# Boundedness of the solutions of a kind of nonlinear parabolic systems

Emilia Anna Alfano  Department of Mathematics University of Salerno, Italy
e-mail<EMAIL_ADDRESS>,  Luisa Fattorusso  Department of Information
Engineering, Infrastructure and Sustainable Energy, Mediterranea University,
of Reggio Calabria, Italy
e-mail<EMAIL_ADDRESS>and  Lubomira Softova  Department of
Mathematics University of Salerno, Italy
e-mail<EMAIL_ADDRESS>
###### Abstract.

We deal with nonlinear systems of parabolic type satisfying componentwise
structural conditions. The nonlinear terms are Carathéodory maps having
controlled growth with respect to the solution and the gradient and the data
are in anisotropic Lebesgue spaces. Under these assumptions we obtain
essential boundedness of the weak solutions.

###### Key words and phrases:

Parabolic systems, Coercivity condition, Weak solutions, Essential boundedness

††2010 _Mathematics Subject Classification_ : Primary 35K40; Secondary 35B50

##  1\. Introduction

This paper aims the boundedness of the weak solutions to the following
nonlinear divergence form systems

${{\bm{u}}}_{t}-{\rm
div\,}{\bm{A}}(x,t,{{\bm{u}}},D{{\bm{u}}})+{{\bm{b}}}(x,t,{{\bm{u}}},D{{\bm{u}}})=0\qquad(x,t)\in
Q_{T}$ (1.1)

where $\Omega\subset\mathbb{R}^{n},$ $n\geq 1,$ is a bounded domain,
$Q_{T}=\Omega\times(0,T)$, $T>0,$ is a cylinder in
${\mathbb{R}}^{n}\times{\mathbb{R}}_{+}$ and the nonlinear operators

$\displaystyle{\bm{A}}(x,t,{\bm{u}},{\bm{z}})$ $\displaystyle:\
Q_{T}\times{\mathbb{R}}^{N}\times{\mathbb{R}}^{N\times
n}\rightarrow{\mathbb{R}}^{N\times n}$
$\displaystyle{\bm{b}}(x,t,{\bm{u}},{\bm{z}})$ $\displaystyle:\
Q_{T}\times{\mathbb{R}}^{N}\times{\mathbb{R}}^{N\times
n}\rightarrow{\mathbb{R}}^{N}$

where $\displaystyle{\bm{A}}=\\{A_{i}^{\alpha}\\}_{1\leq i\leq
n}^{1\leq\alpha\leq N}$ and ${\bm{b}}=(b^{1},\cdots,b^{N})$ are Carathéodory
maps, that is, these are measurable in $(x,t)\in Q_{T}$ for every
$({\bm{u}},{\bm{z}})\in{\mathbb{R}}^{N}\times{\mathbb{R}}^{N\times n}$ and
continuous in $({\bm{u}},{\bm{z}})$ for almost all (a.a.) $(x,t)\in Q_{T}.$
Concerning linear divergence form equations of elliptic type, it is well known
by the result of De Giorgi and Nash [15, 28] that any weak solution is locally
Hölder continuous assuming that the leading coefficients are only
$L^{\infty}.$ Almost ten years were necessary to show, via a counterexample,
that this is not true for divergence form elliptic systems under the same
minimal conditions on the coefficients because of the lack of maximum
principle (cf. [16]). Notice that Giusti and Miranda in [21] proved that the
De Giorgi-Nash result does not hold for quasi-linear systems, even if the
coefficients are analytic functions.

In general, in order to obtain essential boundedness of the solutions, it is
necessary to add some more restrictions on the structure of the operators
considered. The simplest example is given by systems in diagonal form, or the
so-called decoupled systems.

######  Example 1.1.

Let ${\bm{u}}:\Omega\rightarrow\mathbb{R}^{N}$ be a weak solution of ${\rm
div\,}({\bm{A}}(x,D{\bm{u}}))=~{}0$ in $\Omega$ with

$A_{i}^{\alpha}(x,D{\bm{u}})=\sum_{j=1}^{n}\sum_{\beta=1}^{N}\delta_{\alpha\,\beta}A_{ij}^{\alpha\,\beta}(x)D_{j}u^{\beta}$

where $\delta_{\alpha\,\beta}$ is the Kronecker delta. Then each component
$u^{\alpha}$ solves a single elliptic equation and
$\sup_{\Omega}u^{\alpha}\leq\sup_{\partial\Omega}u^{\alpha}$ for each
$\alpha=1,\ldots,N.$

As it concerns nonlinear elliptic type systems, in order to study the
regularity properties of the operator, more structural conditions are
required. For instance, divergence form operators

${\rm div\,}{\bm{A}}(x,{\bm{u}},D{\bm{u}})={\bm{b}}(x,{\bm{u}},D{\bm{u}})$
(1.2)

have been studied under various componentwise conditions.

In [29] Nečas and Stará analyzed quasi-linear systems which are diagonal for
large values of $u^{\alpha}$. Precisely, for each $\alpha=1,\ldots,N$, it is
assumed that

$0<\theta_{\alpha}\leq u^{\alpha}\Longrightarrow
A_{i}^{\alpha}(x,{\bm{u}},D{\bm{u}})=\sum_{j=1}^{n}\sum_{\beta=1}^{N}\delta_{\alpha\,\beta}A_{ij}^{\alpha\,\beta}(x,{\bm{u}})D_{j}u^{\beta}$

where the operator $\big{\\{}A_{ij}^{\alpha\beta}\big{\\}}_{1\leq i,j\leq
n}^{1\leq\alpha,\beta\leq N}$ is supposed to be bounded and elliptic. Then
each solution of the system ${\rm div\,}{\bm{A}}(x,{\bm{u}},D{\bm{u}})=0$
verifies

$\sup_{\Omega}u^{\alpha}\leq\max\\{\theta_{\alpha};\,\sup_{\partial\Omega}u^{\alpha}\\}.$

In [24] Leonetti and Petricca analyzed nonlinear elliptic systems

$-{\rm
div\,}{\bm{A}}(x,{\bm{u}},D{\bm{u}})+{\bm{b}}(x,{\bm{u}},D{\bm{u}})={\bm{f}}(x)\quad
x\in\Omega$

where they impose componentwise coercivity condition on the principal part and
positivity of the lower order terms operator for large values of $u^{\alpha}$
that is there exists $\theta_{\alpha}>0$ such that for each
$u^{\alpha}\geq\theta_{\alpha},$ $\alpha=1,\ldots,N$ one has

$\begin{cases}\nu|{{\bm{z}}}^{\alpha}|^{p}-M_{\alpha}\leq\displaystyle\sum_{i=1}^{n}A_{i}^{\alpha}(x,{\bm{u}},{{\bm{z}}})z_{i}^{\alpha}\qquad
x\in\Omega\\\\[5.0pt] 0\leq b^{\alpha}(x,{\bm{u}},{{\bm{z}}}).\end{cases}$
(1.3)

A bound for $\|{\bm{u}}\|_{L^{\infty}(\Omega,\mathbb{R}^{N})}$ has been
obtained under that hypotheses when the $x$-behavior of the nonlinear terms
${\bm{A}}(x,{\bm{u}},D{\bm{u}})$ and ${\bm{b}}(x,{\bm{u}},D{\bm{u}})$ is
controlled in Lebesgue ([24]) and Morrey ([18, 31, 33, 34]) spaces.

Concerning nonlinear elliptic and parabolic divergence form operators
satisfying quite general growth conditions, there are known results on local
boundedness of the solution, i.e. in any open set $B\Subset\Omega,$ without
any assumptions on the boundary. In this context we can mention the results of
Marcellini et al. treating as the question of local boundedness so the
Lipscitz regularity of the solutions under $p,q$-growth conditions on the
nonlinear operator, see [11, 12, 13, 25, 26] and the references therein.

In our case we interested to obtain global regularity, up to the boundary of
the cylinder $Q_{T}.$ For this goal, following the technique developed in
[23], we assume a priori that the solution is bounded on the parabolic
boundary $\partial Q=\Omega\cup S_{T}.$ Further, estimating the level sets of
the solution in cylinders with small high $Q_{\tau}=\Omega\times(t_{1},t_{2})$
with $0\leq t_{1}<t_{2}\leq T,t_{2}-t_{1}=\tau$ we can estimate the
$L^{\infty}$ norm of ${\bm{u}}$ in $Q_{\tau}.$ Then covering $Q_{T}$ with a
finite number of such sub-cylinders and applying the global boundedness result
in any of them we can proceed up to the upper boundary
$\Omega\times\\{t=T\\}.$

Some more results concerning boundedness and regularity of the solutions for
nonlinear elliptic and parabolic operators can be found in [4, 5, 6, 7, 8, 9,
10, 14, 19, 20, 22, 30, 32]. There is a vast number of regularity results,
obtained via techniques of the variational calculus that we cannot quote here
and for this we direct the reader to the exposition works of Mingione et al.
[17, 27] (see also the references therein).

As it concerns non-coercive operators, we can mention the works of Boccardo
[3] that treat nonlinear elliptic equations and [1, 2] in case of non-coercive
nonlinear equations in unbounded domain.

In what follows we use the standard notation

  * •

$x=(x_{1},\ldots,x_{n})\in{\mathbb{R}}^{n},$
$(x,t)\in{\mathbb{R}}^{n}\times{\mathbb{R}}_{+};$
${\mathcal{B}}_{\rho}(x)=\\{y\in{\mathbb{R}}^{n}:|x-y|<\rho\\}$ is a ball in
${\mathbb{R}}^{n}$ centered in $x$ and of radius $\rho>0,$
$|{\mathcal{B}}_{\rho}|\sim C\rho^{n};$

  * •

$\partial\Omega$ is the boundary of $\Omega,$ $\overline{\Omega}$ is the
closure of $\Omega,$ and $|\Omega|$ stands for the Lebesgue measure of
$\Omega;$

  * •

$Q_{T}=\Omega\times(0,T)$ denotes a cylinder with base $\Omega$ and high
$T>0;$
$S_{T}=\partial\Omega\times[0,T]$ is the lateral boundary of $Q_{T},$
$\partial Q_{T}=S_{T}\cup\\{\Omega\times\\{t=0\\}\\}$ stands for the parabolic
boundary of $Q_{T},$ and $\overline{Q}_{T}=\overline{\Omega}\times[0,T]$;

  * •

$\chi_{D}$ is the characteristic function of any set
$D\subset{\mathbb{R}}^{n+1};$

  * •

${\bm{u}}=(u^{1},\ldots,u^{N})$ is a vector function in
${\mathbb{R}}^{N},N\geq 2$,
where $u^{\alpha}(x,t):Q_{T}\rightarrow{\mathbb{R}}$,
$\forall\,\alpha=1,\dots,N;$

  * •

$u^{\alpha,k}=\max\\{u^{\alpha}(x,t)-k;0\\}$ for any $k>0$ and all $(x,t)\in
Q_{T};$

  * •

$D_{i}u^{\alpha}=\partial u^{\alpha}/\partial x_{i}$ and
$D{\bm{u}}=\\{D_{i}u^{\alpha}\\}_{i\leq n}^{\alpha\leq
N}\in\mathbb{R}^{N\times n}$ is the vector of the space derivatives of
${\bm{u}},$ $D_{t}u^{\alpha}=u_{t}^{\alpha}=\partial u^{\alpha}/\partial t;$

  * •

for any vector-function ${\bm{f}}\in L^{p}(Q,{\mathbb{R}}^{N}),p>1$ we write
$\|{\bm{f}}\|_{p;Q}$ instead of $\|{\bm{f}}\|_{L^{p}(Q;{\mathbb{R}}^{N})}.$

The letter $C$ denotes a positive constant depending on known quantities that
can vary from one occurrence to another. In addition the summation convention
on the repeated indexes is adopted.

##  2\. Definitions and auxiliary results

We say that the domain $\Omega$ satisfies the (A) condition if there exists a
positive constant $A_{\Omega}<1$ such that, for each ball
${\mathcal{B}}_{\rho}$ centered in some point $x\in\partial\Omega$ and
$\rho\leq{\rm diam\,}\Omega$, there holds

$|\Omega\cap{\mathcal{B}}_{\rho}|\leq(1-A_{\Omega})|{\mathcal{B}}_{\rho}|\,.$
(A)

For the function spaces that we are going to use we adopt the notations
introduced in [23].

  1. 1.

We say that ${{\bm{u}}}=(u^{1},\ldots,u^{N}):Q_{T}\rightarrow\mathbb{R}^{N}$
belongs to $L^{2}(Q_{T};{\mathbb{R}}^{N})$ if $u^{\alpha}\in L^{2}(Q_{T})$ for
each $\alpha=1,\ldots,N$ and

$\|{{\bm{u}}}\|_{2;Q_{T}}=\left(\int_{Q_{T}}|{{\bm{u}}}|^{2}\,dxdt\right)^{1\over
2}=\left(\int_{0}^{T}\int_{\Omega}\sum_{\alpha=1}^{N}|u^{\alpha}(x,t)|^{2}\,dxdt\right)^{1\over
2}.$

  2. 2.

For any $q,r\geq 1$ we consider the anisotropic Lebesgue space
$L^{q,r}(Q;\mathbb{R}^{N})$ equipped with mixed norm

$\|{{\bm{u}}}\|_{q,r;Q_{T}}=\left[\int_{0}^{T}\left(\int_{\Omega}\sum_{\alpha=1}^{N}|u^{\alpha}(x,t)|^{q}\,dx\right)^{r\over
q}dt\right]^{1\over r}.$

In the particular case $q=2$, $r=\infty$, we have

$\|{{\bm{u}}}\|_{2,\infty;Q_{T}}=\underset{t\in[0,T]}{{\rm
esssup\,}}\|{{\bm{u}}}(\,\cdot\,,t)\|_{2;\Omega}$

and for $q=r$ we obtain the parabolic Lebesgue space
$L^{q}(Q;\mathbb{R}^{N}).$

  3. 3.

$W_{2}^{1,0}(Q_{T};{\mathbb{R}}^{N})$ and
$W_{2}^{1,1}(Q_{T};{\mathbb{R}}^{N})$ are the Banach spaces consisting of all
${{\bm{u}}}\in L^{2}(Q_{T};{\mathbb{R}}^{N})$ for which the following norms
are finite

$\displaystyle\|{{\bm{u}}}\|_{W_{2}^{1,0}(Q_{T})}=\,$
$\displaystyle\|{{\bm{u}}}\|_{2;Q_{T}}+\|D{{\bm{u}}}\|_{2;Q_{T}}$
$\displaystyle\|{{\bm{u}}}\|_{W_{2}^{1,1}(Q_{T})}=\,$
$\displaystyle\|{{\bm{u}}}\|_{2;Q_{T}}+\|D{{\bm{u}}}\|_{2;Q_{T}}+\|{{\bm{u}}}_{t}\|_{2;Q_{T}}.$

The space $\overset{\circ}{W}{}^{1,1}_{2}(Q_{T};{\mathbb{R}}^{N})$ is the
closure of $C_{0}^{\infty}(Q_{T};\mathbb{R}^{N})$ with respect to the last
norm.

  4. 4.

$V_{2}(Q_{T};\mathbb{R}^{N})$ is a subspace of
$W_{2}^{1,0}(Q_{T};\mathbb{R}^{N})$ such that

$\|{{\bm{u}}}\|_{V_{2}(Q_{T})}=\|{{\bm{u}}}\|_{2,\infty;Q_{T}}+\|D{{\bm{u}}}\|_{2;Q_{T}}<\infty.$
(2.1)

Namely,

$V_{2}(Q_{T};{\mathbb{R}}^{N})=L^{\infty}(0,T;L^{2}(\Omega;{\mathbb{R}}^{N}))\cap
L^{2}(0,T;W_{2}^{1}(\Omega;{\mathbb{R}}^{N}))$

endowed with the norm (2.1).

  5. 5.

$V_{2}^{1,0}(Q_{T};{\mathbb{R}}^{N})$ consists of all ${{\bm{u}}}\in
V_{2}(Q_{T};\mathbb{R}^{N})$ continuous in $t$ with respect to the norm of
$L^{2}(Q_{T};{\mathbb{R}}^{N})$, that is

$\lim_{\Delta t\rightarrow 0}\|{{\bm{u}}}(\cdot,t+\Delta
t)-{{\bm{u}}}(\cdot,t)\|_{2;\Omega}=0$ (2.2)

endowed with the norm (2.1). As usual, we denote by
$\overset{\circ}{V}{}_{2}(Q_{T};{\mathbb{R}}^{N})$ the closure of
$C_{0}^{\infty}(Q_{T};{\mathbb{R}}^{n})$ with respect to the norm (2.1).

For reader’s convenience, we recall the Hölder inequality on anisotropic
Lebesgue spaces and some of its modifications (cf. [23, Ch. II]). Precisely,
for any exponents $q\geq q_{1}\geq 1,$ $r\geq r_{1}\geq 1,$ there hold

$\displaystyle\Big{|}\int_{Q_{T}}u(x,t)\,v(x,t)\,dxdt\Big{|}$
$\displaystyle\leq\|{u}\|_{q,r;Q_{T}}\|{v}\|_{\frac{q}{q-1},\frac{r}{r-1};Q_{T}}$
(2.3) $\displaystyle\|uv\|_{q_{1},r_{1};Q_{T}}$
$\displaystyle\leq\|u\|_{q,r;Q_{T}}\|v\|_{\frac{qq_{1}}{q-q_{1}},\frac{rr_{1}}{r-r_{1}};Q_{T}}.$
(2.4)

Now, we give also some properties of the spaces $V_{2}(Q_{T})$ and
$\overset{\circ}{V}{}_{2}(Q_{T}).$ Let $u\in\overset{\circ}{V}{}_{2}(Q_{T})$,
then from the Gagliardo–Nirenberg interpolation inequality for the
$\overset{\circ}{W}{}^{1}_{2}(\Omega)$-functions there exists a constant
$C_{q}=C_{q}(n,q)$ such that

$\|u(\cdot,t)\|_{q,\Omega}\leq
C_{q}\|u(\cdot,t)\|^{1-\alpha}_{2,\Omega}\|Du(\cdot,t)\|^{\alpha}_{2,\Omega}\quad\text{
for a.a. }t\in[0,T]$

with $\alpha=\frac{n}{2}-\frac{n}{q},$ and where $q$ is as in (2.6). Hence, we
have the following imbedding type inequality in
$\overset{\circ}{V}{}_{2}(Q_{T})$

$\|u\|_{q,r;Q_{T}}\leq
C_{q}\|u\|_{2,\infty;Q_{T}}^{1-\alpha}\,\|Du\|_{2,\alpha r;Q_{T}}^{\alpha}.$

Taking $\alpha={2/r},$ where $r$ is as in (2.6), we get a series of
inequalities and the first one is

$\|u\|_{q,r;Q_{T}}\leq C_{q}\|u\|_{2,\infty;Q_{T}}^{1-{2\over
r}}\,\|Du\|_{2;Q_{T}}^{2\over r}$ (2.5)

where $\displaystyle{1\over r}+{n\over 2q}={n\over 4},$ and $r$ and $q$
satisfy

$\begin{cases}r\in[2,\infty]\quad&q\in[2,{2n\over n-2}],\quad\mbox{ for }n\geq
3\\\\[4.0pt] r\in(2,\infty)\quad&q\in[2,\infty),\quad\ \,\mbox{ for
}n=2\\\\[4.0pt] r\in[4,\infty]\quad&q\in[2,\infty],\quad\ \ \mbox{ for
}n=1.\end{cases}$ (2.6)

Estimating the right-hand side of (2.5) by Young’s inequality

$ab\leq\frac{2}{r}\ a^{r\over 2}+\frac{r-2}{r}\ b^{r\over r-2},\qquad r>2$

we obtain the second inequality

$\|u\|_{q,r;Q_{T}}\leq
C_{q}{\frac{2}{r}}\,\|Du\|_{2;Q_{T}}+C_{q}\frac{r-2}{r}\,\|u\|_{2,\infty;Q_{T}}\,\leq\beta\,\|u\|_{V_{2}(Q_{T})}$
(2.7)

where $\beta=\beta(n,q,r)$ and $q$ and $r$ are as in (2.6).

One more notion that we need is that of the Steklov average. For a function
$\zeta(x,t)\in\overset{\circ}{W}{}^{1,1}_{2}(Q_{[-h,T]})$ that is zero for
$t\leq 0$ and $t\geq T-h,$ we define the Steklov average in the future as

$\zeta_{h}(x,t)={1\over h}\int_{t}^{t+h}\zeta(x,\tau)\,d\tau$ (2.8)

and the Steklov average in the past as

$\zeta_{\overline{h}}(x,t)={1\over h}\int_{t-h}^{t}\zeta(x,\tau)\,d\tau.$
(2.9)

We notice that, for each fixed $h$, $\zeta_{h}(x,t)$ is defined in
$\overline{Q}_{T-h}=\overline{\Omega}\times[0,T-h]$ and
$\zeta_{\overline{h}}(x,t)$ is defined in
$\overline{Q}_{[-h,T]}=\overline{\Omega}\times[-h,T],$ where $0<h<T.$ It is
well known that if $\zeta\in L^{q,r}(Q_{T})$ with $q,r\geq 1$, then
$\zeta_{h}$ approximates $\zeta$ with respect to the norm in $L^{q,r}.$
Furthermore, if $\zeta\in\overset{\circ}{V}{}_{2}(Q_{T})$, then
$\zeta_{h}\in\overset{\circ}{W}{}^{1,1}_{2}(Q_{T-h})$ and $\zeta_{h}$
approximates $\zeta$ with respect to the norm in $V_{2}$ (cf. [23, Ch. II]).
Moreover, if we take $\zeta\in\overset{\circ}{V}{}_{2}(Q_{T}),$ then
$\zeta_{\overline{h}}\in\overset{\circ}{W}{}^{1,1}_{2}(Q_{[h,T]}).$

We recall that the Steklov averages satisfy these two properties:

$\begin{split}\left(\zeta_{\overline{h}}\right)_{t}&=\left(\zeta_{t}\right)_{\overline{h}}\\\
\int_{0}^{T}u(x,t)\,\zeta_{\overline{h}}(x,t)\,dt&=\int_{0}^{T-h}u_{h}(x,t)\,\zeta(x,t)\,dt.\end{split}$
(2.10)

The last equation holds for any square summable function, such that
$\zeta(x,t)=0$ for $t\leq 0$ and for $t\geq T-h.$

The following result permits to give a total estimate for the maximum of the
modulus of the solutions in the whole domain of the definition. Let $u$ in
$V_{2}(Q_{T})$. Denote by $u^{k}=\max\\{u(x,t)-k;0\\},$ $k>0$ for all
$(x,t)\in Q_{T}$ and define the set

$A^{k}(t)=\\{x\in\Omega:\,u(x,t)>k,\,t\in[0,T]\\}.$

######  Theorem 2.1 ([23], Ch. II, Theorem 6.1).

Suppose that $\|u\|_{\infty;S_{T}}\leq M_{0}$, $M_{0}\geq 0$, and

$\|u^{k}\|_{V_{2}^{1,0}(Q_{T})}\leq Ck\mu^{{1+\varkappa}\over r}(k)$ (2.11)

hold for $k\geq M_{0}$, where $\mu(k)=\int_{0}^{T}|A^{k}(t)|^{r\over q}\,dt$,
$q$ and $r$ as in (2.6) and $0<\varkappa<1$. Then

$\|u\|_{\infty;Q_{T}}\leq
2M_{0}\left[1+2^{{2\over\varkappa}+{1\over\varkappa^{2}}}(\beta
c)^{1+{1\over\varkappa}}T^{{1+\varkappa\over r}}|\Omega|^{{1+\varkappa\over
q}}\right]$

where $\beta$ is the constant from (2.7) and $C$ is an arbitrary constant.

##  3\. Statement of the problem

Recall that we are going to study the boundedness of the weak solutions to the
system

$u^{\alpha}_{t}-\sum_{i=1}^{n}D_{i}\big{(}A_{i}^{\alpha}(x,t,{{\bm{u}}},D{{\bm{u}}})\big{)}+{b^{\alpha}}(x,t,{{\bm{u}}},D{{\bm{u}}})=0$
(3.1)

in $Q_{T}$ with $\alpha=1,\cdots,N$. In order to obtain our results we impose
the following assumptions regarding the structure and the asymptotic behaviour
of the nonlinear operators ${\bm{A}}$ and ${\bm{b}}.$

  * $(H_{1})$

Controlled growth conditions: there exist a positive constant $\Lambda$ and
functions $\varphi_{1}\in L^{2}(Q_{T})$ and $\varphi_{2}\in
L^{q_{0},r_{0}}(Q_{T}),$ with $r_{0},q_{0}\geq 1,$ ${1\over r_{0}}+{n\over
2q_{0}}=1+{n\over 4},$ such that

$\displaystyle|{{\bm{A}}}(x,t,{\bm{u}},{\bm{z}})|$
$\displaystyle\leq\Lambda\left(\varphi_{1}(x,t)+|{{\bm{u}}}|^{\frac{n+2}{n}}+|{{\bm{z}}}|\right)$
(3.2) $\displaystyle|{{\bm{b}}}(x,t,{\bm{u}},{\bm{z}})|$
$\displaystyle\leq\Lambda\left(\varphi_{2}(x,t)+|{\bm{u}}|^{1+\frac{4}{n}}+|{\bm{z}}|^{\frac{n+4}{n+2}}\right)$
(3.3)

for a.a. $(x,t)\in Q_{T}$ and all
$({\bm{u}},{\bm{z}})\in{\mathbb{R}}^{N}\times{\mathbb{R}}^{N\times n}.$

  * $(H_{2})$

Componentwise coercivity of the differential operator: there exist positive
constants $\nu,\mu$ and $k_{0}$ such that for all $\alpha\in\\{1,\ldots,N\\}$
and for $|{u^{\alpha}}|\geq k_{0}$, it holds

$\sum_{i=1}^{n}A^{\alpha}_{i}(x,t,{\bm{u}},{\bm{z}})z^{\alpha}_{i}\geq\nu|{{\bm{z}}}^{\alpha}|^{2}-\mu|u^{\alpha}|^{\delta}-{|u^{\alpha}|}^{2}\psi(x,t)$
(3.4)

for a.a. $(x,t)\in Q_{T}.$ The function $\psi\in L^{q_{2},r_{2}}(Q_{T})$ is as
in (3.11).

  * $(H_{3})$

Componentwise sign condition: for $|{u^{\alpha}}|\geq k_{0}$ we have

$-b^{\alpha}(x,t,{{\bm{u}}},{{\bm{z}}})u^{\alpha}\leq|u^{\alpha}|^{2}\psi(x,t)+\mu|u^{\alpha}|^{\delta}+\Lambda|{\bm{z}}^{\alpha}|^{2}$
(3.5)

for a.a. $(x,t)\in Q_{T}$ and for all ${{\bm{z}}}\in{\mathbb{R}}^{N\times n}.$

The vector function ${\bm{u}}\in V^{1,0}_{2}(Q_{T};{\mathbb{R}}^{N})$ is a
weak solution of (3.1) if, for any $0\leq t_{0}\leq t_{1}\leq T$ and each
$\alpha=1,\ldots,N$ it holds

$\begin{split}-\int_{t_{0}}^{t_{1}}\int_{\Omega}&u^{\alpha}(x,t)\eta_{t}^{\alpha}(x,t)\,dxdt\\\
&+\int_{t_{0}}^{t_{1}}\int_{\Omega}\Big{\\{}\sum_{i=1}^{n}A^{\alpha}_{i}(x,t,{\bm{u}},D{\bm{u}})D_{i}\eta^{\alpha}(x,t)\\\
&+b^{\alpha}(x,t,{{\bm{u}}},D{{\bm{u}}})\eta^{\alpha}(x,t)\Big{\\}}\,dxdt\\\
&+\int_{\Omega}u^{\alpha}(x,t)\eta^{\alpha}(x,t)\Big{|}_{t_{0}}^{t_{1}}\,dx=0\end{split}$
(3.6)

for any
$\bm{\eta}(x,t)\in{\overset{\circ}{W}}{}^{1,1}_{2}(Q_{T};{\mathbb{R}}^{N}).$
The definition (3.6) has sense if the integrals are convergent and this is
possible because of the growth conditions $(H_{1}).$ Now, we take as test
function in (3.6) the Steklov average

$\zeta^{\alpha}_{\overline{h}}={1\over
h}\int_{t-h}^{t}\zeta^{\alpha}(x,\tau)\,d\tau$

where $\zeta^{\alpha}\in\overset{\circ}{V}{}_{2}^{1,0}(Q_{T-h}).$ Then, (3.6)
becomes

$\begin{split}-\int_{t_{0}}^{t_{1}}\int_{\Omega}&u^{\alpha}(x,t)({\zeta_{\overline{h}}^{\alpha}})_{t}(x,t)\,dxdt\\\
&+\int_{t_{0}}^{t_{1}}\int_{\Omega}\Big{\\{}\sum_{i=1}^{n}A_{i}^{\alpha}(x,t,{\bm{u}},D{\bm{u}})\,D_{i}\,\zeta_{\overline{h}}^{\alpha}(x,t)\\\
&+b^{\alpha}(x,t,{{\bm{u}}},D{{\bm{u}}})\,\zeta_{\overline{h}}^{\alpha}(x,t)\Big{\\}}\,dxdt\\\
&+\int_{\Omega}u^{\alpha}(x,t)\zeta^{\alpha}_{\overline{h}}(x,t)\bigg{|}_{t_{0}}^{t_{1}}\,dx=0\end{split}$
(3.7)

where $0\leq t_{0}\leq t_{1}\leq T-h.$ By integration by parts and (2.10) the
first term in (3.7) becomes

$\displaystyle-\int_{t_{0}}^{t_{1}}\\!\\!\int_{\Omega}$ $\displaystyle
u^{\alpha}(x,t)\left(\zeta_{\overline{h}}^{\alpha}\right)_{t}(x,t)\,dxdt=-\int_{t_{0}}^{t_{1}}\\!\\!\int_{\Omega}u^{\alpha}(x,t)\left(\zeta_{t}^{\alpha}\right)_{\overline{h}}(x,t)\,dxdt$
$\displaystyle=-\int_{t_{0}}^{t_{1}}\\!\\!\int_{\Omega}u^{\alpha}_{h}(x,t)\zeta_{t}^{\alpha}(x,t)\,dxdt$
$\displaystyle=-\int_{\Omega}u^{\alpha}_{h}(x,t)\zeta^{\alpha}(x,t)\bigg{|}_{t_{0}}^{t_{1}}\,dx+\int_{t_{0}}^{t_{1}}\\!\\!\int_{\Omega}{u_{ht}^{\alpha}}(x,t)\zeta^{\alpha}(x,t)\,dxdt$
$\displaystyle=-\int_{\Omega}u^{\alpha}(x,t)\zeta_{\overline{h}}^{\alpha}(x,t)\bigg{|}_{t_{0}}^{t_{1}}\,dx+\int_{t_{0}}^{t_{1}}\\!\\!\int_{\Omega}{u_{ht}^{\alpha}}(x,t)\zeta^{\alpha}(x,t)\,dxdt$

where $u_{h}^{\alpha}$ is the Steklov averege of $u^{\alpha}$ (see (2.8)).
Hence,

$\begin{split}\int_{t_{0}}^{t_{1}}\\!\\!\int_{\Omega}&\Big{[}u_{ht}^{\alpha}(x,t)\zeta^{\alpha}(x,t)+\sum_{i=1}^{n}A_{i}^{\alpha}(x,t,{\bm{u}},D{\bm{u}})D_{i}\zeta_{\overline{h}}^{\alpha}(x,t)\\\
&+b^{\alpha}(x,t,{\bm{u}},D{\bm{u}})\zeta_{\overline{h}}^{\alpha}(x,t)\Big{]}\,dxdt=0.\end{split}$
(3.8)

For some positive constant $k$ we take

$\zeta^{\alpha}(x,t)=\xi^{2}(x,t)\max\\{u_{h}^{\alpha}(x,t)-k;0\\}=:\xi^{2}u_{h}^{\alpha,k}$
(3.9)

where $\xi(x,t)$ is an arbitrary nonnegative continuous piecewise-smooth
function that is equal to zero on the lateral boundary $S_{T}.$ Observing that

$D_{t}u^{\alpha,k}=D_{t}u^{\alpha},\quad D_{i}u^{\alpha,k}=D_{i}u^{\alpha}$

we transform the first term in (3.8) as follows

$\displaystyle\int_{t_{0}}^{t_{1}}\int_{\Omega}u_{ht}^{\alpha}\,u_{h}^{\alpha,k}\,\xi^{2}\,dxdt$
$\displaystyle={1\over
2}\int_{\Omega}\big{(}u_{h}^{\alpha,k}\big{)}^{2}\xi^{2}(x,t)\Big{|}_{t_{0}}^{t_{1}}\,dx$
$\displaystyle-\int_{t_{0}}^{t_{1}}\int_{\Omega}\big{(}u_{h}^{\alpha,k}\big{)}^{2}\xi(x,t)\xi_{t}(x,t)\,dxdt.$

Now, we pass to the limit in (3.8) as $h\rightarrow 0$, obtaining

$\begin{split}{1\over
2}\,&\|u^{\alpha,k}(\,\cdot,t)\xi(\,\cdot,t)\|_{2,\Omega}^{2}\Big{|}_{t_{0}}^{t_{1}}+\int_{t_{0}}^{t_{1}}\int_{\Omega}\Big{[}-\left(u^{\alpha,k}\right)^{2}\xi(x,t)\,\xi_{t}(x,t)\\\
&+\sum_{i=1}^{n}A_{i}^{\alpha}(x,t,{\bm{u}},D{\bm{u}})\,D_{i}(u^{\alpha,k}\,\xi^{2}(x,t))\\\
&+b^{\alpha}(x,t,{{\bm{u}}},D{{\bm{u}}})\,u^{\alpha,k}\,\xi^{2}(x,t)\Big{]}\,dxdt=0.\end{split}$
(3.10)

The validity of the limit in all of the terms follows by the properties of the
Steklov average (see [23]).

In what follows, we assume that the weak solution ${\bm{u}}$ of (3.1) belongs
to $L^{q_{1},r_{1}}(Q_{T},{\mathbb{R}}^{N})$ where the exponents
$q_{1},r_{1}>1$ and the other parameters to be used in the sequel, satisfy

$\begin{cases}\text{ for }n\geq 2&\\\\[5.0pt]
(\delta-2)\left(\dfrac{1}{r_{1}}+\dfrac{n}{2q_{1}}\right)<1,\
&\delta>2,\quad\dfrac{1}{r_{2}}+\dfrac{n}{2q_{2}}<1,\\\\[10.0pt]
\dfrac{q_{1}}{\delta-2},\,q_{2}\in\left(\dfrac{n}{2},\infty\right],&\dfrac{r_{1}}{\delta-2},\,r_{2}\in\left(1,\infty\right]\\\\[10.0pt]
\psi\in
L^{q_{2},r_{2}}(Q_{T})&q_{2}\in\left(\dfrac{n}{2},\infty\right],\,r_{2}\in(1,\infty]\\\\[10.0pt]
\text{ for }n=1&\\\\[5.0pt]
\dfrac{q_{1}}{\delta-2},\,q_{2}\in[1,\infty],&\dfrac{r_{1}}{\delta-2},\,r_{2}\in(1,\infty)\\\\[10.0pt]
\psi\in
L^{q_{2},r_{2}}(Q_{T})&q_{2}\in[1,\infty],\,r_{2}\in(1,\infty).\end{cases}$
(3.11)

Suppose that the solution ${\bm{u}}$ is essentially bounded on the parabolic
boundary $\partial Q_{T}$ with some constant $M_{0}>0,$ that is
$\|{\bm{u}}\|_{\infty;\,\partial Q_{T}}\leq M_{0};$ we take
$k\geq\max\\{k_{0},M_{0}\\},$ where $k_{0}>0$ is the constant from $(H_{2}),$
and observe that $u^{\alpha,k}$ is equal to zero on the parabolic boundary
$\partial Q_{T}$. Thus, in (3.10) we can choose $t_{0}=0$ and $\xi=1,$ and
hence $\xi_{t}=0,$ that gives

$\begin{split}\frac{1}{2}\|{u}^{\alpha,k}(\,\cdot,t_{1})\|^{2}_{2,A^{\alpha,k}(t_{1})}&+\int_{Q^{\alpha,k}_{t_{1}}}\Big{[}\sum_{i=1}^{n}A^{\alpha}_{i}(x,t,{{\bm{u}}},D{{\bm{u}}})\,D_{i}u^{\alpha,k}\\\
&+b^{\alpha}(x,t,{{\bm{u}}},D{{\bm{u}}})\,u^{\alpha,k}\Big{]}\,dxdt=0\end{split}$
(3.12)

where the integration is taken on the sets

$\displaystyle A^{\alpha,k}(t_{1})$
$\displaystyle=\\{x\in\Omega:u^{\alpha}(x,t_{1})>k,\;t_{1}\in[0,T]\\}$
$\displaystyle Q^{\alpha,k}_{t_{1}}$
$\displaystyle=\\{(x,t)\in\Omega\times[0,t_{1}]:u^{\alpha}(x,t)>k\\}.$

##  4\. Main result

The conditions imposed above permit us to obtain a componentwise maximum
principle and, hence, also essential boundedness of the solution of (3.1).

######  Theorem 4.1.

Let ${\bm{u}}\in V_{2}^{1,0}(Q_{T},{\mathbb{R}}^{N})\cap
L^{q_{1},r_{1}}(Q_{T},{\mathbb{R}}^{N})$ be a weak solution of the system
(3.1) which is essentially bounded on the boundary $\partial Q_{T}.$ Assume
that the terms ${{\bm{A}}}(x,t,{{\bm{u}}},{{\bm{z}}})$ and
${{\bm{b}}}(x,t,{{\bm{u}}},{{\bm{z}}})$ satisfy conditions (3.2)-(3.5) with
$\nu-\Lambda>0.$ Then

$\|u^{\alpha}\|_{\infty;Q_{T}}\leq K$ (4.1)

with a constant $K$ depending on known quantities and
$\|{u^{\alpha}}\|_{q_{1},r_{1};Q_{T}}.$ If we take $\mu=0$ in (3.4)-(3.5),
then the constant $K$ does not depend on the last norm.

###### Proof.

By (3.12), taking into account $\displaystyle{u^{\alpha,k}\over
u^{\alpha}}<1$, (3.4) and (3.5), we get

$\begin{split}\frac{1}{2}&\|u^{\alpha,k}\|^{2}_{2,A^{\alpha,k}(t_{1})}+\int_{Q_{t_{1}}^{\alpha,k}}\left(\nu|Du^{\alpha}|^{2}-\mu|{u^{\alpha}}|^{\delta}-{|u^{\alpha}|}^{2}\psi\right)\,dxdt\\\
&\leq-\int_{Q_{t_{1}}^{\alpha,k}}b^{\alpha}(x,t,{{\bm{u}}},D{{\bm{u}}})\,u^{\alpha}(x,t)\displaystyle\frac{u^{\alpha,k}(x,t)}{u^{\alpha}(x,t)}\,dxdt\\\
&\leq\int_{Q_{t_{1}}^{\alpha,k}}\left(|u^{\alpha}|^{2}\psi+\mu|u^{\alpha}|^{\delta}+\Lambda|Du^{\alpha}|^{2}\right)\,dxdt.\end{split}$
(4.2)

Hence

$\begin{split}\frac{1}{2}&\|u^{\alpha,k}\|^{2}_{2,A^{\alpha,k}(t_{1})}+(\nu-\Lambda)\int_{Q_{t_{1}}^{\alpha,k}}|Du^{\alpha}|^{2}\,dxdt\\\
&\leq
2\int_{Q_{t_{1}}^{\alpha,k}}\left(\mu|u^{\alpha}|^{\delta}+|u^{\alpha}|^{2}\psi\right)\,dxdt.\end{split}$
(4.3)

Keeping in mind that $\nu-\Lambda>0$, we obtain

$\|u^{\alpha,k}\|^{2}_{2;A^{\alpha,k}(t_{1})}+\|Du^{\alpha}\|^{2}_{2;Q^{\alpha,k}_{t_{1}}}\leq
C\int_{Q^{\alpha,k}_{t_{1}}}\left(|u^{\alpha}|^{\delta}+|u^{\alpha}|^{2}\psi\right)\,dxdt$
(4.4)

where $C$ depends on $\mu$ and $\nu-\Lambda.$ Let us estimate the integrals in
the right-hand side. For the first one, using the inequalities in §2, we
obtain

$\begin{split}I_{1}&=\int_{Q_{t_{1}}^{\alpha,k}}|u^{\alpha}|^{\delta-2}\cdot|u^{\alpha}|^{2}\,dxdt\\\
&\leq\left[\int_{0}^{t_{1}}\left(\int_{A^{\alpha,k}(t_{1})}|u^{\alpha}|^{q_{1}}dx\right)^{\frac{r_{1}}{q_{1}}}dt\right]^{\frac{\delta-2}{r_{1}}}\\\
&\quad\times\left[\int_{0}^{t_{1}}\left(\int_{A^{\alpha,k}(t_{1})}|u^{\alpha}|^{\overline{q}_{1}}dx\right)^{\frac{\overline{r}_{1}}{\overline{q}_{1}}}dt\right]^{\frac{2}{\overline{r}_{1}}}\\\
&=\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\|u^{\alpha}\|^{2}_{\overline{q}_{1},\overline{r}_{1},Q_{t_{1}}^{\alpha,k}}\\\
&\leq
2\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\biggl{[}\|u^{\alpha}-k\|^{2}_{\overline{q}_{1},\overline{r}_{1},Q_{t_{1}}^{\alpha,k}}\\\
&\qquad\qquad+k^{2}\left(\int_{0}^{t_{1}}(|A^{\alpha,k}(t)|)^{\frac{\overline{r}_{1}}{\overline{q}_{1}}}\,dt\right)^{\frac{2}{\overline{r}_{1}}}\bigg{]}\end{split}$
(4.5)

where $\displaystyle\overline{q}_{1}={2q_{1}\over q_{1}-(\delta-2)}$,
$\displaystyle\overline{r}_{1}={2r_{1}\over r_{1}-(\delta-2)}$. We notice that

$\begin{cases}\displaystyle 2\leq\overline{r}_{1}<\infty,\quad
2\leq\overline{q}_{1}<{2n\over n-2}\quad&\mbox{ for }n\geq 2\\\ \displaystyle
2<\overline{r}_{1}<\infty,\quad 2\leq\overline{q}_{1}\leq\infty\quad&\mbox{
for }n=1\end{cases}$ (4.6)

and $\displaystyle{1\over\overline{r}_{1}}+{n\over 2\overline{q}_{1}}>{n\over
4}.$ If $n>2$ we can find an opportune $0<\varkappa_{1}<1,$ such that
$\displaystyle{1\over\overline{r}_{1}}+{n\over 2\overline{q}_{1}}={n\over
4}(1+\varkappa_{1}).$ In case of $n=2$ we consider
$\overline{q}_{1}\in[2,\infty).$ Introduce

$\widehat{q}_{1}=\overline{q}_{1}\left(1+\varkappa_{1}\right)\quad\text{and}\quad\widehat{r}_{1}=\overline{r}_{1}\left(1+\varkappa_{1}\right).$

Direct calculations show that $\widehat{q}_{1}$ and $\widehat{r}_{1}$ verify
(2.6) and by (2.4) we obtain

$\begin{split}\|u^{\alpha,k}\|^{2}_{\overline{q}_{1},\overline{r}_{1};Q_{t_{1}}^{\alpha,k}}&\leq\|u^{\alpha,k}\|^{2}_{\widehat{q}_{1},\widehat{r}_{1};Q_{t_{1}}^{\alpha,k}}\,\|\chi_{Q_{t_{1}}^{\alpha,k}}\|^{2}_{{\widehat{q}_{1}\over\varkappa_{1}},{\widehat{r}_{1}\over\varkappa_{1}};Q^{\alpha,k}_{t_{1}}}\\\
&\leq\|u^{\alpha,k}\|^{2}_{\widehat{q}_{1},\widehat{r}_{1};Q_{t_{1}}^{\alpha,k}}\,\left(t_{1}^{{1\over\widehat{r}_{1}}}|\Omega|^{{1\over\widehat{q}_{1}}}\right)^{2\varkappa_{1}}.\end{split}$
(4.7)

In the case $q_{1}=\delta-2,$ which is possible if $n=1,$ we obtain
$\overline{q}_{1}=\infty$ and hence also $\widehat{q}_{1}=\infty.$ Then (4.7)
becames

$\|u^{\alpha,k}\|^{2}_{\infty,\overline{r}_{1};Q_{t_{1}}^{\alpha,k}}\leq\|u^{\alpha,k}\|^{2}_{\infty,\widehat{r}_{1};Q_{t_{1}}^{\alpha,k}}\,t_{1}^{{2\varkappa_{1}\over\widehat{r}_{1}}}.$
(4.8)

This permits us to apply (2.7). In the case $n\geq 2$ we obtain

$\begin{split}I_{1}&\leq
2\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\Biggl{[}\beta^{2}\|u^{\alpha,k}\|_{V_{2}^{1,0}(Q_{t_{1}}^{\alpha,k})}^{2}\\\
&\quad\times\left(\int_{0}^{t_{1}}(|A^{\alpha,k}(t)|)^{\frac{\widehat{r}_{1}}{\widehat{q}_{1}}}dt\right)^{\frac{2}{\overline{r}_{1}}-\frac{2}{\widehat{r}_{1}}}\\\
&\left.+k^{2}\left(\int_{0}^{t_{1}}(|A^{\alpha,k}(t)|)^{\frac{\widehat{r}_{1}}{\widehat{q}_{1}}}dt\right)^{\frac{2(1+\varkappa_{1})}{\widehat{r}_{1}}}\right]\\\
&\leq
2\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\Bigg{\\{}\beta^{2}\|u^{\alpha,k}\|_{V_{2}^{1,0}(Q_{t_{1}}^{\alpha,k})}^{2}\big{(}t_{1}^{\frac{1}{\widehat{r}_{1}}}|\Omega|^{\frac{1}{\widehat{q}_{1}}}\big{)}^{2\varkappa_{1}}\\\
&+k^{2}\left(\int_{0}^{t_{1}}\,|A^{\alpha,k}(t)|^{\frac{\widehat{r}_{1}}{\widehat{q}_{1}}}\,dt\right)^{\frac{2(\varkappa_{1}+1)}{\widehat{r}_{1}}}\Bigg{\\}}\end{split}$
(4.9)

where $\beta=\beta(n,\overline{q}_{1},\overline{r}_{1})$ and
$\displaystyle{\widehat{q}_{1}\over\overline{q}_{1}}={\widehat{r}_{1}\over\overline{r}_{1}}=\varkappa_{1}+1.$

If $q_{1}=\delta-2,$ in view of (4.8), we have

$\begin{split}I_{1}&\leq
2\|u^{\alpha}\|^{\delta-2}_{\delta-2,r_{1},Q_{t_{1}}^{\alpha,k}}\left(\beta^{2}\|u^{\alpha,k}\|_{V_{2}^{1,0}(Q_{t_{1}}^{\alpha,k})}^{2}t_{1}^{\frac{2\varkappa_{1}}{\widehat{r}_{1}}}+k^{2}\,t_{1}^{\frac{2(\varkappa_{1}+1)}{\widehat{r}_{1}}}\right).\end{split}$
(4.10)

We trate $I_{2}$ analogously. By the Hölder inequality (2.3), we obtain

$\begin{split}I_{2}=\int_{Q_{t_{1}}^{\alpha,k}}|u^{\alpha}|^{2}\psi\,dxdt&\leq\|\psi\|_{q_{2},r_{2},Q_{t_{1}}^{\alpha,k}}\,\|u^{\alpha}\|_{\overline{q}_{2},\overline{r}_{2},Q^{\alpha,k}_{t_{1}}}^{2}\end{split}$
(4.11)

where $\displaystyle\overline{q}_{2}=\frac{2q_{2}}{q_{2}-1}$,
$\displaystyle\overline{r}_{2}=\frac{2r_{2}}{r_{2}-1}$. We notice that

$\begin{cases}\displaystyle{1\over\overline{r}_{2}}+{n\over
2\overline{q}_{2}}>{n\over 4}&\\\\[6.0pt]
2\leq\overline{q}_{2}<\dfrac{2n}{n-2},\quad
2\leq\overline{r}_{2}<\infty,\quad&\mbox{ for }n\geq 2\\\\[6.0pt]
2\leq\overline{q}_{2}\leq\infty,\quad\quad\
\,2<\overline{r}_{2}<\infty,\quad&\mbox{ for }n=1.\end{cases}$

As before, we can find $0<\varkappa_{2}<1$ such that
$\displaystyle{1\over\overline{r}_{2}}+{n\over 2\overline{q}_{2}}={n\over
4}(1+\varkappa_{2})$ and $\widehat{q}_{2}=\overline{q}_{2}(1+\varkappa_{2})$
and $\widehat{r}_{2}=\overline{r}_{2}(1+\varkappa_{2})$ verify (2.6) and by
(2.4)

$\displaystyle\|u^{\alpha,k}\|^{2}_{\overline{q}_{2},\overline{r}_{2};Q_{t_{1}}^{\alpha,k}}$
$\displaystyle\leq\|u^{\alpha,k}\|^{2}_{\widehat{q}_{2},\widehat{r}_{2};Q_{t_{1}}^{\alpha,k}}\,\|\chi_{Q_{t_{1}}^{\alpha,k}}\|^{2}_{{\widehat{q}_{2}\over\varkappa_{2}},{\widehat{r}_{2}\over\varkappa_{2}};Q^{\alpha,k}_{t_{1}}}$
$\displaystyle\leq\|u^{\alpha,k}\|^{2}_{\widehat{q}_{2},\widehat{r}_{2};Q_{t_{1}}^{\alpha,k}}\,\left(t_{1}^{{1\over\widehat{r}_{2}}}|\Omega|^{{1\over\widehat{q}_{2}}}\right)^{2\varkappa_{2}}.$

Hence, using (2.7), we get

$\begin{split}I_{2}\leq
2\|\psi\|_{q_{2},r_{2};Q_{T}}&\,\bigg{\\{}\beta^{2}\|u^{\alpha,k}\|^{2}_{V_{2}^{1,0}(Q^{\alpha,k}_{t_{1}})}\big{(}t_{1}^{\frac{1}{\widehat{r}_{2}}}|\Omega|^{\frac{1}{\widehat{q}_{2}}}\big{)}^{2\varkappa_{2}}\\\
&+k^{2}\left[\int_{0}^{t_{1}}|A^{\alpha,k}(t)|^{\frac{\widehat{r}_{2}}{\widehat{q}_{2}}}\,dt\right]^{2(\varkappa_{2}+1)\over\widehat{r}_{2}}\bigg{\\}}.\end{split}$
(4.12)

In case of $q_{2}=1$ and hence $\overline{q}_{2}=\infty,$ $n=1,$ we obtain

$\begin{split}I_{2}\leq
2\|\psi\|_{q_{2},r_{2};Q_{T}}&\,\left(\beta^{2}\|u^{\alpha,k}\|^{2}_{V_{2}^{1,0}(Q^{\alpha,k}_{t_{1}})}\,t_{1}^{\frac{{2\varkappa_{2}}}{\widehat{r}_{2}}}+k^{2}\,t_{1}^{2(\varkappa_{2}+1)\over\widehat{r}_{2}}\right).\end{split}$
(4.13)

Combine (4.9), (4.12) and (4.4), we obtain

$\begin{split}\|u^{\alpha,k}&\|^{2}_{V_{2}^{1,0}(Q_{t_{1}}^{\alpha,k})}\\\
&\leq
C\Biggl{\\{}\|u^{\alpha,k}\|_{V_{2}^{1,0}(Q_{t_{1}}^{\alpha,k})}^{2}\Bigg{[}\beta^{2}\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\big{(}t_{1}^{\frac{1}{\widehat{r}_{1}}}|\Omega|^{\frac{1}{\widehat{q}_{1}}}\big{)}^{2\varkappa_{1}}\\\
&+\beta^{2}\|\psi\|_{q_{2},r_{2},Q_{T}}\big{(}t_{1}^{\frac{1}{\widehat{r}_{2}}}|\Omega|^{\frac{1}{\widehat{q}_{2}}}\big{)}^{2\varkappa_{2}}\Bigg{]}\\\
&+\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\,k^{2}\left(\int_{0}^{t_{1}}\,|A^{\alpha,k}(t)|^{\frac{\widehat{r}_{1}}{\widehat{q}_{1}}}\,dt\right)^{\frac{2(\varkappa_{1}+1)}{\widehat{r}_{1}}}\\\
&+k^{2}\|\psi\|_{q_{2},r_{2},Q_{T}}\left(\int_{0}^{t_{1}}|A^{\alpha,k}(t)|^{\frac{\widehat{r}_{2}}{\widehat{q}_{2}}}\,dt\right)^{2(\varkappa_{2}+1)\over\widehat{r}_{2}}\Biggl{\\}}.\end{split}$
(4.14)

Taking $t_{1}$ such small that

$C\beta^{2}\left(\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\left(t_{1}^{\frac{1}{\widehat{r}_{1}}}|\Omega|^{\frac{1}{\widehat{q}_{1}}}\right)^{2\varkappa_{1}}+\|\psi\|_{q_{2},r_{2},Q_{T}}\left(t_{1}^{\frac{1}{\widehat{r}_{2}}}|\Omega|^{\frac{1}{\widehat{q}_{2}}}\right)^{2\varkappa_{2}}\right)<{1\over
2}$

we obtain

$\begin{split}\|u^{\alpha,k}\|^{2}&{}_{V_{2}^{1,0}(Q_{t_{1}}^{\alpha,k})}\\\
&\leq
C\,k^{2}\Bigg{[}\|u^{\alpha}\|^{\delta-2}_{q_{1},r_{1},Q_{t_{1}}^{\alpha,k}}\,\left(\int_{0}^{t_{1}}\,|A^{\alpha,k}(t)|^{\frac{\widehat{r}_{1}}{\widehat{q}_{1}}}\,dt\right)^{\frac{2(\varkappa_{1}+1)}{\widehat{r}_{1}}}\\\
&+\|\psi\|_{q_{2},r_{2},Q_{T}}\left(\int_{0}^{t_{1}}|A^{\alpha,k}(t)|^{\frac{\widehat{r}_{2}}{\widehat{q}_{2}}}\,dt\right)^{2(\varkappa_{2}+1)\over\widehat{r}_{2}}\Bigg{]}.\end{split}$
(4.15)

Analogously in case of $n=1$.

Since (4.15) is equivalent to (2.11) in Theorem 2.1 we deduce that
$u^{\alpha}$ is essentially bounded from above in a cylinder with high $t_{1}$
which is small enough. In order to extend this result in the whole cylinder we
consider $u^{\alpha}$ successively in $Q_{2}=\Omega\times(t_{1},2t_{1})$,
$Q_{3}=\Omega\times(2t_{1},3t_{1})$ and so on, covering in such way the whole
cylinder $Q_{T}.$

Moreover, it is possible to estimate $u^{\alpha}(x,t)$ from below. To this
aim, we apply the result just obtained to the function
$\tilde{u}^{\alpha}(x,t)=-u^{\alpha}(x,t)$ which verifies equations similar to
$u^{\alpha}(x,t)$, i.e.

${\tilde{u}^{\alpha}}_{t}-{\rm
div\,}\tilde{A}^{\alpha}(x,t,\tilde{{\bm{u}}},D\tilde{{\bm{u}}})+\tilde{b}^{\alpha}(x,t,\tilde{{\bm{u}}},D\tilde{{\bm{u}}})=0\qquad(x,t)\in
Q_{T},$

where

$\tilde{A}^{\alpha}(x,t,\tilde{{\bm{u}}},D\tilde{{\bm{u}}})=-{A}^{\alpha}(x,t,-\tilde{{\bm{u}}},-D\tilde{{\bm{u}}})$

and

$\tilde{b}^{\alpha}(x,t,\tilde{{\bm{u}}},D\tilde{{\bm{u}}})=-{b}^{\alpha}(x,t,-\tilde{{\bm{u}}},-D\tilde{{\bm{u}}})$

satisfy the conditions (3.11)-(3.5). Thus, by Theorem 2.1, we obtain

$\|u^{\alpha}\|_{\infty;Q_{T}}\leq K,$

where
$K=K(M_{0},k_{0},q_{i},r_{i},n,\varkappa_{i},|\Omega|,T,\|u^{\alpha}\|_{q_{1},r_{1};Q_{T}}).$
∎

### Acknowledgments.

All the authors are members of INDAM - GNAMPA. The research of L. Softova is
partially supported by the project GNAMPA 2020 ”Elliptic operators with
unbounded and singular coefficients on weighted $L^{p}$ spaces”.
The research of E.A. Alfano is partially supported by the project ALPHA-MENTE,
Lotto1/Ambito AV01.

The authors are very indebted to the referee for the valuable remarks that
improve the exposition of the paper.

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# Perturbative aspects of mass dimension one fermions non-minimally coupled to
electromagnetic field

Willian Carvalho<EMAIL_ADDRESS>Faculdade de Física,
Universidade Federal do Pará, 66075-110, Belém, Pará, Brazil.  M. Dias
<EMAIL_ADDRESS>Universidade Federal de São Paulo, Departamento de
Física, Rua São Nicolau 210, 09913-030, Diadema, São Paulo, Brazil.  A. C.
Lehum<EMAIL_ADDRESS>Faculdade de Física, Universidade Federal do Pará,
66075-110, Belém, Pará, Brazil.  J. M. Hoff da Silva<EMAIL_ADDRESS>Departamento de Física, Universidade Estadual Paulista, UNESP, Av. Dr.
Ariberto Pereira da Cunha, 333, Guaratinguetá, SP, Brazil.

###### Abstract

This paper addresses perturbative aspects of the renormalization of a fermion
with mass dimension one non-minimally coupled to the electromagnetic field.
Specifically, we calculate the one-loop corrections to the propagators and
vertex functions of the model and determine the one-loop beta function of the
non-minimal electromagnetic coupling. Additionally, we perform calculations of
the two-loop corrections to the gauge field propagator, demonstrating that it
remains massless and transverse up to this order. We also find that the non-
minimal electromagnetic coupling can exhibit asymptotic freedom if a certain
condition is satisfied. As a potential dark matter candidate, these findings
suggest that the field may decouple at high energies. This aspect holds
significance for calculating the relic abundance and freeze-out temperature of
the field, particularly in relation to processes involving the ordinary
particles of the Standard Model.

##  I Introduction

It dates from over two decades ago, the first version of fermionic mass
dimension one fields as candidates to dark matter Ahluwalia:2004ab . Since
then, the field has undergone modifications in the formulation to conciliate
it with Lorentz symmetries (for a broad discussion and physical consequences,
see physrev).

Recent advancements in the theory of fermionic fields characterized by mass
dimension one have been made Ahluwalia:2022yvk . These developments have
revealed that the field possesses a two-fold Wigner degeneracy wig ,
effectively doubling its degrees of freedom. Consequently, the field exhibits
complete Poincarè symmetry. The resulting construction provides a first-
principle candidate for dark matter, considering that constraints
significantly impact the feasible interactions with standard matter fields,
necessitating perturbative renormalizability. While certain potential
couplings can still be achieved through a Higgs portal (refer to Alves:2014kta
for an analysis based on the earlier version of the field), it is conceivable
that an additional field associated with a hidden dark sector’s $U(1)$ gauge
symmetry may exist dark_universe .

The renormalization of the previous version of the model, specifically in the
absence of gauge interactions, was examined in Ref. deBrito:2019hih . The
findings from that investigation indicate that the obtained physical results
also apply to the current scenario. Building upon this prior work, the
objective of our study is to extend the analysis by incorporating a gauge
interaction. To accomplish this, we employ dimensional regularization to
evaluate the one-loop renormalization of fermions with mass dimension one that
are non-minimally coupled to the electromagnetic field. Notably, we consider
the presence of a renormalizable non-minimal coupling term
$\tilde{e}\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}[\gamma_{\mu},\gamma_{\nu}]\lambda
F^{\mu\nu}$, which is allowed by gauge symmetry and has been proposed as a
potential source for an effective mass term for the photon Ahluwalia:2004ab .
The primary result presented in this paper is the demonstration that photon
propagation remains massless and transverse up to the two-loop order. This
achievement corroborates the transverse aspect of photon self-energy tensor
found very recently, via symmetry arguments, in the comprehensive study of
Ref. GNR .

The structure of this paper is as follows: In the subsequent section, we begin
by outlining the fundamental characteristics of the mass dimension one field
with Wigner degeneracy and establish the framework of the model. Section III
investigates the one-loop self-energy of the mass dimension one fermion,
photon self-energies, vertex function renormalization, and the beta function.
Section IV presents the results for the two-loop gauge field self-energy.
Certain integrals are provided in the appendix, alongside the main text, for
the sake of brevity. Lastly, we offer concluding remarks in the final section.
It is important to emphasize that throughout this paper, we adopt natural
units in which the values of $c$ and $\hbar$ are set to unity, as well as the
convention for the spacetime signature of $(+---)$.

##  II Definition of the model

There are specific $\mathcal{R}\oplus\mathcal{L}$ spin $1/2$ representations
of the Lorentz group given in the rest frame by

$\displaystyle\xi({\bf 0},1)$
$\displaystyle=\sqrt{m}\left(\begin{array}[]{cc}0\\\ i\\\ 1\\\
0\end{array}\right),\quad\xi({\bf 0},2)=\sqrt{m}\left(\begin{array}[]{cc}-i\\\
0\\\ 0\\\ 1\end{array}\right),$ (9) $\displaystyle\xi({\bf 0},3)$
$\displaystyle=\sqrt{m}\left(\begin{array}[]{cc}1\\\ 0\\\ 0\\\
-i\end{array}\right),\quad\xi({\bf 0},4)=\sqrt{m}\left(\begin{array}[]{cc}0\\\
1\\\ i\\\ 0\end{array}\right),$ (18)

and

$\displaystyle\zeta({\bf 0},1)$
$\displaystyle=\sqrt{m}\left(\begin{array}[]{cc}i\\\ 0\\\ 0\\\
1\end{array}\right),\quad\zeta({\bf
0},2)=\sqrt{m}\left(\begin{array}[]{cc}0\\\ i\\\ -1\\\ 0\end{array}\right),$
(27) $\displaystyle\zeta({\bf 0},3)$
$\displaystyle=\sqrt{m}\left(\begin{array}[]{cc}0\\\ 1\\\ -i\\\
0\end{array}\right),\quad\zeta({\bf
0},4)=\sqrt{m}\left(\begin{array}[]{cc}-1\\\ 0\\\ 0\\\ -i\end{array}\right),$
(36)

with the property of being eigenspinors (with eigenvalues $\pm 1$) of the
charge conjugation operator

$\mathcal{C}=\left(\begin{array}[]{cc}\mbox{\boldmath$\displaystyle{\bf
O}$}&\sigma_{2}\\\ -\sigma_{2}&\mbox{\boldmath$\displaystyle{\bf
O}$}\end{array}\right)K,$ (37)

where $K$ complex conjugates functions and spinors to its right. Therefore,
the two above sets of spinors are neutral in this sense. The arbitrary
momentum spinors can be derived by applying a boost operator $D(L(p))$
belonging to the direct sum of right-handed and left-handed representations,
denoted as $\mathcal{R}\oplus\mathcal{L}$. For further details, please refer
to Refs. Ahluwalia:2022yvk  and physrep . Consequently, we define a quantum
field whose expansion coefficients are determined by the aforementioned
spinors as follows:

$\displaystyle\lambda(x)=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2mE({\bf p})}}$
$\displaystyle\times\sum_{\sigma}\Big{[}a({\bf p},\sigma)\xi({\bf
p},\sigma)e^{-ip\cdot x}+b^{\dagger}({\bf p},\sigma)\zeta({\bf
p},\sigma)e^{ip\cdot x}\Big{]}.$ (38)

It can be verified that boost and translations are naturally accommodated by
the field definition and, in Ref. Ahluwalia:2022yvk , it is shown that
rotational symmetry is also accomplished as far as the doubling of degrees of
freedom, encompassed by the Wigner degeneracy, is present. In this case, the
annihilation and creation operators respect fermionic statistics, leading to
the so-called mass dimension one fermionic (MDOF) field. It is important to
refer to Refs Ahluwalia:2022yvk ; GNR  for a complete account on the darkness
of such a field.

The dual of Elko spinors cannot be identical to the standard one, as it is
evident from the observation that Elkos have zero norms under the Dirac dual.
For a formal account on the possibility of dual definitions, one can refer to
Appendix A.1 of the Ref. physrep . Introducing the quantity
$D=\gamma_{\mu}p^{\mu}/m$, the Elko dual can be defined by

$\displaystyle\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}({\bf
p},\sigma)$ $\displaystyle=\left[+\mathcal{D}\,\xi({\bf
p},\sigma)\right]^{\dagger}\gamma_{0},$ (39)
$\displaystyle\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\zeta}}({\bf
p},\sigma)$ $\displaystyle=\left[-\mathcal{D}\,\zeta({\bf
p},\sigma)\right]^{\dagger}\gamma_{0},$ (40)

and hence we are led to

$\displaystyle\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}(x)=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2mE({\bf p})}}$
$\displaystyle\times\sum_{\sigma}\Big{[}a^{\dagger}({\bf
p},\sigma)\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}({\bf
p},\sigma)e^{ip\cdot x}+b({\bf
p},\sigma)\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\zeta}}({\bf
p},\sigma)e^{-ip\cdot x}\Big{]},$

as the field adjoint. We can obtain, after applying $\lambda$
and$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}$ upon in or
out states, the following Feynman rules for the external legs for the MDOF
particle and antiparticle, respectively

$\displaystyle\frac{\xi({\bf
p},\sigma)}{\sqrt{m}}=\vbox{\hbox{\includegraphics[scale={.15}]{LEG1.png}}}\qquad\qquad\qquad\frac{\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}({\bf
p},\sigma)}{\sqrt{m}}=\vbox{\hbox{\includegraphics[scale={.15}]{LEG2.png}}}$
$\displaystyle\frac{\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\zeta}}({\bf
p},\sigma)}{\sqrt{m}}=\vbox{\hbox{\includegraphics[scale={.15}]{LEG3.png}}}\qquad\qquad\qquad\frac{\zeta({\bf
p},\sigma)}{\sqrt{m}}=\vbox{\hbox{\includegraphics[scale={.15}]{LEG4.png}.}}$

It can be verified that the Feynman-Dyson propagator for the MDOF field is
proportional to the Klein-Gordon one, from which the mass dimension one can be
read and the free Hamiltonian is positive definite.

We now present the Lagrangian that describes the coupling between fermions
with mass dimension one ($\lambda$) and a $U(1)$ gauge field, as given in Ref.
Ahluwalia:2022yvk

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{(1+\delta_{3})}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}(1+\delta_{2})\partial_{\mu}\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}\partial^{\mu}\lambda-\frac{m^{2}}{2}(1+\delta_{m^{2}})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}\lambda$
(41)
$\displaystyle+\tilde{e}(1+\delta_{1})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}[\gamma_{\mu},\gamma_{\nu}]\lambda
F^{\mu\nu}-g(1+\delta_{g})(\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}\lambda)^{2}+\mathcal{L}_{GF},$

where $\tilde{e}$ represents the MDOF/gauge-field non-minimal coupling
constant and $\mathcal{L}_{GF}$ is the gauge-fixing Faddeev-Popov Lagrangian.
The counterterms $\delta_{i}$ are obtained from field redefinitions
$\lambda\rightarrow Z_{2}^{1/2}\lambda$ and $A^{\mu}\rightarrow
Z_{3}^{1/2}A^{\mu}$, along with the following relationships:

$\displaystyle Z_{2}$ $\displaystyle=$ $\displaystyle 1+\delta_{2};$
$\displaystyle Z_{3}$ $\displaystyle=$ $\displaystyle 1+\delta_{3};$
$\displaystyle m^{2}Z_{m^{2}}$ $\displaystyle=$
$\displaystyle(1+\delta_{m^{2}})m^{2}=m_{0}^{2}Z_{2};$
$\displaystyle\tilde{e}Z_{1}$ $\displaystyle=$
$\displaystyle\tilde{e}(1+\delta_{1})=\tilde{e}_{0}Z_{2}Z_{3}^{1/2};$
$\displaystyle gZ_{g}$ $\displaystyle=$ $\displaystyle
g(1+\delta_{g})=g_{0}Z_{2}^{2},$ (42)

where $\tilde{e}_{0}$ and $g_{0}$ represent the bare coupling constants.

The propagators of the model in terms of the renormalized quantities are given
by

$\displaystyle S_{F}(p)$ $\displaystyle=$ $\displaystyle\langle
T\lambda(p)\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}(-p)\rangle=\frac{2i\leavevmode\nobreak\
\bf{1}}{p^{2}-m^{2}};$ (43a) $\displaystyle\Delta^{\mu\nu}(p)$
$\displaystyle=$ $\displaystyle\langle
TA^{\mu}(p)A^{\nu}(-p)\rangle=-\frac{i}{p^{2}}\left(\eta^{\mu\nu}-(1-\xi)\frac{p^{\mu}p^{\nu}}{p^{2}}\right).$
(43b)

##  III The one-loop renormalization

Let us show the obtained results, case by case, starting from the MDOF self-
energy.

###  III.1 The one-loop MDOF self-energy

The one-loop contributions to the MDOF self-energy are depicted in Figure 1.
We use a set of MATHEMATICA packages to generate and manipulate the
amplitudesfeyncalc ; feynarts ; feynrules ; feynhelpers . The corresponding
expressions for the first diagram are given by

$\displaystyle\Sigma_{1}(p)=2ig\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\text{tr}{\bf{1}}-1}{k^{2}-m^{2}}=-\frac{3g}{8\pi^{2}}\text{A}_{0}\left(m^{2}\right)=-\frac{3gm^{2}}{8\pi^{2}\epsilon}+\mathrm{finite},$
(44)

where $\epsilon=(4-D)/2$ and $\text{A}_{0}\left(m^{2}\right)$ is a Passarino-
Veltman (PaVe) integral. We use the same notations and conventions for PaVe
integrals as the Ref.feyncalc .

For the second diagram, we have

$\displaystyle\Sigma_{2}(p)$ $\displaystyle=$ $\displaystyle
8i\tilde{e}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\left[(\gamma\cdot(p-k)).\gamma^{\nu}-\gamma^{\nu}.(\gamma\cdot(p-k))\right]\left[(\gamma\cdot(k-p)).\gamma^{\mu}-\gamma^{\mu}.(\gamma\cdot(k-p))\right]}{(k^{2}-m^{2})(k-p)^{2}}$
(45)
$\displaystyle\times\left(g^{\mu\nu}+\frac{\left(1-\xi\right)(k-p)^{\mu}(p-k)^{\nu}}{(k-p)^{2}}\right)$
$\displaystyle=$
$\displaystyle\frac{2\tilde{e}^{2}\left[\text{A}_{0}\left(m^{2}\right)\left((D-2)(\gamma\cdot
p)^{2}+p^{2}\right)-(D-2)\left(m^{2}+p^{2}\right)\left((\gamma\cdot
p)^{2}-p^{2}\right)\text{B}_{0}\left(p^{2},0,m^{2}\right)\right]}{\pi^{2}p^{2}}$
$\displaystyle=$
$\displaystyle\frac{6\tilde{e}^{2}m^{2}}{\pi^{2}\epsilon}+\mathrm{finite}.$

Summing these diagrams and adding the counterterm contribution, Figure 1-(3),
we have

$\displaystyle\Sigma(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\delta_{2}p^{2}-m^{2}\delta_{m^{2}}\right)-\frac{3m^{2}\left(g-16\tilde{e}^{2}\right)}{8\pi^{2}\epsilon}+\mathrm{finite}.$
(46)

Finally, imposing finiteness over minimal subtraction renormalization scheme
(MS), we find the counterterms

$\displaystyle\delta_{m^{2}}=\frac{3(16\tilde{e}^{2}-g)}{4\pi^{2}\epsilon},\hskip
28.45274pt\delta_{2}=0.$ (47)

Notice that, unlike the Dirac fermions coupled to photons, the wave-function
counterterm is vanishing for the MDOF field at one-loop order. Besides, Ref.
deBrito:2019hih  demonstrated that the MDOF four-point function is finite, and
this property persists even in gauge interactions.

###  III.2 The photon self-energy

The diagrams of the photon self-energy are depicted in Fig. 2. The
corresponding expression is given by

$\displaystyle\Pi^{\mu\nu}(p)$ $\displaystyle=$
$\displaystyle-8i\tilde{e}^{2}\leavevmode\nobreak\
\text{tr}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\left[(\gamma\cdot
p)\gamma^{\nu}-\gamma^{\nu}(\gamma\cdot p)\right]\left[(\gamma\cdot
p).\gamma^{\mu}-\gamma^{\mu}(\gamma\cdot
p)\right]}{\left(k^{2}-m^{2}\right)\left((k-p)^{2}-m^{2}\right)}$ (48)
$\displaystyle=$
$\displaystyle-\frac{16\tilde{e}^{2}}{\pi^{2}}\text{B}_{0}\left(p^{2},m^{2},m^{2}\right)\left(p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}\right)$
$\displaystyle=$
$\displaystyle\frac{16\tilde{e}^{2}}{\pi^{2}\epsilon}\left(p^{2}g^{\mu\nu}-p^{\mu}p^{\nu}\right)+\mathrm{finite}.$

Adding the couterterm and imposing finiteness, we find

$\displaystyle\delta_{3}=\frac{16\tilde{e}^{2}}{\pi^{2}\epsilon}.$ (49)

This is a remarkable result. In Ref. Ahluwalia:2004ab , it was mentioned by
the authors that the interaction term
$\tilde{e}\bar{\lambda}[\gamma_{\mu},\gamma_{\nu}]\lambda F^{\mu\nu}$ could
impact the propagation of the photon by inducing an effective mass term.
However, as indicated by Eq.(48), it appears that this is not the case, as the
one-loop photon propagation remains transverse and massless. We shall revisit
this point in the next section, studying two-loop effects.

###  III.3 The vertex function renormalization

The diagrams of the electromagnetic vertex function are depicted in Fig. 3.
The corresponding expression is given by

$\displaystyle\Gamma^{\mu}_{1}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle-128\tilde{e}^{3}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\left(\gamma^{\mu}\leavevmode\nobreak\
\gamma\cdot(p_{1}+p_{2})-\gamma\cdot(p_{1}+p_{2})\gamma^{\mu}\right)}{\left((k+p_{2})^{2}-m^{2}\right)\left((k-p_{1})^{2}-m^{2}\right)}$
(50) $\displaystyle=$
$\displaystyle-\frac{8i\tilde{e}^{3}}{\pi^{2}}\left[\gamma^{\mu}\leavevmode\nobreak\
\gamma\cdot(p_{1}+p_{2})-\gamma\cdot(p_{1}+p_{2})\gamma^{\mu}\right]\text{B}_{0}\left((p_{1}+p_{2})^{2},m^{2},m^{2}\right)$
$\displaystyle=$
$\displaystyle-\frac{8i\tilde{e}^{3}}{\pi^{2}\epsilon}\left[\gamma^{\mu}\leavevmode\nobreak\
\gamma\cdot(p_{1}+p_{2})-\gamma\cdot(p_{1}+p_{2})\gamma^{\mu}\right]+\mathrm{finite},$

where we have used the momentum conservation $p_{3}=p_{1}+p_{2}$.

The second diagram gives

$\displaystyle\Gamma^{\mu}_{2}(p_{1},p_{2})$ $\displaystyle=$ $\displaystyle
4\tilde{e}g\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\left(\gamma^{\mu}\leavevmode\nobreak\
\gamma\cdot(p_{1}+p_{2})-\gamma\cdot(p_{1}+p_{2})\gamma^{\mu}\right)}{\left((k+p_{2})^{2}-m^{2}\right)\left((k-p_{1})^{2}-m^{2}\right)}$
(51) $\displaystyle=$
$\displaystyle\frac{i\tilde{e}g}{2\pi^{2}}\left[\gamma^{\mu}\leavevmode\nobreak\
\gamma\cdot(p_{1}+p_{2})-\gamma\cdot(p_{1}+p_{2})\gamma^{\mu}\right]\text{B}_{0}\left((p_{1}+p_{2})^{2},m^{2},m^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{i\tilde{e}g}{2\pi^{2}\epsilon}\left[\gamma^{\mu}\leavevmode\nobreak\
\gamma\cdot(p_{1}+p_{2})-\gamma\cdot(p_{1}+p_{2})\gamma^{\mu}\right]+\mathrm{finite}.$

Adding these contributions to the vertex counterterm, we have

$\displaystyle\Gamma^{\mu}(p_{1},p_{2})$ $\displaystyle=$
$\displaystyle-i\left(\frac{\tilde{e}(16\tilde{e}^{2}-g)}{2\pi^{2}\epsilon}+2\tilde{e}\delta_{1}\right)\left[\gamma^{\mu}\leavevmode\nobreak\
\gamma\cdot(p_{1}+p_{2})-\gamma\cdot(p_{1}+p_{2})\gamma^{\mu}\right]+\mathrm{finite},$
(52)

from which imposing finiteness we find

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle\frac{g-16\tilde{e}^{2}}{4\pi^{2}\epsilon}.$ (53)

###  III.4 The one-loop beta function

The Lagrangian given in Eq. (41) shows that the relation between bare and
renormalized couplings is expressed as
$Z_{1}\tilde{e}=\mu^{-2\epsilon}\tilde{e}_{0}Z_{2}Z_{3}^{1/2}$. The MDOF wave-
function renormalization constant $Z_{2}$, equal to $(1+\delta_{2})$, is
unity. As a result, the beta function for the electric charge can be expressed
as

$\displaystyle\beta(\tilde{e})=\lim_{\epsilon\rightarrow
0}\mu\frac{d\tilde{e}}{d\mu}=\frac{\tilde{e}}{2\pi^{2}}\left(g-48\tilde{e}^{2}\right),$
(54)

where we can see that the non-minimal electromagnetic coupling can be
asymptotically free if $g<48\tilde{e}^{2}$.

##  IV The two-loop gauge field self-energy

This section focuses on obtaining the two-loop corrections to the gauge-field
propagation. This process involves the two-loop diagrams depicted in Fig.4 and
the one-loop diagrams with a counterterm insertion shown in Fig.5.

Our approach to computing the two-loop diagrams is similar to that described
in Ref.Bevilaqua:2021uzk . Naturally, each diagram $i$ should have the most
general (Lorentz Invariant) form
$\Pi_{i}^{\mu\nu}(p)=\eta^{\mu\nu}p^{2}\leavevmode\nobreak\
A_{i}(p)+p^{\mu}p^{\nu}B_{i}(p)$, from which $A_{i}(p)$ and $B_{i}(p)$ can be
obtained through the projections:

$\displaystyle A_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{(D-1)p^{2}}\left(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\Pi_{i}^{\mu\nu},$
$\displaystyle B_{i}$ $\displaystyle=$
$\displaystyle-\frac{1}{(D-1)p^{2}}\left(\eta_{\mu\nu}-D\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\Pi_{i}^{\mu\nu}.$

By summing over the diagrams, we obtain the expected result
$\Pi(p)=\sum_{i}A_{i}(p)=-\sum_{i}B_{i}(p)$, which implies that the photon
polarization tensor has the transverse form. In these calculations, we reduced
the scalar two-loop integrals using the Tarasov algorithmTarasov:1997kx , with
the assistance of the computational package TARCER Mertig:1998vk . The
resulting scalar two-loop integrals table can be found in Martin:2003qz .

The two-loop gauge field self-energy, depicted in Fig.4, is given by

$\displaystyle\Pi^{\mu\nu}_{2l}(p)=(\eta^{\mu\nu}p^{2}-p^{\mu}p^{\nu})\Pi_{2l}(p),$
(55)

where

$\displaystyle\Pi_{2l}(p)$ $\displaystyle=$
$\displaystyle\frac{1024(D-2)\tilde{e}^{2}\leavevmode\nobreak\
(\text{A}_{0}(m^{2}))^{2}}{(4\pi)^{D}(D-1)m^{2}p^{2}\left(4m^{2}-p^{2}\right)}\left[16\tilde{e}^{2}\left(\left(-D^{2}+D+3\right)p^{2}+4(D-4)m^{2}\right)+3(D-1)gp^{2}\right]$
(56) $\displaystyle+\frac{16384\tilde{e}^{2}\leavevmode\nobreak\
\text{A}_{0}(m^{2})\text{B}_{0}(p^{2},m^{2},m^{2})}{(4\pi)^{D}(D-1)m^{2}\left(4m^{2}-p^{2}\right)}\Big{[}\tilde{e}^{2}\left(2(D((D-7)D+19)-19)m^{2}+(D-4)(D-2)p^{2}\right)$
$\displaystyle-\frac{3}{8}(D-3)(D-1)gm^{2}\Big{]}+\frac{1024\tilde{e}^{2}(\text{B}_{0}(p^{2},m^{2},m^{2}))^{2}}{(4\pi)^{D}}\left[\frac{16((D-8)D+13)\tilde{e}^{2}}{D-1}+g\right]$
$\displaystyle-\frac{32768(D-4)\tilde{e}^{4}}{(4\pi)^{D}(D-1)p^{2}}\left[(D-3)\text{J}^{(D)}_{\\{1,m\\},\\{1,m\\},\\{1,0\\}}+\left(p^{2}-4m^{2}\right)\text{J}^{(D)}_{\\{2,m\\},\\{1,m\\},\\{1,0\\}}\right],$

with the integrals $J$ defined as in Ref.Mertig:1998vk . Expanding above
expression around $D=4$ and $p^{2}\approx 0$, we have

$\displaystyle\Pi_{2l}(p)=-\frac{4\tilde{e}^{2}\left(16\tilde{e}^{2}-g\right)}{\pi^{4}\epsilon^{2}}+\frac{4\tilde{e}^{2}\left(16\tilde{e}^{2}-g\right)}{\pi^{4}\epsilon}\left(2\log\left(\frac{m^{2}}{\mu^{2}}\right)+2\gamma_{E}-3-2\log{(4\pi)}\right)+\mathcal{O}(p^{2}),$
(57)

where $\epsilon=(4-D)/2$ and $\mu$ is a mass scale introduced by the
dimensional regularization.

The one-loop diagrams with insertions of counterterms are given by

$\displaystyle\Pi_{CT}(p)$ $\displaystyle=$
$\displaystyle-\frac{32\tilde{e}^{2}}{\pi^{2}}\left[(\delta_{1}-\delta_{2})\text{B}_{0}\left(p^{2},m^{2},m^{2}\right)+m^{2}(\delta_{m^{2}}-\delta_{2})\text{C}_{0}\left(0,p^{2},p^{2},m^{2},m^{2},m^{2}\right)\right]$
(58) $\displaystyle=$
$\displaystyle\frac{16\tilde{e}^{2}}{\pi^{2}}\left[(4\delta_{1}-5\delta_{2}+\delta_{m^{2}})+2\frac{(\delta_{1}-\delta_{2})(\epsilon(\gamma_{E}-2-\log(4\pi))-1)}{\epsilon}+2(\delta_{1}-\delta_{2})\log\left(\frac{m^{2}}{\mu^{2}}\right)\right]$
$\displaystyle+\mathcal{O}(p^{2}).$

Substituting the counterterms Eqs. (47) and (53) into (58), and adding it to
the Eq. (57), we finally have the UV divergent part of the two-loop
corrections to the gauge field propagation as

$\displaystyle\Pi^{\mu\nu}_{2l}(p^{2})=\left(\frac{4\tilde{e}^{2}\left(16\tilde{e}^{2}-g\right)}{\pi^{4}\epsilon^{2}}+\mathcal{O}(p^{2})\right)\left(\eta^{\mu\nu}p^{2}-p^{\mu}p^{\nu}\right).$
(59)

It is important to emphasize that the interaction term
$\tilde{e}\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\lambda}}[\gamma_{\mu},\gamma_{\nu}]\lambda
F^{\mu\nu}$ does not lead to an effective mass term, contrary to what is
suggested in Ahluwalia:2004ab , as it can be seen from Eqs. (57) and (58), the
photon propagation remains massless and transverse up to the two-loop order.

##  V Final Remarks

Our study focuses on investigating the role of the MDOF field in Quantum
Electrodynamics (QED) within the framework of renormalization at one and two-
loop orders. Notably, we observe a distinction between Dirac and MDOF field
regarding the vanishing wave-function counterterm. Additionally, our results
reveal that the photon propagation remains massless and transverse up to the
two-loop order. An important result of our study is that the non-minimal
electromagnetic coupling can exhibit asymptotic freedom if the condition
$g<48\tilde{e}^{2}$ is satisfied.

This finding has profound implications for the behavior of the MDOF field at
high energies. It suggests that the MDOF field has the potential to decouple
from other particles and interactions at these energy scales. This aspect is
relevant when considering the MDOF field as a candidate for dark matter. It
implies that the MDOF field may behave differently from other matter fields at
high energies, which has implications for calculating its relic abundance and
freeze-out temperature. In particular, it impacts processes involving the
interactions between the MDOF field and the ordinary particles of the Standard
Model.

Furthermore, future investigations can expand upon our calculations by
incorporating gravitational corrections and exploring higher energy scales.
Such analyses would shed light on more intriguing aspects of the primordial
universe and cosmological applications involving the MDOF particle as a
constituent of dark matter.

###### Acknowledgements.

We would like to thank D. V. Ahluwalia for useful discussion and comments.
W.C. is partially supported by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior (CAPES). J.M.H.S. thanks CNPq (grant No. 307641/2022-8) for
financial support.

##  Appendix A Integrals notations

We are adopting the same notations and conventions for Passarino-Veltman
(PaVe) one-loop integrals as the Ref.feyncalc . The integrals appearing in our
work are

$\displaystyle\text{A}_{0}(m^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{D}k}{i\pi^{2}}\frac{1}{k^{2}-m^{2}};$ (60)
$\displaystyle\text{B}_{0}(p^{2},m^{2},m^{2})$ $\displaystyle=$
$\displaystyle\int\frac{d^{D}k}{i\pi^{2}}\frac{1}{(k^{2}-m^{2})((k-p)^{2}-m^{2})};$
(61) $\displaystyle\text{C}_{0}\left(0,p^{2},p^{2},m^{2},m^{2},m^{2}\right)$
$\displaystyle=$
$\displaystyle\int\frac{d^{D}k}{i\pi^{2}}\frac{1}{(k^{2}-m^{2})^{2}((k+p)^{2}-m^{2})}.$
(62)

The two-loop integrals in this work are labeled in accordance with Ref.
Mertig:1998vk . We will now provide a quotation of the two-loop integrals that
appear in our calculations:

$\displaystyle\text{J}^{(D)}_{\\{1,m\\},\\{1,m\\},\\{1,0\\}}$ $\displaystyle=$
$\displaystyle\iint\frac{d^{D}k_{1}\leavevmode\nobreak\
d^{D}k_{2}}{\pi^{D}}\frac{1}{(k_{1}^{2}-m^{2})((k_{1}-k_{2})^{2}-m^{2})(k_{2}-p)^{2}};$
(63) $\displaystyle\text{J}^{(D)}_{\\{2,m\\},\\{1,m\\},\\{1,0\\}}$
$\displaystyle=$ $\displaystyle\iint\frac{d^{D}k_{1}\leavevmode\nobreak\
d^{D}k_{2}}{\pi^{D}}\frac{1}{(k_{1}^{2}-m^{2})^{2}((k_{1}-k_{2})^{2}-m^{2})(k_{2}-p)^{2}}.$
(64)

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Figure 1: The one-loop ELKO self-energy. Continuous, dashed and wavy lines
represent ELKO, auxiliary and photon propagators, respectively.  Figure 2: The
one-loop photon self-energy. Figure 3: The one-loop corrections to the vertex
function. Figure 4: The two-loop corrections to the gauge field propagation.
Figure 5: One-loop corrections to the propagation of the gauge field with the
insertion of a single counterterm.