:

(1.1) $\mathcal{G}_{q,k}=\left\\{g\in[L^{q}(Q)]^{n}:\,g\geq 0\text{ s.t.
}\|g\|_{L^{q}(Q)}\leq k\|g\|_{L^{1}(Q)}\right\\}.$

In this context, the main result which gives $L^{q}$ estimates for the sources
assuming that the sources belong to some class $\mathcal{G}_{q,k}$ is the
following:

######  Theorem 1.

Consider the system (S) with hypotheses $(H1)-(H5)$. Then for $2\leq
q<\infty,k>0$, $g\in\mathcal{G}_{q,k}$ and the corresponding solution $y\in
W^{2,1}_{q}(Q)$, there exists $C=C(q,k)>0$ such that

(1.2) $\left\|g\right\|_{L^{q}(Q)}\leq C_{1}\|\zeta\|_{L^{2}(\Sigma_{1})}.$

Concerning $q=\infty$, for sources $g\in\mathcal{G}_{\infty,k}$ and
corresponding solutions $y\in W^{2,1}_{q}(Q)$ for all $q<\infty$, there exists
$C=C(k)>0$ such that

(1.3) $\left\|g\right\|_{L^{\infty}(Q)}\leq
C_{1}\|\zeta\|_{L^{2}(\Sigma_{1})}.$

##  2\. $L^{q}-L^{2}$-Carleman estimates with boundary observations

The key ingredient to prove the above Theorem is a family of $L^{q}$ Carleman
estimates with observations on the boundary. To derive these estimates, it is
necessary to use the classical mechanism but with auxiliary functions
satisfying supplementary technical properties.These properties, in turn,
dictate the special choice of the shape of the domain.

The choice of auxiliary functions is the following:

  * •

$\psi_{0}\in C^{2}(\overline{\Omega})$ s.t.
$\psi_{0}|_{\Gamma_{0}}=0,\quad\psi_{0}|_{\Gamma_{1}}=1,\quad|\nabla\psi_{0}|>0$
in $\overline{\Omega}$,
$\quad\frac{\partial\psi_{0}}{\partial\nu}|_{\Gamma_{0}}<0$,

  * •

$\varphi_{0}(t,x):=\frac{e^{\lambda\psi_{0}(x)}}{t(T-t)},\quad\alpha_{0}(t,x):=\frac{e^{\lambda\psi_{0}(x)}-e^{1.5\lambda\|\psi_{0}\|_{C(\overline{\Omega})}}}{t(T-t)}$

  * •

$\psi=\psi_{0}+K$, $K$ big enough such that

$\frac{\sup\psi}{\inf\psi}\leq\frac{8}{7}$

and the weight functions

$\varphi(t,x):=\frac{e^{\lambda\psi(x)}}{t(T-t)}\,,\quad\alpha(t,x):=\frac{e^{\lambda\psi(x)}-e^{1.5\lambda\|\psi\|_{C(\overline{\Omega})}}}{t(T-t)}.$

######  Remark 1.

The choice of annular domains is imposed by technical restrictions in
obtaining the proper Carleman estimates described in the following and more
precisely by the conditions imposed to $\psi_{0}$ which imply necessarily the
existence of two connected components of the boundary with the same topology.
So, other choices of the domain is possible with restrictions imposed by
existence of such a function.

The Carleman estimate we obtain is the following

######  Proposition 1.

Let $g\in L^{q}(Q)$, with $2\leq q<\infty$. Then there exist $s_{0}=s_{0}(q)$,
$\lambda_{0}=\lambda_{0}(q)$, s.t. for $\lambda>\lambda_{0}$,
$s^{\prime},s>s_{0}$, $\frac{s^{\prime}}{s}>\bar{\gamma}>1$, there exists
$C=C(q,\bar{\gamma})$:

(2.1)
$\displaystyle\|ye^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(Dy)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D^{2}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D_{t}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}$
$\displaystyle\leq C\left[\|ge^{s{\alpha}}\|_{L^{q}(Q)}+\|\zeta
e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right]$

where the constant $C$ depends on $\lambda$ but independent of $s$.

The result in Proposition 1 relies on the regularizing effect of the parabolic
flow combined with a bootstrap argument applied to the linear parabolic system
and using the following $L^{2}$ Carleman estimate that was obtained in [7]
using the classical techniques from [3]:

######  Proposition 2.

For $g\in L^{2}(Q)$, there exist constants $\lambda_{0}=\lambda_{0}(\Omega),$
$s_{0}=s_{0}(\Omega)$ such that, for any $\lambda\geq\lambda_{0}$, $s\geq
s_{0}$ and some $C=C(T,\Omega)$, the following inequality holds:

(2.2)
$\displaystyle\int_{Q}\left[(s\varphi)^{-1}\left(|D_{t}y|^{2}+|D^{2}y|^{2}\right)+s\lambda^{2}\varphi|Dy|^{2}+s^{3}\lambda^{4}\varphi^{3}|y|^{2}\right]e^{2s\alpha}dxdt+\int_{[0,T]\times\Gamma_{0}}s^{3}\lambda^{3}\varphi^{3}|y|^{2}e^{2s\alpha}d\sigma$
$\displaystyle\leq
C\left(\int_{Q}|g|^{2}e^{2s\alpha}dxdt+\int_{[0,T]\times\Gamma_{1}}s^{3}\lambda^{3}\varphi^{3}|\zeta|^{2}e^{2s\alpha}d\sigma\right),\qquad\quad$

for $y\in H^{1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$ solution of
(S).

In the following we denote the operators entering the system by

$B=(B_{i})_{i=\overline{1,n}}$, $B_{i}y_{i}=\beta_{i}(x)\frac{\partial
y_{i}}{\partial n_{A_{i}}}+\eta_{i}(x)y_{i}$ , $L=(L_{i})_{i=\overline{1,n}}$
, $L_{i}y_{i}=-\sum\limits_{j,k=1}^{N}D_{j}(a_{i}^{jk}D_{k}y_{i})$,
$L^{1}=(L^{1}_{i})_{i=\overline{1,n}}$,
$L^{0}=(L^{0}_{i})_{i=\overline{1,n}}$, where the lower-order operators are
given by ($w$ is a scalar function, $y$ is vector valued function):

(2.3)
$L^{1}_{i}w=\sum\limits_{{\begin{subarray}{c}k=\overline{1,N}\end{subarray}}}b_{i}^{k}D_{k}w,\quad
L^{0}_{i}y=\sum\limits_{l=\overline{1,n}}c^{l}_{i}y_{l},\quad
i=\overline{1,n}.$

Using these notations the system (S) is written more compactly

(2.4) $\begin{cases}&D_{t}y+Ly+L^{1}y+L^{0}y=g,\,(0,T)\times\Omega,\\\
&By=0,\,(0,T)\times\partial\Omega.\end{cases}$

For the proof of $L^{q}$ estimates we need Sobolev embedding results for
anisotropic Sobolev spaces ( see [5], Lemma 3.3):

######  Lemma 1.

Consider $u\in W^{2,1}_{p}(Q)$.

Then $u\in Z_{1}$ where

$Z_{1}=\left\\{\begin{array}[]{lll}L^{q}(Q)&\text{ with
}q\leq\frac{(n+2)p}{n+2-2p}&\text{ when }p<\frac{N+2}{2}\\\ L^{q}(Q)&\text{
with }q\in[1,\infty),&\text{ when }p=\frac{N+2}{2}\\\
C^{\alpha,\alpha/2}(Q)&\text{ with }0<\alpha<2-\frac{N+2}{p},&\text{ when
}p>\frac{N+2}{2}\end{array}\right.$

and there exists $C=C(Q,p,N)$ such that

$\|u\|_{Z_{1}}\leq C\|u\|_{W^{2,1}_{p}(Q)}.$

Moreover, $Du\in Z_{2}$ where

$Z_{2}=\left\\{\begin{array}[]{lll}L^{q}(Q)&\text{ with
}q\leq\frac{(N+2)p}{N+2-p}&\text{ when }p<{N+2}\\\ L^{q}(Q)&\text{ with
}q\in[1,\infty),&\text{ when }p={N+2}\\\ C^{\alpha,\alpha/2}(Q)&\text{ with
}0<\alpha<1-\frac{N+2}{p},&\text{ when }p>{N+2}\end{array}\right.$

and there exists $C=C(p,N)$ such that

$\|Du\|_{Z_{2}}\leq C\|u\|_{W^{2,1}_{p}(Q)}.$

We introduce the following auxiliary functions which do not depend on space
variable $x$:

$\overline{\varphi}:=\frac{e^{\lambda(K+1)}}{t(T-t)},\quad\underline{\varphi}:=\frac{e^{\lambda
K}}{t(T-t)},\quad\overline{\alpha}:=\frac{e^{\lambda(K+1)}-e^{1.5\lambda(K+1)}}{t(T-t)},\quad\underline{\alpha}:=\frac{e^{\lambda
K}-e^{1.5\lambda(K+1)}}{t(T-t)}.$

######  Remark 2.

Observe that for some $\sigma>\sigma_{0}$ and for some
$\tilde{\lambda}_{0}(\sigma_{0})>0$ we have for $\lambda>\tilde{\lambda}_{0}$
that

$-(\sigma-1)e^{1.5\lambda(K+1)}+\sigma e^{\lambda(K+1)}-e^{\lambda
K}\leq-\sigma\lambda e^{\lambda(K+1)}.$

Which gives

$\displaystyle\frac{e^{m\lambda(K+1)}\sigma^{m}s_{1}^{m}\lambda^{m}}{t^{m}(T-t)^{m}}e^{\frac{-(\sigma-1)s_{1}e^{1.5\lambda(K+1)}+\sigma
s_{1}e^{\lambda(K+1)}-s_{1}e^{\lambda
K}}{t(T-t)}}\leq\frac{e^{m\lambda(K+1)}\sigma^{m}s_{1}^{m}\lambda^{m}}{t^{m}(T-t)^{m}}e^{\frac{-\sigma
s_{1}\lambda e^{\lambda(K+1)}}{{t(T-t)}}}$
$\leq\sup_{\mu\in[0,\infty)}\mu^{m}e^{-\mu}=C(m),$

which gives

(2.5)
$\overline{\varphi}^{m}s_{2}^{m}\lambda^{m}e^{s_{2}\overline{\alpha}}\leq
C(m)e^{s_{1}\underline{\alpha}}.$

In conclusion, for all $m>0$ and $\sigma_{0}>1$, there exist
$\tilde{\lambda}_{0}=\tilde{\lambda}_{0}(\sigma_{0})>0$ and $C=C(m)$ such that
if $\lambda>\lambda_{0}$ and $s_{1},s_{2}>0$ with
$\frac{s_{2}}{s_{1}}=\sigma>\sigma_{0}$, one has

(2.6) $\varphi^{m}s_{2}^{m}\lambda^{m}e^{s_{2}\alpha}\leq
C(m)e^{s_{1}\alpha},$

with $\varphi,\alpha$ as above.

The auxiliary functions are built on $\psi_{0}$ in the same way as in the case
of internal observations with similar estimates on the weights (see [10]).

Proof of Proposition 1.  For a given $\gamma>1$ and $j\in\mathbb{N}$ denote by

$w^{j}:=ye^{\gamma^{j}s\overline{\alpha}}=w^{j-1}e^{\gamma^{j-1}s\overline{\alpha}(\gamma-1)}.$

Notice that since $\gamma>1$, for fixed $j$ there exists
$\bar{\lambda}_{0}(j)>0$ such that

(2.7)
$e^{\gamma^{j}s\alpha}<e^{\gamma^{j}s\overline{\alpha}}<e^{s\alpha}\text{ for
all }\lambda\geq\bar{\lambda}_{0}(j).$

Each $w^{j}$ verifies the initial boundary value problem

(2.8) $\left\\{\begin{aligned}
&D_{t}w^{j}+Lw^{j}=ge^{\gamma^{j}s\overline{\alpha}}+O[s\gamma^{j}\overline{\varphi}^{2}e^{\gamma^{j-1}s\overline{\alpha}(\gamma-1)}]w^{j-1},\text{
in }(0,T)\times\Omega\\\ &Bw^{j}=0,\text{ on }(0,T)\times\partial\Omega,\\\
&w^{j}(0,\cdot)=0\text{ in }\Omega.\end{aligned}\right.\quad$

Observe that the boundary conditions satisfied by $w^{j}$ remain the same as
those satisfied by $y^{j}$ as we have used in the definition of $w^{j}$ the
space independent weight $\overline{\alpha}$.

We describe the bootstrap argument: based on the the regularity argument from
Lemma 1, we construct the sequence $\\{q_{j}\\}_{j\in\mathbb{N}}$:

(2.9) $q_{0}=2,\quad
q_{j}:=\begin{cases}\dfrac{(N+2)q_{j-1}}{N+2-q_{j-1}},\text{ if
}q_{j-1}<N+2,\\\ \frac{3}{2}q_{j-1},\text{ if }q_{j-1}\geq N+2.\end{cases}$

Observe that the sequence $\\{q_{j}\\}_{j\in\mathbb{N}}$ is increasing to
infinity. Now, since $g\in L^{q}(Q)$, we can take $m$ such that $q_{m-1}\leq
q<q_{m}$ and we get, by standard Sobolev embedding, that:

(2.10) $W^{1,q_{j-1}}(Q)\subset L^{q_{j}}(Q),\,j=1,\ldots,m,$

since the Sobolev exponent $q_{j}^{*}:=\frac{(N+1)q_{j-1}}{N+1-q_{j-1}}$ is
greater than $q_{j}$.

Now, an argument like the one in Remark 2 gives that there exist
$S_{0},\Lambda_{0}\geq 0$ and $C=C(j)>0$ such that for $s\geq
S_{0},\lambda\geq\Lambda_{0}$ and $j=\overline{1,m}$ we have

(2.11)
$s\gamma^{j}\overline{\varphi}^{2}e^{\gamma^{j-1}s\overline{\alpha}(\gamma-1)}\leq
C(j).$

Since the initial data $w^{j}(0,\cdot)$ is zero and using the previous
estimate (2.11), the parabolic regularity gives that for $\lambda$ big enough
($\lambda>\max\\{\lambda_{0},\Lambda_{0},\max_{j=\overline{1,m}}\\{\tilde{\lambda}_{j},\bar{\lambda}_{j}\\}\\}$)
we have

(2.12) $\|w^{j}\|_{W^{2,1}_{q_{j-1}}(Q)}\leq
C\left(\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j-1}}(Q)}+\|w^{j-1}\|_{L^{q_{j-1}}(Q)}\right).$

Using Lemma 1 and the embedding (2.10) we get that $w^{j}\in L^{{q}_{j}}(Q)$
and, from the previous inequality, we obtain the estimate

(2.13) $\|w^{j}\|_{L^{q_{j}}(Q)}\leq
C\left(\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j-1}}(Q)}+\|w^{j-1}\|_{L^{q_{j-1}}(Q)}\right)\text{
for }j=1,\ldots,m,$

and so

(2.14) $\|w^{m}\|_{L^{q_{m}}(Q)}\leq
C\left(\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|w^{0}\|_{L^{q_{0}}(Q)}\right).$

Regarding the first order terms, Lemma 1 implies that
$\|Dw^{m}\|_{L^{q_{m}}(Q)}\leq\|w^{m}\|_{W^{2,1}_{q_{m-1}}(Q)}$, meaning that
we have estimates also for first order derivatives

(2.15) $\|w^{m}\|_{L^{q_{m}}(Q)}+\|Dw^{m}\|_{L^{q_{m}}(Q)}\leq
C\left(\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|w^{0}\|_{L^{q_{0}}(Q)}\right).$

Since

$\|ye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}\leq\|ye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q}(Q)}\leq
C\|ye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q^{m}}(Q)}$

and

$\|Dye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}\leq\|Dye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q}(Q)}\leq
C\|Dye^{\gamma^{m}s\overline{\alpha}}\|_{L^{q^{m}}(Q)},$

using (2.15) we can obtain a partial estimate,

(2.16)
$\|ye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}+\|Dye^{\gamma^{m}s\alpha}\|_{L^{q}(Q)}\leq
C\left(\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|w^{0}\|_{L^{q_{0}}(Q)}\right).$

Now we use (2.7), the fact that ${q_{0}=2}$, $w^{0}=ye^{s\overline{\alpha}}$,
and the $L^{2}$ Carleman inequality (3.9) to properly bound
$\|w^{0}\|_{L^{q_{0}}(Q)}$:

(2.17) $\|ye^{s\overline{\alpha}}\|_{L^{2}(Q)}\leq
C\|ye^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(Q)}\leq
C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(Q)}+\|s^{\frac{3}{2}}\lambda^{2}\varphi^{\frac{3}{2}}\zeta
e^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$

Using (2.7), Remark 2 and the fact that $q>q_{j}$ forall
$q_{j}\in\overline{1,m-1}$, there exists $C>0$ such that the right hand-side
of (2.15) is bounded as follows

(2.18)
$\displaystyle\sum_{j=1}^{m-1}\|ge^{\gamma^{j}s\overline{\alpha}}\|_{L^{q_{j}}(Q)}+\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(Q)}+\|s^{\frac{3}{2}}\lambda^{2}\varphi^{\frac{3}{2}}\zeta
e^{\frac{1}{\gamma}s\alpha}\|_{L^{2}(\Sigma_{1})}$ $\displaystyle\leq
C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{\frac{1}{\gamma^{2}}s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$

In order to obtain estimates for time derivatives and second order space
derivatives, we return to the parabolic problem verified by $w^{m+1}$:

(2.19) $\left\\{\begin{aligned}
&D_{t}w^{m+1}+Lw^{m+1}=ge^{\gamma^{m+1}s\overline{\alpha}}+O[s\gamma^{m+1}\overline{\varphi}^{2}e^{\gamma^{m}s\overline{\alpha}(\gamma-1)}]w^{m},\text{
in }(0,T)\times\Omega\\\ &Bw^{m+1}=0,\text{ on }(0,T)\times\partial\Omega,\\\
&w^{m+1}(0,\cdot)=0\text{ in }\Omega\end{aligned}\right.\quad$

Parabolic regularity along with (2.11) gives

(2.20) $\|w^{m+1}\|_{W^{2,1}_{q}(Q)}\leq
C\left(\|ge^{\gamma^{m+1}s\overline{\alpha}(\gamma-1)}\|_{L^{q}(Q)}+\|w^{m}\|_{L^{q}(Q)}\right)$

which implies, using again (2.7), (2.15) and (2.18) that

(2.21)
$\displaystyle\|D^{2}ye^{\gamma^{m+1}s\alpha}\|_{L^{q}(Q)}\leq\|D^{2}ye^{\gamma^{m+1}s\overline{\alpha}}\|_{L^{q}(Q)}=\|D^{2}w^{m+1}\|_{L^{q}(Q)}$
$\displaystyle\leq C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{\frac{1}{\gamma^{2}}s\alpha}\|_{L^{2}(\Sigma_{1})}\right)$

and using (2.11), (2.7), (2.15), (2.18) that

(2.22)
$\displaystyle\|(D_{t}y)e^{\gamma^{m+1}s\alpha}\|_{L^{q}(Q)}\leq\|D_{t}w^{m+1}\|_{L^{q}(Q)}+C\|w^{m}\|_{L^{q}(Q)}$
$\displaystyle\leq C\left(\|ge^{\frac{1}{\gamma}s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{\frac{1}{\gamma^{2}}s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$

From (2.15),(2.18),(2.21) and (2.22) and taking
$\gamma=\bar{\gamma}^{\frac{1}{m+3}}$, $s$ changed into
$\frac{1}{\gamma^{2}}s$, we obtain the conclusion,

(2.23)
$\displaystyle\|ye^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(Dy)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D^{2}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D_{t}y)e^{s^{\prime}\alpha}\|_{L^{q}(Q)}\leq$
$\displaystyle\leq C\left(\|ge^{s{\alpha}}\|_{L^{q}(Q)}+\|\zeta
e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$

######  Remark 3.

Observe that if $q>N+1$, then the Morrey embedding theorem gives $L^{\infty}$
estimates for $y$ and $Dy$: there exist $s_{0},\lambda_{0}>0$, s.t. for
$\lambda>\lambda_{0}$, $s^{\prime},s>s_{0}$,
$\frac{s^{\prime}}{s}>\bar{\gamma}>1$, there exists $C=C(\bar{\gamma})$:

(2.24)
$\|ye^{s^{\prime}\alpha}\|_{L^{\infty}(Q)}+\|Dye^{s^{\prime}\alpha}\|_{L^{\infty}(Q)}\leq
C\left(\|ge^{s\alpha}\|_{L^{q}(Q)}+\|\zeta
e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right).$

##  3\. Source stability for linear systems. Proof of Theorem 1

In the following, the strong solutions $y\in W^{2,1}_{q}(Q)$ that we work with
are also variational solutions of the system (S).

For this purpose, we denote by
$\Gamma_{D}^{i}=\\{x\in\partial\Omega|\beta_{i}=0\\}$ the Dirichlet boundary
corresponding to $y_{i}$ and consider the Hilbert spaces $V_{i}=\\{v\in
H^{1}(\Omega):v=0\text{ on }\Gamma_{D}^{i}\\}$ and let
$V=V_{1}\times\cdots\times V_{n}$.

Then, for some initial data $y_{0}\in L^{2}(\Omega)$, $y\in L^{2}(0,T;V)\cap
H^{1}(0,T;V^{\prime})\subset C([0,T];H)$ is a weak solution of system (S) if:

(3.1) $\displaystyle\sum\limits_{i=1}^{n}(\langle
y_{i}(t),v_{i}\rangle_{L^{2}(\Omega)}-\langle
y_{0,i},v_{i}\rangle_{L^{2}(\Omega)})+\sum\limits_{i=1}^{n}\int_{0}^{t}\langle\mathbb{A}_{i}\nabla
y_{i}(\tau,\cdot),\nabla v\rangle_{L^{2}(\Omega)}d\tau$
$\displaystyle+\sum\limits_{i=1}^{n}\int_{0}^{t}\langle b_{i}\cdot\nabla
y_{i}(\tau,\cdot),v_{i}\rangle_{L^{2}(\Omega)}d\tau+\sum\limits_{l=1}^{n}\int_{0}^{t}\langle
c_{i}^{l}y_{l}(\tau,\cdot),v_{i}\rangle_{L^{2}(\Omega)}d\tau$
$\displaystyle+\sum\limits_{i=1}^{n}\int_{0}^{t}\langle\eta_{i}y_{i}(\tau,\cdot),v_{i}\rangle_{L^{2}(\partial\Omega\setminus\Gamma_{D}^{i})}d\tau=\sum\limits_{i=1}^{n}\int_{0}^{t}\langle
g_{i}(\tau,\cdot),v_{i}\rangle_{L^{2}(\Omega)}d\tau,\quad\forall
t\in(0,T),\quad\forall v_{i}\in V_{i},$

where $\mathbb{A}_{i}$ are the matrix coefficients matrix
$(a_{i}^{jk})_{j,k\in\overline{1,N}}$ and
$b_{i}=(b_{i}^{k})^{\top}_{k\in\overline{1,N}}$.

Proof of Theorem 1. We have to prove that there exists $C=C(k)$ such that for
$g\in\mathcal{G}_{q,k}$,

(3.2) $\displaystyle\left\|g\right\|_{L^{q}(Q)}\leq
C\|\zeta(y)\|_{L^{2}(\Sigma_{1})}.$

We argue by contradiction. Then there exists a sequence of sources denoted as
$(g^{m})_{m}\subset\mathcal{G}_{q,k}$, and the corresponding solutions
$(y^{m})_{m}\subset W^{2,1}_{q}(Q)$, such that:

(3.3)
$\displaystyle\left\|g^{m}\right\|_{L^{q}(Q)}>m\|\zeta(y^{m})\|_{L^{2}(\Sigma_{1})}.$

With no loss of generality we may assume that
$\left\|g^{m}\right\|_{L^{q}(Q)}=1$. For the case when $2\leq q<\infty$, up to
a subsequence denoted again $(g^{m})_{m}$, $g^{m}\rightharpoonup g$ weakly
$L^{q}(Q)$ for some $g\in L^{q}(Q)$. For the case when $q=\infty$ we have that
up to a subsequence $g^{m}\rightharpoonup g$ weak-* in $L^{\infty}(Q)$ for
some $g\in L^{\infty}(Q)$.

This means that the above equation (3.3) gives for the sequence of
observations that

(3.4) $\zeta(y^{m})\rightarrow 0\text{ in }L^{2}(\Sigma)\text{ as
}m\rightarrow\infty.$

Observe that in both cases the weak limit $g$ of $(g^{m})_{m}$ is not zero.
Indeed, for the case when $2\leq q<\infty$, since $g^{m}\in\mathcal{G}_{q,k}$,
for $1\in L^{q^{\prime}}(Q)$ we have

$\langle
g^{m},1\rangle_{L^{q},L^{q^{\prime}}}=\int_{Q}g^{m}\geq\frac{1}{k}\|g^{m}\|_{L^{q}(Q)}=\frac{1}{k}.$

Because $(g^{m})_{m}$ converges weakly in $L^{q}(Q)$ we have that
$\int_{Q}g\geq\frac{1}{k}>0$ and thus $g\not\equiv 0$.

Also for the case when $q=\infty$, the weak-* limit $g$ of $(g^{m})_{m}$ is
not zero. Since $g^{m}\in\mathcal{G}_{k}$ and $1\in L^{1}(Q)$,

$\langle
g^{m},1\rangle_{L^{\infty},L^{1}}=\int_{Q}g^{m}\geq\frac{1}{k}\|g^{m}\|_{L^{\infty}(Q)}=\frac{1}{k}.$

Because $(g^{m})_{m}$ converges weak-* in $L^{\infty}(Q)$ we have that
$\int_{Q}g\geq\frac{1}{k}>0$ and thus $g\not\equiv 0$.

Now, we pass to the limit in the weak formulation of the problem (3.1) for
some initial data $y_{0}\in L^{2}(\Omega)$.

Consider $(y^{m})_{m}$ a sequence of solutions for the system (S) with
corresponding sources $(g^{m})_{m}\subset L^{q}(Q)$. For $2\leq q<\infty$,
$(g^{m})_{m}$ is bounded in $L^{q}(Q)$, $(\zeta^{m})_{m}=(\zeta(y^{m}))_{m}$
is bounded in $L^{2}(\Sigma_{1})$ and using the Carleman estimate for the
solution $y^{m}$ corresponding to the sources $g^{m}$, we have that

(3.5)
$\displaystyle\|y^{m}e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(Dy^{m})e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D^{2}y^{m})e^{s^{\prime}\alpha}\|_{L^{q}(Q)}+\|(D_{t}y^{m})e^{s^{\prime}\alpha}\|_{L^{q}(Q)}$
$\displaystyle\leq
C\left[\|g^{m}e^{s{\alpha}}\|_{L^{q}(Q)}+\|\zeta^{m}e^{s\alpha}\|_{L^{2}(\Sigma_{1})}\right],$

which gives for a cylinder of form
$Q^{\epsilon}=(\epsilon,T-\epsilon)\times\Omega$ with $\epsilon>0$ fixed
$0<\epsilon<\frac{T}{2}$, arbitrarily small, that $(y^{m})_{m}$ is bounded in
${W^{2,1}_{q}(Q^{\epsilon})}$.

Thus $(y^{m})_{m}$ is bounded in $L^{q}(\epsilon,T-\epsilon;W^{2,q}(\Omega))$
and $D_{t}y^{m}$ is bounded in $L^{q}(\epsilon,T-\epsilon;L^{q}(\Omega))$, for
$2\leq q<\infty$. Since $W^{2,q}(\Omega)\subset W^{1,q}(\Omega)\subset
L^{q}(\Omega)$ with compact embeddings, by Aubin-Lions lemma applied to
sequence $(y^{m})_{m}$, there exists $y\in L^{q}_{loc}(0,T;W^{1,q}(\Omega))$
and a subsequence also denoted $(y^{m})_{m}$ such that:

(3.6) $y^{m}\longrightarrow y\text{ in
}L^{q}(\epsilon,T-\epsilon;W^{1,q}(\Omega))\text{ as
}m\longrightarrow\infty,\forall\epsilon>0.$

Strong convergence of $(y^{m})_{m}$ in
$L^{q}(\epsilon,T-\epsilon;W^{1,q}(\Omega))$ and weak convergence of
$(g^{m})_{m}$ in $L^{q}(Q)$, $2\leq q<\infty$ allow to pass to the limit in
the variational formulation of the problem (3.1) to conclude that $y$ is
solution to (S) corresponding to $g$. We also have in this situation , for all
$\varepsilon$ as above, weak convergence to $y$ in
$L^{q}(\epsilon,T-\epsilon;W^{2,q}(\Omega))$.

When $q=+\infty$ we have, up to a subsequence, weak-* convergence of
$(g^{m})_{m}$ to $g$ in $L^{\infty}(Q)$ and weak $L^{p}(Q)$ for all $2\leq
p<+\infty$. By the above argument we find in fact strong convergence, up to a
subsequence, of $(y^{m})_{m}$ to $y$ in
$L^{p}(\epsilon,T-\epsilon;W^{1,p}(\Omega))$, as well as weak convergence in
$L^{p}(\epsilon,T-\epsilon;W^{2,p}(\Omega))$, for $2\leq p<\infty$, and $y$ is
solution corresponding to the source $g$.

Considering that for $2\leq p<\infty$ the observation $\zeta$ is linear
continuous operator in $L(W^{2,p}(\Omega),L^{p}(\Gamma_{1}))$ we have that

$\zeta(y)=\lim\zeta(y^{m})$

weakly in $L^{q}(\epsilon,T-\epsilon;L^{q}(\Gamma_{1}))$ if $q<\infty$ and
weakly in $L^{p}(\epsilon,T-\epsilon;L^{p}(\Gamma_{1})$ for all $2\leq
p<\infty$ if $q=\infty$. By the lower semicontinuity of the norm in $L^{p}$ we
find from (3.3) that $\zeta(y)=0$.

Now we use (H5) to see that both $y$ and $\displaystyle\frac{\partial
y}{\partial n}$ are zero on $\Sigma_{1}$. At this point we consider a slightly
larger domain, $\tilde{\Omega}=\Omega\cup\\{\Gamma_{1}+B_{\delta}(0)\\}$.
Consider $\tilde{y}$ the extension with $0$ of $y$ to
$\tilde{Q}=(0,T)\times\tilde{\Omega}$ and $\tilde{g}$ also the extension with
$0$ of $g$. Observe that $\tilde{y}$ is variational solution corresponding to
$\tilde{g}$ and with null boundary conditions on the new piece of boundary. We
are now in position to apply the strong maximum principle for weak solutions
to parabolic systems obtained in [8] we get that $\tilde{y}$ is zero in
$\tilde{Q}$ and thus the source $\tilde{g}$ must be zero in $\tilde{Q}$, which
contradicts the fact that $g\not=0$.

### Comparison with the results of O. Yu Imanuvilov and M. Yamamoto

We mention here the result of O. Yu Imanuvilov and M. Yamamoto [4] obtained in
$L^{2}$ for the source of one equation. It is important to see that the shape
of $\Omega\subset\mathbb{R}^{N}$ is not restricted and the homogeneous
boundary conditions are prescribed on a part of the boundary $\Gamma_{0}$ and
the observation is made on $\Gamma_{1}=\partial\Omega\setminus\Gamma_{0}$:

(I-Y)
$\left\\{\begin{array}[]{ll}D_{t}y-\sum\limits_{j,k=1}^{N}D_{j}(a^{jk}D_{k}y)+\sum\limits_{k=1}^{N}b^{k}D_{k}y+cy=g&(0,T)\times\Omega,\\\
\beta(x)\frac{\partial y}{\partial n_{A}}+\eta(x)y=0&(0,T)\times\Gamma_{0},\\\
\end{array}\quad\right.$

For a fixed $\theta\in(0,T)$ an observation instant of time, they have
obtained $L^{2}$ boundary estimates in the following class of sources:

(3.7) $\mathcal{G}_{2,\tilde{c}}=\left\\{g\in
W^{1,1}((0,T);L^{2}(\Omega)):\,\left|\frac{\partial g(t,x)}{\partial
t}\right|\leq\tilde{c}|g(\theta,x)|,{a.e.}(t,x)\in Q\right\\}.$

In this context, their result provides estimates in terms of more measured
quantities involving the gradient and the time derivative of the solution on
the observed boundary:

######  Theorem 2.

Let $g\in\mathcal{G}_{2,\tilde{c}}$ and $y\in W^{2,1}_{2}(Q)$ a solution to
(I-Y) corresponding to $g$. Then

(3.8) $\|g\|_{L^{2}}\leq
C(\|y(\theta,\cdot)\|_{W^{2}_{2}(\Omega)}+\|y\|_{L^{2}(\Sigma_{1})}+\|\nabla
D_{t}y\|_{L^{2}(\Sigma_{1})}+\|\nabla y\|_{L^{2}(\Sigma_{1})}),$

where $\Sigma_{1}=(0,T)\times\Gamma_{1}$.

The result is based on the two following Carleman estimates:

######  Proposition 3.

For $g\in L^{2}(Q)$, there exist constants $\lambda_{0}=\lambda_{0}(\Omega),$
$s_{0}=s_{0}(\Omega)$ such that, for any $\lambda\geq\lambda_{0}$, $s\geq
s_{0}$ and some $C=C(T,\Omega)$, the following inequality holds:

(3.9)
$\displaystyle\int_{Q}\left[(s\varphi)^{p-1}\left(|D_{t}y|^{2}+|D^{2}y|^{2}\right)+(s\varphi)^{p+1}|Dy|^{2}+(s\varphi)^{p+3}|y|^{2}\right]e^{2s\alpha}dxdt$
$\displaystyle\leq C\int_{Q}(s\varphi)^{p}|g|^{2}e^{2s\alpha}dxdt+$
$\displaystyle+C\int_{[0,T]\times\Gamma_{1}}\left((s\varphi)^{p}|D_{t}y|^{2}+(s\varphi)^{p+1}|\nabla
y|^{2}+(s\varphi)^{p+3}|y|^{2}\right)e^{2s\alpha}d\sigma,\quad p\in\\{0,1\\}$

for $y\in H^{1}(0,T;L^{2}(\Omega))\cap L^{2}(0,T;H^{2}(\Omega))$ solution of
(I-Y).

Observe that if in our situation we consider homogeneous Dirichlet boundary
conditions on $\Gamma_{1}$ the Carleman estimates of O. Yu Imanuvilov and M.
Yamamoto gives our Carleman estimate as ob $\Gamma_{1}$ we would have in this
case $D_{t}y|_{\partial\Omega}=0$ and $\nabla
y|_{\partial\Omega}=\frac{\partial y}{\partial n}$. As a consequence our
result remains true for any domain without restrictions on the topology if we
consider homogeneous boundary conditions. However this is not the case if
Neumann or Robin are prescribed on $\Gamma_{1}$ as the estimate (3.8) needs
supplementary measurements on the tangential and time derivatives of the
solution on $\Gamma_{1}$.

## References

  * [1] V. Barbu.  Exact controllability of the superlinear heat equation.  Appl. Math. Optim., 42(1):73–89, 2000.
  * [2] Jean-Michel Coron, Sergio Guerrero, and Lionel Rosier.  Null controllability of a parabolic system with a cubic coupling term.  SIAM J. Control Optim., 48(8):5629–5653, 2010.
  * [3] A.V. Fursikov and O.Yu. Imanuvilov.  Controllability of evolution equations. Seoul: Seoul National Univ., 1996.
  * [4] Oleg Yu Imanuvilov and Masahiro Yamamoto.  Lipschitz stability in inverse parabolic problems by the Carleman estimate.  Inverse Probl., 14(5):1229–1245, 1998.
  * [5] O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva.  Linear and quasilinear equations of parabolic type.  Vol. 23. American Mathematical Society, Providence, R.I., 1968.
  * [6] Kévin Le Balc’h.  Controllability of a $4\times 4$ quadratic reaction-diffusion system.  J. Differential Equations, 266(6):3100–3188, 2019.
  * [7] C.-G. Lefter and E.-A. Melnig.  Reaction-diffusion systems in annular domains: source stability estimates with boundary observations.
  * [8] C.-G. Lefter and E.-A. Melnig.  Remarks on the strong maximum principle and strong invariance results for weak solutions to parabolic equations and systems.
  * [9] E.-A. Melnig.  Stability in ${L}^{q}$-norm for inverse source parabolic problems.  Journal of Inverse and Ill-posed Problems, 28(6):797–814, 2020\.
  * [10] E.-A. Melnig.  Stability in inverse source problems for nonlinear reaction-diffusion systems.  NoDEA Nonlinear Differential Equations Appl., 28(45), 2021.
  * [11] Murray H. Protter and Hans F. Weinberger.  Maximum principles in differential equations.  Springer-Verlag, New York, 1984.  Corrected reprint of the 1967 original.
  * [12] Hans F. Weinberger.  Invariant sets for weakly coupled parabolic and elliptic systems.  Rend. Mat. (6), 8:295–310, 1975.  Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday.