# A uniform bound for solutions to a thermo-diffusive system

Joonhyun La111Department of Mathematics, Stanford University, Stanford, CA
94305,USA<EMAIL_ADDRESS>Lenya Ryzhik222Department of Mathematics,
Stanford University, Stanford, CA 94305,USA<EMAIL_ADDRESS>
###### Abstract

We obtain uniform in time bounds for the solutions to a class of thermo-
diffusive systems with classical and fractional diffusions. In the classical
diffusion case, the nonlinearities are assumed to be at most exponentially
growing, while for the fractional diffusions they should grow slightly sub-
exponentially. The proofs are based on the strategy of [24].

##  1 Introduction

In this paper, we investigate the global boundedness of the solutions to the
initial value problem for reaction-diffusion systems of the form

$\displaystyle\partial_{t}u_{i}=\eta_{i}\Delta
u_{i}-p_{i}(U,V),~{}~{}i=1,\cdots,M$ (1.1)
$\displaystyle\partial_{t}v_{j}=\kappa_{j}\Delta
v_{j}+f_{j}(U,V),~{}~{}j=1,\cdots,N,$ $\displaystyle
U=(u_{1},\cdots,u_{M}),V=(v_{1},\cdots,v_{N}),$

posed in the whole space $x\in{\mathbb{R}}^{n}$, as well as for its non-local
dispersal versions. The initial conditions

$u_{i,0}(x):=u_{i}(0,x),~{}~{}v_{j,0}(x):=v_{j}(0,x),$ (1.2)

are assumed to be nonnegative and bounded, so that there exists $K_{1,2}>0$
such that

$0\leq u_{i,0}(x)\leq K_{1},~{}~{}0\leq v_{j,0}(x)\leq K_{2},~{}~{}\hbox{ for
$i=1,\cdots,M$, $j=1,\cdots,N$, and $x\in{\mathbb{R}}^{n}$.}$ (1.3)

In this model, we think of $u_{i}(t,x)$, $i=1,\dots,M$, as fuels that are
consumed in the chemical reaction, and $v_{j}(t,x)$, $j=1,\dots,N$, as the
products of the reaction.

We assume that the nonlinearities $p_{i}(U,V)$ and $f_{j}(U,V)$ satisfy the
following set of assumtpions. First, we suppose that

$\displaystyle\text{if }U,V\geq 0\text{ and }v_{j}=0,\text{ then
}f_{j}(U,V)\geq 0,$ (1.4) $\displaystyle\text{if }U,V\geq 0\text{ and
}u_{i}=0,\text{ then }p_{i}(U,V)=0.$

This ensures that both $u_{i}(t,x)$ and $u_{j}(t,x)$ remain non-negative if
they are non-negative at $t=0$ for all $x\in{\mathbb{R}}^{n}$. Here, we use
the notation $U\geq 0$ if all the components of a vector $U$ are non-negative.

Second, we assume that the consumption rates are non-negative:

$p_{i}(U,V)\geq 0,~{}~{}\text{ for all
}i=1,\cdots,M,~{}U\in{\mathbb{R}}^{M},~{}V\in{\mathbb{R}}^{N}.$ (1.5)

and the production rates are non-negative and grow at most exponentially:

$0\leq f_{j}(U,V)\leq C\exp(Z_{j}\cdot V)\text{ for all
}j=1,\cdots,N,~{}U\in{\mathbb{R}}^{M},~{}V\in{\mathbb{R}}^{N}.$ (1.6)

Here, $C$ is a positive constant and $Z_{j}\in{\mathbb{R}}^{N}$ is a constant
$N$-dimensional real vector with positive entries. The positivity of
$p_{i}(U,V)$ agrees with the intuition that all $u_{i}$ are fuels consumed in
the reaction. It also guarantees a uniform bound for $u_{i}(t,x)$. More,
precisely, if (1.5) holds then the maximum principle for the standard heat
equation implies that

$0\leq u_{i}(t,x)\leq K_{1},\hbox{ for all $t\geq 0$ and
$x\in{\mathbb{R}}^{n}$},$ (1.7)

as long as (1.3) holds at $t=0$. The positivity of $f_{j}(U,V)$ means that
this term promotes the production, while the exponential bound on $f_{j}(U,V)$
in (1.5) is needed for technical reasons but is likely to be optimal for
reasons we outline below.

The last assumption we need is that the production rates are bounded by a
multiple of the total consumption: there exist $A_{ji}\geq 0$, so that

$\displaystyle f_{j}(U,V)\leq\sum_{i=1}^{M}A_{ji}p_{i}(U,V),~{}~{}\text{ for
each }U\in\mathbb{R}^{M},V\in\mathbb{R}^{N}.$ (1.8)

The first main result of this paper is the following uniform bound.

######  Theorem 1.1.

Let $U(t,x),V(t,x)$ be a solution to (1.1) with the initial conditions (1.2),
that satisfy (1.3), with some $K_{1}>0$ and $K_{2}>0$. Suppose that
$p_{i}(U,V)$ and $f_{j}(U,V)$ satisfy assumptions (1.4), (1.5), (1.6) and
(1.8). Then, $U(t,x)$ and $V(t,x)$ are globally bounded in time: there exists
$C>0$ that depends on the constants in the above assumptions, so that

$0\leq u_{i}(x,t),\leq C,~{}~{}0\leq v_{j}(x,t)\leq C$ (1.9)

for all $t>0$ and $x\in{\mathbb{R}}^{n}$.

As we have mentioned, the maximum principle and positivity of the initial
conditions $u_{i,0}(x)$ imply immediately that $u_{i}(t,x)$ satisfies the
uniform bound (1.7). Thus, the only non-trivial claim in (1.9) is the bound on
$v(t,x)$.

#### A generalization to fractional diffusions

The strategy in the proof of Theorem 1.1 can be applied to other underlying
dispersal processes. To avoid long formulations, we only consider the one
fuel-one product situation, and assume that both the fuel and the reaction
product disperse via a fractional diffusion of the same order $s\in(0,1)$:

$\begin{gathered}\partial_{t}u=-\eta(-\Delta)^{s}u-f_{1}(u,v),\\\
\partial_{t}v=-\kappa(-\Delta)^{s}v+f_{2}(u,v).\\\ \end{gathered}$ (1.10)

The nonlinearity $f_{1}(u,v)$ is the fuel consumption rate, and $f_{2}(u,v)$
is the production rate. Once again, we assume that $f_{1,2}(u,v)$ satisfy
(1.4), in the form

$\begin{gathered}f_{1}(u,v)\geq 0,~{}~{}\hbox{for all $u,v\geq 0$,}\\\
f_{1}(0,v)=0,~{}\hbox{ for all $v\geq 0$,}~{}~{}f_{2}(u,0)\geq
0,~{}~{}\hbox{for all $u\geq 0$.}\end{gathered}$ (1.11)

Thus, solutions to (1.10) are positive at any time $t>0$ if the initial
conditions $u_{0}(x)$ and $v_{0}(x)$ are non-negative and not identically
equal to zero. Assumption (1.8) takes the form

$f_{2}(u,v)\leq Cf_{1}(u,v),$ (1.12)

so that the production rate is controlled by a multiple of the consumption
rate. One difference with the diffusive case is that the exponential bound
(1.5) needs to be slightly strengthened: we assume that there exist
$\rho\in(0,1)$ and $C>0$ such that

$f_{2}(u,v)\leq Ce^{Cv^{\rho}}.$ (1.13)

We have the following result.

######  Theorem 1.2.

Under the above assumptions, let $u(t,x)$, $v(t,x)$ be the solution to (1.10)
with initial conditions $u_{0},v_{0}\geq 0\in L^{\infty}(\mathbb{R}^{n})$ that
are non-negative. Then, there exists a constant $K>0$ such that

$0\leq u(t,x),v(t,x)\leq K,~{}~{}\hbox{ for all $t\geq 0$ and
$x\in{\mathbb{R}}^{n}$.}$ (1.14)

#### Related works

The literature on reaction-diffusion systems is truly vast, as they arise in a
wide range of applied sciences, from chemistry to atmospheric sciences. The
following discussion of some of the related work is far from being complete,
and we refer to [34] for an extensive literature review.

There are numerous types of reaction-diffusion systems, characterized by the
nature of nonlinearities of the system. The systems we discuss in this paper
appear naturally, for example, in the exothermic combustion modeling and
ecology [10, 31, 32, 33]. One of the simplest examples of combustion models is

$\begin{split}\partial_{t}u=\eta\Delta u-u^{m}\phi(v),\\\
\partial_{t}v=\kappa\Delta v+u^{m}\phi(v),\end{split}$ (1.15)

with the nonlinearities of the form $u^{m}\phi(v)$, where $\phi(v)$ is
positive and grows as $v\rightarrow\infty$.

The question of existence of global existence and uniform bound for combustion
models was originally posed by R. Martin. For the polynomial nonlinearities
$\phi(v)$, this was resolved in [1] in the case $m=1$ and $\phi(v)=v^{\beta}$,
$1\leq\beta<{(n+2)}/{n}$, and [30], for $m=1$, and $\phi(v)=v^{\beta}$, with
an arbitrary $\beta>0$. In [23], this question has been answered for
nonlinearities of the form $ue^{v^{\rho}}$, with $\rho\in(0,1)$, using a
Lyapunov functional introduced in [30]. Then, [2] considered exponential
nonlinearities with $L^{\infty}$-small initial conditions, based on the
Lyapunov functional method and semigroup theory. All of these papers
considered the problem on a bounded domain $\Omega\subset\mathbb{R}^{n}$. For
a particular class of super-exponential nonlinearity, global existence and
uniform bound of solution for $L^{\infty}$ small initial data in a bounded
domain have been proved in [39].

The uniform bounds on the whole space are more difficult because there is,
potentially, an infinite supply of fuel, unlike in a bounded domain. In the
case of whole space $\mathbb{R}^{n}$ with $L^{\infty}$ initial data, a
localized version of Lyapunov functional method was adopted in [9] to obtain
the global existence of solution and at most $O(\log\log t)$ growth as
$t\to+\infty$. The authors of [9] considered nonlinearities of the form
$uv^{m}$, but their argument extends to $u^{m}e^{v}$ with appropriate
modifications. The Lyapunov functional method was used to prove a global bound
in a more elaborate system with more variables with a polynomial nonlinearity
as well [28].

Our work is based on a careful analysis of the solutions to forced heat
equations with identical forcing but different diffusivities. Along this line
of thought, [29] pointed out an elegant argument that global existence of the
solution to (1.15) with $\eta\geq\kappa$ on ${\mathbb{R}}^{n}$ can be proved
by comparing the heat kernels with diffusivity coefficients $\eta$ and
$\kappa$. The idea was adopted and further extended in [24], and global bounds
of the solution in whole space with large $L^{\infty}$ initial data for
exponential nonlinearities for any $\eta,\kappa$ has been established. Their
work is directly related to this work. We should point out that closely
related models, including combustion models with exponential nonlinearities,
have been studied on bounded domain in [25] and [26], also based on a precise
estimates of kernels. Related bounds for thermo-diffusive systems and
traveling waves in cylinders in systems with advection have been considered in
[3, 5, 8, 13, 20, 21].

It is known that unlike in combustion models, in general reaction-diffusion
systems one cannot expect global existence or uniform bounds on solutions: in
[35] and [36] existence of reaction-diffusion systems with a finite time blow
up (with inhomogeneous Dirichlet boundary condition) have been proved. Weak
solutions for such systems have been considered in [6, 11, 12, 14, 16, 17, 37]

We should mention another important class of reaction-diffusion equations
coming from chemical reactions, such as

$\partial_{t}a_{i}-d_{i}\Delta
a_{i}=(-1)^{i+1}(a_{2}a_{4}-a_{1}a_{3}).i=1,\cdots,4.$ (1.16)

This system describes a reversible chemical reaction

$A_{1}+A_{3}\rightleftharpoons A_{2}+A_{4}.$

Compared to the combustion models we consider in the present paper, the
nonlinearities of chemical reaction models may grow slower but do not have a
sign. On the other hand, the system (1.16) conserves mass

$\frac{d}{dt}\sum_{i}\int a_{i}dx=0$ (1.17)

and, moreover, it dissipates the entropy at each point:

$-(a_{2}a_{4}-a_{1}a_{3})\log\left(\frac{a_{2}a_{4}}{a_{1}a_{3}}\right)\leq
0.$ (1.18)

It turns out that this pointwise dissipation structure is useful in the
mathematical study of chemical reaction models. This was used in, among
others, [7, 15, 19, 27, 38, 40, 42] to obtain global in time existence of the
solutions. On the other hand, to the best of authors’ knowledge, an analogue
of (1.18) for combustion models is not known.

#### Outline of the proof and organization of the paper

We should stress that the case $M=N=1$ in Theorem 1.1 has been already proved
in [24]. For the most part, the arguments in the proof of this theorem are not
original and are guided by the proof in [24]. One reason to present the full
details here is that the proof of Proposition 2.1 below, which is a crucial
step in the proof of Theorem 1.1, is different from that in [24] and seems to
us to be easier to follow. Some other steps have also been streamlined. Let us
now explain the strategy that comes from [24]. It is not difficult to see
that, given the uniform bound (1.7) and the bound (1.8) on the production
rates, in terms of the consumption rates, the essential issue is the
following. Let $f(t,x)$ and $h(t,x)$ be the solutions to the standard heat
equation

$\partial_{t}f=\eta\Delta f+G(t,x),$ (1.19)

and

$\partial_{t}h=\kappa\Delta h+G(t,x),$ (1.20)

with the initial conditions $g(0,x)=h(0,x)=0$. Note that (1.19) and (1.20)
have two different diffusivities $\kappa\neq\eta$ but the forcing term
$G(t,x)$ in the two equations is the same. Suppose that we know that
$G(t,x)\geq 0$ for all $t\geq 0$ and $x\in{\mathbb{R}}^{d}$ and that $f(t,x)$
satisfies a uniform bound

$0\leq f(t,x)\leq K,~{}~{}\hbox{for all $t\geq 0$ and
$x\in{\mathbb{R}}^{d}$.}$ (1.21)

The question is what kind of bound on the function $h(t,x)$ one can infer from
(1.21). The main observation of [24] is that one can not deduce a uniform
bound on $h(t,x)$ but it does obey a parabolic BMO bound: see Proposition 2.1
in Section 2 below. This bound does not use essentially any information on the
forcing $G(t,x)$ except for (1.21). As we have mentioned, the proof here is
completely different from that in [24].

The next step is to deduce, also solely from (1.21) a local space-time bound
on the averages of the non-negative function $h(t,x)$: for any unit space-time
cube $Q_{1}$ we have

$\int_{Q_{1}}h(t,x)dxdt\leq C(K).$ (1.22)

this is the content of Proposition 3.1 in Section 3. This bound can be
improved, once again, using solely (1.21), to an exponential moment bound
stated in Proposition 4.1 in Section 4. For any $k>0$ we have

$\int_{Q_{1}}e^{kh(t,x)}dxdt\leq C(K).$ (1.23)

This seems to be as far as one can get without any assumptions on the forcing
$G(t,x)$. However, the exponential bound (1.6) on the reaction rate, together
with (1.23) and the John-Nirenberg inequality allows one to finish the proof
of Theorem 1.1. This is done in Section 5.

The proof of Theorem 1.2 is contained in Section 6. It follows a similar
blueprint. In Section 6.1, we establish an $s$-parabolic BMO bound on the
solution: see Corollary 6.2. The uniform bound on the space-time averages of
the solution is presented in Proposition 6.3 in Section 6.2. However, in the
fractional case we can not get the full exponential bound (1.23) but rather
its slightly sub-exponential version: for any $k>0$ and $0<\rho<1$ we have

$\int_{Q_{1}}e^{kv^{\rho}(t,x)}dxdt\leq C(K).$ (1.24)

This bound is proved in Proposition 6.5 in Section 6.3. It allows us to finish
the proof of Theorem 1.2 in Section 6.4.

Notation. We denote by $g_{t}(x)$ the standard heat kernel:

$g_{t}(x)=\frac{1}{(4\pi t)^{n/2}}e^{-|x|^{2}/(4t)}.$ (1.25)

We will use the convention that $g_{t}(x)=0$ for $t\leq 0$. We use the
notation $C$ for constants that may change from line to line but do not depend
on the time $t>0$.

Acknowledgement. LR was partially supported by NSF grant DMS-1910023 and ONR
grant N00014-22-1-2174.

##  2 A parabolic BMO bound on the solution

As in [24], the main step in the proof of Theorem 1.1 is a global in time
parabolic BMO bound on the solution. Before we state the result, let us set
some notation and recall the definition of the parabolic BMO semi-norm. A
parabolic cylinder of size $R>0$ centered at a point
$(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{n}$ is a set of the form

$Q=Q_{R}(t,x)=\\{(s,y):~{}|x-y|<R,~{}|t-s|<R^{2}\\}\subset{\mathbb{R}}\times{\mathbb{R}}^{n}.$
(2.1)

For $f\in L_{loc}^{1}({\mathbb{R}}\times\mathbb{R}^{n})$, we denote the
average of $f$ over a measurable set $Q$ by

$(f)_{Q}=\frac{1}{|Q|}\int_{Q}f(x,t)dxdt.$

Here, $|Q|$ is the $(n+1)$-dimensional Lebesgue measure of $Q$. The parabolic
BMO semi-norm is defined by

$\|f\|_{\operatorname{p-BMO}}=\sup\Big{\\{}\frac{1}{|Q|}\int_{Q}|f-(f)_{Q}|dxdt\Big{\\}},$

with the supremum is taken over all parabolic cylinders
$Q\subset\mathbb{R}^{n}\times\mathbb{R}_{+}.$ Here is the main result of this
section.

######  Proposition 2.1.

Let $u(t,x)$, $v(t,x)$ be the solution to the system (1.1) under the
assumptions (1.3), (1.4), (1.5) and (1.8). There exists a constant $C_{1}$
that depends on the constants $K_{1}$ and $K_{2}$ in (1.3), and also on the
dimension $n$, diffusivities $\kappa_{j},\eta_{i}$, and the constants $A_{ji}$
that appear in (1.8) such that

$\|v\|_{\operatorname{p-BMO}({\mathbb{R}}_{+}\times{\mathbb{R}}^{n})}\leq
C_{1}.$ (2.2)

Let us emphasize that the proof of Proposition 2.1 does not use the
exponential bound (1.6) on the reaction rates. Its conclusion holds as long as
the solution to (1.1) exists and the production rates are bounded by the
consumption rates, as in (1.8).

###  2.1 The proof of Proposition 2.1

The idea behing the proof of Proposition 2.1 is very simple. We may use the
Duhamel formula to represent the solution to (1.1) with the initial condition
(1.2) as

$u_{i}(t,x)=e^{t\eta_{i}\Delta}u_{i,0}(x)-F_{i}(t,x),~{}~{}v_{j}(t,x)=e^{t\kappa_{j}\Delta}v_{j,0}(x)+H_{j}(t,x).$
(2.3)

Here, the functions $P_{i}$ and $H_{j}$ are solutions to the heat equations

$\displaystyle\partial_{t}F_{i}=\eta_{i}\Delta F_{i}+p_{i}(U,V),~{}~{}t>0,$
(2.4) $\displaystyle F_{i}(0,x)\equiv 0,$

and

$\displaystyle\partial_{t}H_{j}=\kappa_{j}\Delta H_{j}+f_{j}(U,V),~{}~{}t>0,$
(2.5) $\displaystyle H_{j}(0,x)\equiv 0.$

The maximum principle for the standard heat equation implies that

$0\leq e^{t\eta_{i}\Delta}u_{0}(x)\leq K_{1},~{}~{}0\leq
e^{t\kappa_{j}\Delta}v_{0}(x)\leq K_{2},~{}~{}\hbox{ for all $t>0$ and
$x\in{\mathbb{R}}^{n}$.}$ (2.6)

Hence, a parabolic BMO bound on $v_{j}(t,x)$ would follow from a bound on
$H_{j}(t,x)$. On the other hand, together with (1.7), the first inequality in
(2.6) implies that

$0\leq F_{i}(t,x)\leq K_{1},i=1,\cdots,M.$ (2.7)

In addition, by assumption (1.8) we have

$H_{j}(t,x)\leq\sum_{i=1}^{M}A_{ji}H_{ij}(t,x).$ (2.8)

Here, the functions $H_{ij}$, $i=1,\cdots,M,j=1,\cdots,N$ are the solutions to

$\displaystyle\partial_{t}H_{ij}=\kappa_{j}\Delta
H_{ij}+p_{i}(U,V),~{}~{}t>0,$ (2.9) $\displaystyle H_{ij}(0,x)\equiv 0.$

Note that (2.4) and (2.9) are heat equations with identical forcing but
different diffusivities. One may then ask: does the uniform bound on
$F_{i}(t,x)$ in (2.7) imply some bound on the solution $H_{ij}(t,x)$ to (2.9)
even though the diffusivities in the two equations are different? The physical
issue here is the following: suppose, for instance, that the diffusivity
$\kappa_{j}$ in the equation (2.9) for $H_{ij}$ is larger than the diffusivity
$\eta_{i}$ in the equation (2.4) for $F_{i}$. On one hand, the larger
diffusivity should help the solution ”cool” faster, so that if $F_{i}$ is
bounded, one may expect that the function $H_{ij}$ should be bounded as well.
On the other, the larger diffusivity in the equation for $H_{ij}$ may take the
underlying Brownian particles in the Feynman-Kac representation of the
solution to an inhomogeneous heat equation

$H_{ij}(t,x)={\mathbb{E}}_{x}\int_{0}^{t}p_{i}(U(t-s,x+X_{s}),V(t-s,x+X_{s}))ds,$
(2.10)

to the regions where the forcing $p_{i}(s,x)$ is large. This would promote the
growth of $H_{ij}$ that is faster than the growth of $F_{i}$. Here, $X_{s}$ is
the Brownian motion with diffusivity $\kappa_{j}$ starting at the position $x$
at the time $s=0$. This non-trivial competition between cooling by faster
spreading and heating by potentially visiting the hot spots is crucial in
understanding the eventual behavior of the solution, and our understanding is
far from complete. In this paper, we show that in the setting of thermo-
diffusive systems the heating is eventually controlled by the consumption of
the fuels $F_{i}$ by the reaction but this control is far from being simple.

Let us drop the subscripts, writing $\kappa=\kappa_{j}$, $\eta=\eta_{i}$,
$F=F_{i}$, $H=H_{ij}$, and $p(t,x)=p_{i}(U(t,x),V(t,x))$. Thus, our goal is to
analyze

$H(t,x)=\int_{0}^{t}e^{\kappa(t-s)\Delta}p(s,\cdot)(x)ds=\int_{0}^{t}\int
g_{\kappa(t-s)}(x-y)p(s,y)dyds$ (2.11)

when we already know that

$0\leq F(t,x)=\int_{0}^{t}e^{\eta(t-s)\Delta}p(s,\cdot)(x)ds=\int_{0}^{t}\int
g_{\eta(t-s)}(x-y)p(s,y)dyds\leq K_{1}.$ (2.12)

The key observation is that $H$ can be written as a space-time singular
integral of the bounded function $F$. This is seen as as follows. The
difference $\Phi=H-F$ satisfies

$\displaystyle\partial_{t}\Phi=\kappa\Delta H-\eta\Delta
F=\kappa\Delta(H-F)+(\kappa-\eta)\Delta F=\kappa\Delta\Phi+(\kappa-\eta)\Delta
F,$ (2.13)

with the initial condition $\Phi(0,x)=0$. Therefore, symbolically, we can
write

$\Phi=(\kappa-\eta)\mathcal{T}_{\kappa}F,$ (2.14)

and

$H=F+(\kappa-\eta)\mathcal{T}_{\kappa}F.$ (2.15)

Here, we have set

$\mathcal{T}_{\kappa}F:=\Delta(\partial_{t}-\kappa\Delta)^{-1}F.$ (2.16)

The key ingredient in the proof of Proposition 2.1 is the following
$\operatorname{p-BMO}$ bound.

######  Lemma 2.2.

There exists a constant $C_{\mathcal{T}}$, which depends on $\kappa$ and $n$
such that for all $F\in L^{\infty}({\mathbb{R}}_{+}\times{\mathbb{R}}^{n})$,
we have

$\|\mathcal{T}_{\kappa}F\|_{\operatorname{p-BMO}}\leq
C_{\mathcal{T}}\|F\|_{L^{\infty}({\mathbb{R}}_{+}\times\mathbb{R}^{n})}.$

This lemma corresponds to Lemma 3.3 of [24]. The proof presented here is
different and is based on the estimates on the singular kernel of the operator
$\mathcal{T}_{\kappa}$. Note that Lemma 2.2 gives immediately the parabolic
BMO bound (2.2) in Proposition 2.1, as we have, from (2.3), (2.8) and (2.15):

$\displaystyle 0\leq v_{j}(t,x)$
$\displaystyle=e^{\kappa_{j}t\Delta}v_{j,0}(x)+H_{j}(t,x)\leq
e^{\kappa_{j}t\Delta}v_{j,0}(x)+\sum_{i=1}^{M}A_{ji}H_{ij}(t,x)$ (2.17)
$\displaystyle\leq
e^{\kappa_{j}t\Delta}v_{j,0}(x)+\sum_{i=1}^{M}A_{ji}(F_{i}(t,x)+(\kappa_{j}-\eta_{i})\mathcal{T}_{\kappa_{j}}F_{i}(t,x)).$

The functions $F_{i}(t,x)$ are uniformly bounded by (2.7), hence also bounded
in the parabolic BMO space, and the terms $\mathcal{T}_{\kappa_{j}}P_{i}$ are
bounded in the parabolic BMO space by Lemma 2.2. This gives (2.2). Thus, to
finish the proof of Proposition 2.1, we only need to prove Lemma 2.2.

###  2.2 The proof of Lemma 2.2

Recall that for $F\in L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})$, the
function $\Psi={\mathcal{T}_{\kappa}}F$ is the solution to

$\displaystyle\partial_{t}\Psi=\kappa\Delta\Psi+\Delta F,$ (2.18)
$\displaystyle\Psi(x,0)=0.$

Extending $F$ by 0 to negative times, the action of $\mathcal{T}_{\kappa}$ can
be written as a singular integral operator

$\mathcal{T}_{\kappa}F(t,x)=\int_{\mathbb{R}^{n}\times\mathbb{R}}K(t-s,x-y)F(s,y)dyds,$
(2.19)

with the kernel

$K(t,x)=\Delta g_{\kappa t}(x)=\frac{1}{(4\pi\kappa
t)^{{n}/{2}}}e^{-{|x|^{2}}/{(4\kappa
t)}}\Big{(}\frac{1}{4\kappa^{{2}}}\frac{|x|^{2}}{{t^{2}}}-\frac{n}{2\kappa
t}\Big{)},~{}~{}t>0,$ (2.20)

and the convention that $K(t,x)=0$ for $t\leq 0$.

We need to show that there exists a constant $C_{\mathcal{T}}$ such that for
any parabolic cylinder

$Q=Q_{R}(x_{0},t_{0})=\\{(t,x)\in{\mathbb{R}}\times\mathbb{R}^{n}\times:~{}|x-x_{0}|<R,|t-t_{0}|<R^{2}\\},$
(2.21)

and any function $F\in L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})$, we
have

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F-(\mathcal{T}_{\kappa}F)_{Q}|dxdt\leq
C_{\mathcal{T}}\|F\|_{L^{\infty}}.$ (2.22)

Note that for any constant $a$, we can write

$a-(\mathcal{T}_{\kappa}F)_{Q}=\frac{1}{Q}\int_{Q}(a-\mathcal{T}_{\kappa}F)dxdt,$

so that

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F-(\mathcal{T}_{\kappa}F)_{Q}|dxdt\leq\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F-a|dxdt+|a-(\mathcal{T}_{\kappa}F)_{Q}|\leq\frac{2}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F-a|dxdt.$
(2.23)

Therefore, it suffices to show that there exists a constant $K_{\mathcal{T}}$,
such that for any parabolic cylinder $Q$ and function $F\in
L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})$ we can find a constant
$a_{Q}\in{\mathbb{R}}$ so that

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F-a_{Q}|dxdt\leq
K_{\mathcal{T}}\|F\|_{L^{\infty}}.$ (2.24)

The constant $K_{\mathcal{T}}$ should not depend on $Q$ or $F$ but the
constant $a_{Q}$ may.

Given a cylinder $Q=Q_{R}(t_{0},x_{0})$, we use the notation
$2Q=Q_{2R}(t_{0},x_{0})$ and decompose

$F=(F-(F)_{Q})\mathbbm{1}_{2Q}+(F-(F)_{Q})\mathbbm{1}_{(2Q)^{c}}+(F)_{Q}=:F^{(1)}+F^{(2)}+(F)_{Q}.$
(2.25)

Note that, as $(F)_{Q}$ is a constant, $\mathcal{T}_{\kappa}(F)_{Q}=0$, so
that

$\mathcal{T}_{\kappa}F=\mathcal{T}_{\kappa}F^{(1)}+\mathcal{T}_{\kappa}F^{(2)}.$
(2.26)

We will treat the two terms in (2.26) separately.

#### A bound on $\mathcal{T}_{\kappa}F^{(1)}$

For the first term in (2.26), we write

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F^{(1)}|dxdt\leq\frac{1}{|Q|}\Big{(}\int_{Q}|\mathcal{T}_{\kappa}F^{(1)}|^{2}dxdt\Big{)}^{{1}/{2}}|Q|^{{1}/{2}}\leq\Big{(}\frac{1}{|Q|}\int_{\mathbb{R}^{n}\times\mathbb{R}}|\mathcal{T}_{\kappa}F^{(1)}|^{2}dxdt\Big{)}^{{1}/{2}}.$
(2.27)

To bound the right side of (2.27), let us show that the operator
$\mathcal{T}_{\kappa}$ is bounded on
$L^{2}({\mathbb{R}}\times{\mathbb{R}}^{n})$. This is a consequence of
Mikhlin’s theorem [41] but let us give a short direct proof. Recall that

$\mathcal{T}_{\kappa}F=\Delta\tilde{\Phi},$ (2.28)

and the function $\tilde{\Phi}$ satisfies

$\partial_{t}\tilde{\Phi}=\kappa\Delta\tilde{\Phi}+F,~{}~{}t>0,$ (2.29)

with $\tilde{\Phi}(x,t)=0$ for $t\leq 0$. Multiplying (2.29) by
$\Delta\tilde{\Phi}$, integrating, we obtain

$\frac{1}{2}\int|\nabla\tilde{\Phi}|^{2}(0,x)dx-\frac{1}{2}\int|\nabla\tilde{\Phi}|^{2}(T,x)dx=\kappa\|\Delta\tilde{\Phi}\|_{L^{2}(\mathbb{R}^{n}\times[0,T])}^{2}+\int_{\mathbb{R}^{n}\times[0,T]}F\Delta\tilde{\Phi}dxdt,$

As $\nabla\tilde{\Phi}(0,x)=0$, this gives

$\displaystyle\kappa\|\Delta\tilde{\Phi}\|_{L^{2}(\mathbb{R}^{n}\times[0,T])}^{2}+\frac{1}{2}\int|\nabla\tilde{\Phi}|^{2}(T,x)dx\leq\|F\|_{L^{2}(\mathbb{R}^{n}\times[0,T])}\|\Delta\tilde{\Phi}\|_{L^{2}(\mathbb{R}^{n}\times[0,T])},$
(2.30)

whence

$\displaystyle\|\Delta\tilde{\Phi}\|_{L^{2}(\mathbb{R}^{n}\times[0,T])}\leq\frac{1}{\kappa}\|F\|_{L^{2}(\mathbb{R}^{n}\times[0,T])}.$
(2.31)

Taking into account (2.28) and passing to the limit $T\to+\infty$, we get an
$L^{2}-L^{2}$ bound on the operator $\mathcal{T}_{\kappa}$:

$\|\mathcal{T}_{\kappa}F\|_{L^{2}(\mathbb{R}^{n}\times\mathbb{R})}\leq\frac{1}{\kappa}\|F\|_{L^{2}(\mathbb{R}^{n}\times\mathbb{R})}.$
(2.32)

Going back to (2.27), we obtain

$\displaystyle\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F^{(1)}|dxdt$
$\displaystyle\leq\Big{(}\frac{1}{|Q|}\int_{\mathbb{R}^{n}\times\mathbb{R}}|\mathcal{T}_{\kappa}F^{(1)}|^{2}dxdt\Big{)}^{{1}/{2}}\leq\frac{1}{\kappa}\frac{1}{|Q|^{{1}/{2}}}\Big{(}\int_{\mathbb{R}^{n}\times\mathbb{R}}|F^{(1)}|^{2}dxdt\Big{)}^{{1}/{2}}$
(2.33)
$\displaystyle=\frac{1}{\kappa}\Big{(}\frac{1}{|Q|}\int_{2Q}|F-(F)_{Q}|^{2}dxdt\Big{)}^{{1}/{2}}\leq\frac{2}{\kappa}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}.$

Thus, taking $a=0$ in (2.23), we see that

$\|\mathcal{T}_{\kappa}F^{(1)}\|_{\operatorname{p-BMO}}\leq\frac{2}{\kappa}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}.$
(2.34)

#### A bound on $\mathcal{T}_{\kappa}F^{(2)}$

Next, we show that if we take

$b=\int K(x_{0}-y,t_{0}-s)F^{(2)}(y,s)dyds,$ (2.35)

then

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F^{(2)}-b|dxdt\leq
C\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})},$ (2.36)

with a constant $C$ that does not depend on $Q$ or $F$. Recalling (2.23) and
the remark below (2.32), we see that this will give the desired bound

$\|\mathcal{T}_{\kappa}F^{(2)}\|_{\operatorname{p-BMO}}\leq
C\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}.$ (2.37)

We first show that

$|b|\leq
C_{1}(\kappa,n)\frac{t_{0}}{R^{2}}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}.$
(2.38)

To this end, we use (2.20) to write

$\displaystyle|b|$
$\displaystyle\leq\int|K(t_{0}-s,x_{0}-y)||F_{2}(s,y)|dyds\leq
I\cdot\|F_{2}\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})},$ (2.39)

with

$\displaystyle I=$ $\displaystyle\int_{0}^{t_{0}}\int_{|x_{0}-y|\geq
2R}\frac{e^{-{|x_{0}-y|^{2}}/{(4\kappa(t_{0}-s))}}}{(4\pi\kappa(t_{0}-s))^{{n}/{2}}}\Big{(}\frac{1}{4\kappa}\frac{|x_{0}-y|^{2}}{{t_{0}-s}}+\frac{n}{2\kappa}\Big{)}\frac{1}{\kappa(t_{0}-s)}dyds$
(2.40) $\displaystyle=\int_{0}^{t_{0}}\int_{|y|\geq
2R}\frac{e^{-{|y|^{2}}/{(4\kappa s)}}}{(4\pi\kappa
s)^{{n}/{2}}}\Big{(}\frac{1}{4\kappa}\frac{|y|^{2}}{s}+\frac{n}{2\kappa}\Big{)}\frac{1}{\kappa
s}dyds\leq C(n,\kappa)\int_{0}^{t_{0}}\int_{|y|\geq
2R}\frac{e^{-{|y|^{2}}/{(8\kappa s)}}}{s^{{n}/{2}+1}}dyds$
$\displaystyle\leq\frac{C(n,\kappa)}{R^{2}}\int_{0}^{t_{0}}\int_{|y|\geq
2R}\frac{|y|^{2}e^{-{|y|^{2}}/{(8\kappa
s)}}}{s^{{n}/{2}+1}}dyds\leq\frac{C(n,\kappa)}{R^{2}}\int_{0}^{t_{0}}\int_{|y|\geq
2R}\frac{e^{-{|y|^{2}}/{(16\kappa
s)}}}{s^{{n}/{2}}}dyds\leq\frac{C(n,\kappa)t_{0}}{R^{2}}.$

Together with (2.39), this gives (2.38), since

$\|F^{(2)}\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}\leq
2\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}.$

Next we show that (2.36) holds. Since $F_{2}(t,x)$ is supported away from
$(t_{0},x_{0})$, we can integrate without taking the principal value, and use
the zero integral property of $K(t,x)$ to write

$\displaystyle\int_{Q}|\mathcal{T}_{\kappa}F^{(2)}-b|dxdt=\int_{Q}\Big{|}\int_{\mathbb{R}^{n}\times\mathbb{R}}(K(t-s,x-y)-K(t_{0}-s,x_{0}-y))F^{(2)}(s,y)dyds\Big{|}dxdt$
(2.41)
$\displaystyle=\int_{Q}\Big{|}\int_{\mathbb{R}^{n}\times\mathbb{R}}(K(t-s,x-y)-K(t_{0}-s,x_{0}-y))(F(s,y)-(F)_{Q})\chi_{(2Q)^{c}}(s,y)dyds\Big{|}dxdt$
$\displaystyle\leq\int_{Q}\Big{(}\int_{\mathbb{R}^{n}\times\mathbb{R}\setminus
2Q}|F(s,y)-(F)_{Q}||K(t-s,x-y)-K(t_{0}-s,x_{0}-y)|dyds\Big{)}dxdt$
$\displaystyle=\int_{\mathbb{R}^{n}\times\mathbb{R}\setminus
2Q}|F(s,y)-(F)_{Q}|\Big{(}\int_{Q}|K(t-s,x-y)-K(t_{0}-s,x_{0}-y)|dxdt\Big{)}dyds$
$\displaystyle\leq
2\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}\sum_{j=1}^{\infty}\int_{2^{j+1}Q\setminus
2^{j}Q}\Big{(}\int_{Q}|K(t-s,x-y)-K(t_{0}-s,x_{0}-y)|dxdt\Big{)}dyds.$

The kernel $K(t,x)$ satisfies the following key bounds:

$\displaystyle|K(t,x)|\leq\frac{B_{2}}{(\sqrt{t}+|x|)^{n+2}},~{}~{}|\nabla_{x}K(t,x)|\leq\frac{B_{2}}{(\sqrt{t}+|x|)^{n+3}},~{}~{}|\partial_{t}K(x,t)|\leq\frac{B_{2}}{\left(\sqrt{t}+|x|\right)^{n+4}},$
(2.42)

with a constant $B_{2}$ that depends on $n$ and $\kappa$. Using (2.42), we
obtain

$\displaystyle|K(x_{1},t_{1})-K(x_{2},t_{2})|\leq|K(x_{1},t_{1})-K(x_{2},t_{1})|+|K(x_{2},t_{1})-K(x_{2},t_{2})|$
(2.43) $\displaystyle\leq|x_{1}-x_{2}||(\nabla_{x}K)(\lambda
x_{1}+(1-\lambda)x_{2},t_{1})|$
$\displaystyle+(|K(x_{2},t_{1})|+|K(x_{2},t_{2})|)^{{1}/{2}}|t_{2}-t_{1}|^{{1}/{2}}|\partial_{t}K(x_{2},\lambda^{\prime}t_{1}+(1-\lambda^{\prime})t_{2})|^{{1}/{2}}$
$\displaystyle\leq\frac{B_{2}|x_{1}-x_{2}|}{(\sqrt{t_{1}}+|\lambda
x_{1}+(1-\lambda)x_{2}|)^{n+3}}$
$\displaystyle+B_{2}\Big{(}\frac{|t_{2}-t_{1}|^{1/2}}{(\sqrt{t_{1}}+|x_{2}|)^{(n+2)/2}}+\frac{|t_{2}-t_{1}|^{1/2}}{\sqrt{t_{2}}+|x_{2}|)^{(n+2)/2}}\Big{)}\frac{1}{((\lambda^{\prime}t_{1}+(1-\lambda^{\prime})t_{2})^{{1}/{2}}+|x_{2}|)^{(n+4)/2}},$

with some $\lambda,\lambda^{\prime}\in[0,1]$.

We will apply (2.43) with

$t_{1}=t-s$, $t_{2}=t_{0}-s$, $x_{1}=x-y$, $x_{2}=x_{0}-y$, (2.44)

as appears in the last integral in the right side of (2.41). Since the point
$(t_{0},x_{0})$ is the center of the parabolic cylinder $Q$, and the point
$(t,x)$ lies in $Q$, while the point $(s,y)$ lies outside the parabolic
cylinder $2Q$, there exists a constant $B_{3}=B_{3}(n)>1$ such that

$\frac{1}{B_{3}}\leq\frac{|\lambda
x+(1-\lambda)x_{0}-y|}{|x_{0}-y|},\Big{(}\frac{|t-s|}{|t_{0}-s|}\Big{)}^{{1}/{2}},\Big{(}\frac{|\lambda^{\prime}t+(1-\lambda^{\prime})t_{0}-s|}{|t_{0}-s|}\Big{)}^{{1}/{2}}\leq
B_{3}.$ (2.45)

Inserting the points given by (2.44) into (2.43) and keeping in mind (2.45),
we get

$\displaystyle|K(x-y,t-s)-K(x_{0}-y,t_{0}-s)|\leq
B_{4}(n,\kappa)\frac{|x-x_{0}|+|t-t_{0}|^{{1}/{2}}}{\left(|t_{0}-s|^{{1}/{2}}+|x_{0}-y|\right)^{n+3}}.$
(2.46)

Therefore, for each $(s,y)\in 2^{j+1}Q\setminus 2^{j}Q$, we have, since
$(t,x)\in Q_{R}(t_{0},x_{0})$:

$\displaystyle\int_{Q}|K(x-y,t-s)-K(x_{0}-y,t_{0}-s)|dxdt\leq\frac{2B_{4}R|Q|}{\left(|t_{0}-s|^{{1}/{2}}+|x_{0}-y|\right)^{n+3}}\leq\frac{B_{5}}{2^{(n+3)j}},$
(2.47)

with the constant $B_{5}$ that depends on $n$ and $\kappa$. Going back to
(2.41) and uisng (2.47) gives

$\displaystyle\int_{Q}|\mathcal{T}_{\kappa}F^{(2)}-b|dxdt\leq
2B_{5}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}\sum_{j=1}^{\infty}\frac{|2^{j+1}Q\setminus
2^{j}Q|}{2^{(n+3)j}}\leq
2B_{5}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}\sum_{j=1}^{\infty}\frac{2^{(j+1)(n+2)}}{2^{(n+3)j}}|Q|$
(2.48)
$\displaystyle=2^{n+3}B_{5}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}|Q|=B_{6}(n,\kappa)\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}|Q|.$

This gives (2.36).

With the bounds (2.33) and (2.36) in hand, we get (2.24) with $a_{Q}=b$ simply
by writing

$\displaystyle\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F-b|dxdt$
$\displaystyle=\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F^{(1)}+\mathcal{T}_{\kappa}F^{(2)}-b|dxdt\\!\leq\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F^{(1)}|dxdt+\\!\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F^{(2)}-b|dxdt$
(2.49)
$\displaystyle\leq\frac{2}{\kappa}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}+B_{6}(n,\kappa)\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})},$

finishing the proof of Lemma 2.2. $\Box$

##  3 A bound on the averages

In this section, we prove the following bound on the local space-time
averages.

######  Proposition 3.1.

Let $u(t,x)$, $v(t,x)$ be the solution to the system (1.1) under the
assumptions (1.3), (1.4), (1.5) and (1.8). There exists a constant $C_{1}$
that depends on the constants $K_{1}$ and $K_{2}$ that appear in (1.3), and
also on the dimension $n$, diffusivities $\kappa_{j},\eta_{i}$, and the
constants $A_{ji}$ in (1.5) and (1.8), such that for any parabolic cylinder
$Q_{1}(t_{0},x_{0})$, with $x_{0}\in{\mathbb{R}}^{n}$ and $t_{0}>1$, we have

$|(v)_{Q_{1}}|\leq C.$ (3.1)

As with Proposition 2.1, the conclusion of Proposition 3.1 does not rely on
the exponential bound (1.6) on the reaction rates. Its conclusion holds as
long as the solution to (1.1) exists and the production rates are bounded by
the consumption rates, as in (1.8). Moreover, in Proposition 4.1 below we will
improve this result to an exponential moment bound, still without using
assumption (1.6).

Let us write, as in (2.3):

$v_{j}(t,x)=e^{t\kappa_{j}\Delta}v_{j.0}(x)+H_{j}(t,x),$ (3.2)

and recall the bound (2.8) together with representation (2.15)

$H_{j}(t,x)\leq\sum_{i=1}^{M}A_{ji}H_{ij}(t,x)\leq\sum_{i=1}^{M}A_{ji}\big{(}F_{i}(t,x)+(\kappa_{j}-\eta_{i})\mathcal{T}_{\kappa}F_{i}(t,x)\big{)}.$
(3.3)

Here, $H_{ij}$ are the solutions to (2.9), and $F_{i}$ are the solutions to
(2.4) that are uniformly bounded due to the bound (2.7). Thus, the conclusion
of Proposition 3.1 follows from the following.

######  Lemma 3.2.

For any $\alpha\in(0,1)$ there exists a constant $C(n,\kappa,\alpha)$ that
depends on $\alpha$ and also on the dimension $n$, and on $\kappa$, so that
for any parabolic cylinder $Q_{R}(t_{0},x_{0})$, with
$x_{0}\in{\mathbb{R}}^{n}$ and $t_{0}\in[R^{2},T-R^{2}]$, and any function
$F\in L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})$, we have

$|(\mathcal{T}_{\kappa}F)_{Q}|\leq
C(n,\kappa)\Big{(}1+\frac{1}{R^{2(1-\alpha)}}\Big{)}\|F\|_{L^{\infty}}({\mathbb{R}}\times{\mathbb{R}}^{n}).$
(3.4)

Let us mention that the dependence on $R$ in the right side of (3.4) is far
from optimal. Since we are going to use this estimate only with $R=1$, this is
not important here.

#### The proof of Lemma 3.2

Note that $\Phi=\mathcal{T}_{\kappa}F$ can be written as

$\Phi=\frac{1}{\kappa}(\partial_{t}J-F),$ (3.5)

where $J$ is the solution to

$\partial_{t}J-\kappa\Delta J=F,$ (3.6)

with the initial condition $J(0,x)=0$. To see that (3.5) holds, observe that
the derivative $\partial_{t}J$ satisfies

$\partial_{t}(\partial_{t}J)-\kappa\Delta(\partial_{t}J)=\partial_{t}F,$ (3.7)

with the initial condition $(\partial_{t}J)(0,x)=F(0,x)$. Hence,
$(\partial_{t}J-F)$ satisfies

$\partial_{t}(\partial_{t}J-F)-\kappa\Delta(\partial_{t}J-F)=\partial_{t}F-\partial_{t}F+\kappa\Delta
F=\kappa\Delta F,$ (3.8)

with $(\partial_{t}J-F)(0,x)=0$. This implies (3.5).

We deduce from (3.5) that to prove (3.4) we only need to bound
$(\partial_{t}J)_{Q}$. We write

$\displaystyle(\partial_{t}J)_{Q}$
$\displaystyle=\frac{1}{|Q|}\int_{|x-x_{0}|<R}\int_{|t-t_{0}|<R^{2}}\partial_{t}J(t,x)dtdx$
(3.9)
$\displaystyle=\frac{1}{R^{n+2}|Q_{1}|}\int_{|x-x_{0}|<R}\left(J(t_{0}+R^{2},x)-J(t_{0}-R^{2},x)\right)dx.$

The standard parabolic regularity theory immediately gives a bound on the
$C^{\alpha}$-regularity of $J(t,x)$ in $t$, for any $\alpha\in(0,1)$. That is,
there exists a constant $D_{\alpha}$ that depends only on $\kappa$ and
$\alpha\in(0,1)$ so that

$|J(t_{1},x)-J(t_{2},x)|\leq
D_{\alpha}\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}|t_{1}-t_{2}|^{\alpha},~{}~{}\hbox{
for all $t_{1},t_{2}>0$ and $x\in{\mathbb{R}}^{n}$.}$ (3.10)

Note that no such uniform bound is possible for the function $J(t,x)$ itself
if we only know that the function $F(t,x)$ is bounded.

Let us recall the proof of (3.10), for the convenience of the reader, since
the proof is quite short and elementary. Assume, without loss of generality,
that $t_{2}\geq t_{1}$, and write

$\displaystyle J(t_{1},x)-J(t_{2},x)=$
$\displaystyle\int_{0}^{t_{1}}\int_{{\mathbb{R}}^{n}}\Big{(}\frac{e^{-|x-y|^{2}/4\kappa(t_{1}-s)}}{(4\pi\kappa(t_{1}-s))^{n/2}}-\frac{e^{-|x-y|^{2}/4\kappa(t_{2}-s)}}{(4\pi\kappa(t_{2}-s))^{n/2}}\Big{)}F(s,y)dyds$
(3.11)
$\displaystyle-\int_{t_{1}}^{t_{2}}\int_{{\mathbb{R}}^{n}}\frac{e^{-|x-y|^{2}/4\kappa(t_{2}-s)}}{(4\pi\kappa(t_{2}-s))^{n/2}}F(s,y)dyds=I_{1}(t_{1},t_{2},x)+I_{2}(t_{1},t_{2},x).$

The second term above satisfies the simple estimate

$|I_{2}(t_{1},t_{2},x)|\leq\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}|t_{2}-t_{1}|.$
(3.12)

As for $I_{1}$, we write, using the Newton-Leibniz formula in the $t$
variable, for a fixed $s\in[0,t_{1}]$

$\frac{e^{-|x-y|^{2}/4\kappa(t_{2}-s)}}{(4\pi\kappa(t_{2}-s))^{n/2}}-\frac{e^{-|x-y|^{2}/4\kappa(t_{1}-s)}}{(4\pi\kappa(t_{1}-s))^{n/2}}=\int_{t_{1}}^{t_{2}}\frac{h(z)}{(4\pi\kappa(\tau-s))^{n/2+1}}d\tau,~{}~{}z=\frac{x-y}{\sqrt{\kappa(\tau-s)}},$
(3.13)

with an integrable function

$h(z)=\Big{(}-\frac{n}{2}+\frac{|z|^{2}}{4}\Big{)}e^{-|z|^{2}}.$

Thus, we have, changing the variables $y\to z$ in the integral over
${\mathbb{R}}^{n}$, and integrating $z$ out, using integrability of $h(z)$:

$\displaystyle|I_{1}(t_{1},t_{2},x)|$ $\displaystyle\leq
C\|F\|_{L^{\infty}}\int_{0}^{t_{1}}\int_{t_{1}}^{t_{2}}\frac{d\tau}{\tau-s}ds=C\|F\|_{L^{\infty}}\int_{0}^{t}{{\log}}\biggl{(}\frac{t_{2}-s}{t_{1}-s}\biggl{)}ds$
(3.14) $\displaystyle=C\|F\|_{L^{\infty}}(t_{2}\log t_{2}-t_{1}\log
t_{1}-(t_{2}-t_{1})\log(t_{2}-t_{1})).$

This proves that

$|I_{1}(t_{1},t_{2},x)|\leq
C(\kappa,\alpha)\|F\|_{L^{\infty}}|t_{2}-t_{1}|^{\alpha},$ (3.15)

for all $\alpha\in(0,1)$, and (3.10) follows.

Going back to (3.9) and using (3.10), we obtain, for any $\alpha\in(0,1)$:

$\displaystyle|(\partial_{t}J)_{Q}|$
$\displaystyle\leq\frac{1}{R^{n+2}|Q_{1}|}\int_{|x-x_{0}|<R}|J(x,t_{0}+R^{2})-J(x,t_{0}-R^{2})|dx\leq\frac{C(n,\kappa,\alpha)}{R^{n+2}}R^{n}R^{2\alpha}\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}$
(3.16)
$\displaystyle=\frac{C(n,\kappa,\alpha)}{R^{2(1-\alpha)}}\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}.$

This implies (3.4) and finishes the proof of Lemma 3.2. $\Box$

##  4 An exponential moment bound

To improve the BMO-bound in Proposition 2.1 and the local averages bound in
Proposition 3.1 to the global in time $L^{\infty}$-bound on $v_{j}(t,x)$ in
Theorem 1.1, we will need the following exponential moment bound.

######  Proposition 4.1.

For any vector $\bar{Z}\in{\mathbb{R}}^{N}$ with positive entries
$\bar{Z}_{k}\geq 0$, $k=1,\dots,N$, there exists a constant $C_{r,Z}>0$ that
depends on $K_{1}$ and $K_{2}$, the constants $A_{ji}$ in (1.8), the dimension
$n$ and the diffusivities $\eta_{i}$ and $\kappa_{j}$, as well as on $r$, so
that for all $t_{0}>1$ and $x_{0}\in{\mathbb{R}}^{n}$, we have

$\frac{1}{|Q_{1}(t_{0},x_{0})|}\int_{Q_{1}(t_{0},x_{0})}e^{(\bar{Z}\cdot
V)}dxdt\leq C_{r},$ (4.1)

for all $j=1,\cdots,N$.

As with Propositions 2.1 and 3.1, the conclusion of Proposition 4.1 does not
rely on the exponential bound (1.6) on the reaction rates. The exponential
bound in this proposition seems to be as far as one can get without using any
assumptions on the reaction rates as long as the production rates are
controlled by the consumption rates as in (1.8).

###  4.1 A decomposition into a bounded function and a small p-BMO function

So far, we have shown that the functions $H_{ij}(t,x)$, the solutions to
(2.9), are bounded in $\operatorname{p-BMO}$ and have bounded averages over
parabolic cylinders of size one. The first step in the proof of Lemma 4.1 is
to prove that they can be decomposed into a sum of a ”large” but bounded
function and a function that has a small $\operatorname{p-BMO}$ norm. Given an
integer $m\geq 1$, let us set

$\displaystyle\alpha(\kappa,\eta,m)=\left(\frac{\eta}{\kappa}\right)^{\frac{n}{2}}e^{{m^{2}(|\kappa-\eta|)}/{\kappa}},~{}~{}\beta(\kappa,m)=2^{{n}/{2}}e^{-{m^{2}}/{(8\kappa)}}.$
(4.2)

The decomposition comes from the following lemma.

######  Lemma 4.2.

Let $H(t,x)$ and $F(t,x)$ be given by

$\displaystyle H(t,x)=\int_{0}^{t}\int g_{\kappa(t-s)}(x-y)p(s,y)dyds,$ (4.3)
$\displaystyle F(t,x)=\int_{0}^{t}\int g_{\eta(t-s)}(x-y)p(s,y)dyds,$

with the same function $p(t,x)\geq 0$. Assume that $0\leq F(s,y)\leq K_{1}$
for all $0\leq s\leq t$ and $y\in{\mathbb{R}}^{n}$. Then, for any $m>1$, we
have

$0\leq
H(t,x)\leq(\alpha(\kappa,\eta,m)+\beta(\kappa,m))K_{1}+(2\kappa-\eta)\beta(\kappa,m)\mathcal{T}_{2\kappa}F(t,x).$
(4.4)

#### The proof of Lemma 4.2

Given $t>0$ and $x\in\mathbb{R}^{n}$, let $m$ be a large positive integer, to
be determined later. We introduce the space-time sets

$\displaystyle{\cal B}(t,x)$
$\displaystyle=\Big{\\{}0<s<t,~{}y\in\mathbb{R}^{n}:{|x-y|^{2}}>m^{2}(t-s)\Big{\\}},$
(4.5) $\displaystyle{\cal G}(t,x)$ $\displaystyle={\cal
B}^{c}(t,x)=\Big{\\{}0<s<t,~{}y\in\mathbb{R}^{n}:{|x-y|^{2}}\leq
m^{2}(t-s)\Big{\\}},$

and decompose $H(t,x)$ as

$H(t,x)=H_{g}(t,x)+H_{b}(t,x),$ (4.6)

with

$\displaystyle
H_{g}(t,x)=\int_{\mathcal{G}(t,x)}g_{\kappa(t-s)}(x-y)p(y,s)dyds,$ (4.7)
$\displaystyle H_{b}(t,x)=\int_{{\cal B}(t,x)}g_{\kappa(t-s)}(x-y)p(y,s)dyds.$

Observe that if $\kappa<\eta$, then we have

$\exp\left(-\frac{|x|^{2}}{4\kappa
t}\right)\leq\exp\Big{(}-\frac{|x|^{2}}{4\eta t}\Big{)},~{}~{}\hbox{ for all
$t>0$ and $x\in{\mathbb{R}}^{n}$,}$ (4.8)

so that

$g_{\kappa(t-s)}(x-y)\leq\Big{(}\frac{\eta}{\kappa}\Big{)}^{\frac{n}{2}}g_{\eta(t-s)}(x-y).$
(4.9)

On the other hand, if $\kappa>\eta$ and $(s,y)\in\mathcal{G}(t,x)$, then

$-\frac{|x-y|^{2}}{4\kappa(t-s)}=-\frac{|x-y|^{2}}{4\eta(t-s)}+\frac{|x-y|^{2}}{4\eta(t-s)}\left(\frac{\kappa-\eta}{\kappa}\right)\leq-\frac{|x-y|^{2}}{4\eta(t-s)}+m^{2}\frac{\kappa-\eta}{\kappa}.$
(4.10)

Thus, if $(s,y)\in\mathcal{G}(t,x)$, then we can bound the kernel
$g_{\kappa(t-s)}(x-y)$ in terms of $g_{\eta(t-s)}(x-y)$, once again:

$g_{\kappa(t-s)}(x-y)=\frac{1}{(4\pi\kappa(t-s))^{{n}/{2}}}e^{-{|x-y|^{2}}/{(4\kappa(t-s))}}\leq\alpha(\kappa,\eta,m)g_{\eta(t-s)}(x-y),$
(4.11)

with $\alpha(\kappa,\eta,m)$ given by (4.2). Therefore, the component
$H_{g}(t,x)$ is controlled by $F(t,x)$:

$\displaystyle H_{g}(t,x)$
$\displaystyle=\int_{\mathcal{G}(t,x)}g_{\kappa(t-s)}(x-y)p(y,s)dyds\leq\alpha(\kappa,\eta,m)\int_{\mathcal{G}(t,x)}g_{\eta(t-s)}(x-y)p(y,s)dyds$
(4.12)
$\displaystyle\leq\alpha(\kappa,\eta,m)\int_{0}^{t}\int_{{\mathbb{R}}^{n}}g_{t-s}(x-y)p(y,s)dyds=\alpha(\kappa,\eta,m)F(t,x)\leq\alpha(\kappa,\eta,m)K_{1}.$

This is natural: the points in the set $\mathcal{G}(t,x)$ are relatively close
to $(t,x)$, hence their effect on the solution at $(t,x)$ is not significantly
enhanced by a larger diffusivity $\kappa>1$.

On the set ${\cal B}(t,x)$ we will sacrifice the diffusivity, increasing it
even larger than $\kappa$ but will gain the small factor $\beta(\kappa,m)$
given by (4.2). This is done by writing, for $(s,y)\in{\cal B}$:

$-\frac{|x-y|^{2}}{4\kappa(t-s)}=-\frac{|x-y|^{2}}{8\kappa(t-s)}-\frac{|x-y|^{2}}{8\kappa(t-s)}\leq-\frac{|x-y|^{2}}{8\kappa(t-s)}-\frac{m^{2}}{8\kappa},$
(4.13)

so that

$g_{\kappa(t-s)}(x-y)=\frac{1}{(4\pi\kappa(t-s))^{{n}/{2}}}e^{-{|x-y|^{2}}/{(4\kappa(t-s))}}\leq\beta(\kappa,m)g_{2\kappa(t-s)}(x-y),$
(4.14)

with $\beta(\kappa,m)$ as in (4.2). This gives

$H_{b}(t,x)=\int_{{\cal
B}(t,x)}g_{\kappa(t-s)}(x-y)p(y,s)dyds\leq\beta(\kappa,m)W(t,x).$ (4.15)

Here, we have set

$W(t,x)=\int_{0}^{t}\int_{{\mathbb{R}}^{n}}g_{2\kappa(t-s)}(x-y)p(y,s)dyds.$
(4.16)

The function $W(t,x)$ is the solution to the forced heat equation

$\displaystyle W_{t}=2\kappa\Delta W+p(t,x),$ (4.17)

with the initial condition $W(0,x)=0$. This problem is is nearly identical to
(2.9) except for the diffusivity $\kappa$ replaced by $2\kappa$. As in (2.15),
we can write

$W=F+(2\kappa-\eta)\mathcal{T}_{2\kappa}F.$ (4.18)

Summarizing (4.6), (4.12), (4.15) and (4.18), and using (2.7) one more time,
we obtain

$H(t,x)\leq\alpha(\kappa,m)K_{1}+\beta(\kappa,m)K_{1}+\beta(\kappa,m)(2\kappa-\eta)\mathcal{T}_{2\kappa}F,$
(4.19)

which is (4.4). $\Box$

###  4.2 The proof of Proposition 4.1

We now prove Proposition 4.1. Note that there exists $C>0$ such that

$\bar{Z}\cdot V(t,x)\leq C(K_{2}+\sum_{i,j}H_{ij}(t,x)),~{}~{}\hbox{for all
$t>0$ and $x\in{\mathbb{R}}^{n}$.}$ (4.20)

Here, $H_{ij}(t,x)$ are the solutions to (2.9). Thus, (4.1) would follow if we
can show that for every $r>0$ there exists a constant $C_{r}>0$ so that for
all unit parabolic cylinders $Q_{1}(t_{0},x_{0})$ we have

$\frac{1}{|Q_{1}(t_{0},x_{0})|}\int_{Q_{1}(t_{0},x_{0})}\exp\Big{\\{}r\sum_{i,j}H_{ij}(t,x)\Big{\\}}dxdt\leq
C_{r}.$ (4.21)

The proof of (4.21) uses the parabolic John-Nirenberg inequality [41] that
bounds the deviation of a $\operatorname{p-BMO}$ function from its local
average.

######  Proposition 4.3.

(John-Nirenberg inequality) There exist constants $A,B>0$, such that for all
parabolic cylinders $Q=Q_{R}(t_{0},x_{0})$ and all $\lambda>0$, we have

$\big{|}\\{(t,x)\in Q:~{}|g(t,x)-(g)_{Q}|\geq\lambda\\}|\leq
B\exp\Big{(}-\frac{A\lambda}{\|g\|_{\operatorname{p-BMO}}}\Big{)}|Q|,$ (4.22)

for all $g\in\operatorname{p-BMO}$. Here, $|Q|$ is the $(n+1)$-dimensional
Lebesgue measure of $Q$.

Let us use (4.4) to bound $H_{ij}(t,x)$ as

$H_{ij}(t,x)\leq(\alpha(\kappa_{j},\eta_{i},m)+\beta(\kappa_{j},m))K_{1}+(2\kappa_{j}-\eta_{i})\beta(\kappa_{j},m)\mathcal{T}_{2\kappa_{j}}F_{i}(t,x).$
(4.23)

This leads to

$\displaystyle\int_{Q}\exp\Big{\\{}r\sum_{i,j}H_{ij}(t,x)\Big{\\}}dxdt\leq\exp\Big{(}e^{Cm^{2}}\Big{)}\int_{Q}\exp\Big{\\{}\sum_{i,j}(2\kappa_{j}-\eta_{i})\beta(\kappa_{j},m)r\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)\Big{\\}}dxdt$
(4.24)
$\displaystyle\leq\exp\Big{(}e^{Cm^{2}}\Big{)}\int_{Q}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}\big{|}(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}\big{|}\Big{\\}}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt.$

Here, we have set $Q=Q_{1}(t_{0},x_{0})$, and denote by $C$ constants that may
vary from line to line and depend on $K_{1}$ and $K_{2}$, as well as the
constants $A_{ji}$ in (1.8), but not on $m$ or the parabolic cylinder $Q$.

The first factor in the integrand in the right side of (4.24) can be estimated
using Lemma 3.2. This gives

$\displaystyle\int_{Q}\exp\Big{\\{}r\sum_{i,j}H_{ij}(t,x)\Big{\\}}dxdt\leq\exp\Big{(}e^{Cm^{2}}\Big{)}\int_{Q}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt.$
(4.25)

Next, we estimate the integral in the right side of (4.25) using the Hölder
and algebraic-arithmetic mean inequalities:

$\displaystyle\int_{Q}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt$
(4.26)
$\displaystyle\leq\prod_{i,j}\Big{(}\int_{Q}\exp\Big{\\{}CMNe^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt\Big{)}^{\frac{1}{MN}}$
$\displaystyle\leq\frac{1}{MN}\sum_{i,j}\int_{Q}\exp\Big{\\{}CMNe^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt.$

We estimate the right side above as

$\displaystyle\int_{Q}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt$
(4.27)
$\displaystyle\leq\frac{1}{MN}\sum_{i,j}\sum_{q=0}^{\infty}\exp\left(CMNe^{-m^{2}/C}(q+1)\right)\big{|}\\{(t,x)\in
Q:~{}|(\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q})\in[q,q+1]\\}\big{|}.$

Now, we use the John-Nirenberg inequality to bound the right side in (4.27) by

$\displaystyle\int_{Q}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt$
(4.28)
$\displaystyle\leq\frac{\exp\big{(}CMNe^{-m^{2}/C}\big{)}}{MN}\sum_{i,j}\sum_{q=0}^{\infty}\exp\big{(}CMNe^{-m^{2}/C}q\big{)}B\exp\Big{(}-\frac{Aq}{\|\mathcal{T}_{2\kappa_{j}}F_{i}\|_{\operatorname{p-BMO}}}\Big{)}|Q|.$

We use Lemma 2.2 to obtain from (4.28) that

$\displaystyle\frac{1}{|Q|}\int_{Q}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt$
(4.29)
$\displaystyle\leq\frac{\exp\big{(}CMNe^{-m^{2}/C}\big{)}}{MN}\sum_{i,j}\sum_{q=0}^{\infty}\exp\big{(}CMNe^{-m^{2}/C}q\big{)}B\exp\Big{(}-\frac{Aq}{C}\Big{)}$
$\displaystyle={B\exp\big{(}CMNe^{-m^{2}/C}\big{)}}\sum_{q=0}^{\infty}\exp\Big{(}\Big{[}MNe^{-m^{2}/C}-\frac{A}{C}\Big{]}q\Big{)}.$

We now choose $m$ sufficiently large, so that

$MNe^{-m^{2}/C}<\frac{A}{2C},$ (4.30)

leading to

$\displaystyle\frac{1}{|Q|}\int_{Q}\exp\Big{\\{}\sum_{i,j}Ce^{-m^{2}/C}|\mathcal{T}_{2\kappa_{j}}F_{i}(t,x)-(\mathcal{T}_{2\kappa_{j}}F_{i})_{Q}|\Big{\\}}dxdt\leq
C.$ (4.31)

Combining this with (4.25) gives

$\frac{1}{|Q|}\int_{Q}e^{r\sum_{i,j}H_{ij}(t,x)}dxdt\leq C,$

finishing the proof of (4.21) and thus that of Proposition 4.1. $\Box$

##  5 The proof of Theorem 1.1

We are now ready to prove the time-independent $L^{\infty}$-bound on
$v_{j}(t,x)$ in Theorem 1.1. It is here that we will finally use the
exponential bound (1.6) on the reaction rates and Proposition 4.1. First, we
remark that the existence of solution for arbitrary $T>0$ can be shown by the
same argument as the upcoming argument; the only difference is the initial
time. Instead of starting from $\tau$, by starting from $t=0$, one obtains a
$L^{\infty}$ bound (depending on time), and that is enough for proving
existence of solution. Let $T>2$ be fixed, and let $\tau\in[T-2,T-1]$. Then,
we have

$\displaystyle v_{j}(T,x)$
$\displaystyle=(e^{\kappa_{j}(T-\tau)\Delta}v_{j}(\tau))(x)+\int_{\tau}^{T}e^{(T-s)\kappa_{j}\Delta}f_{j}(s)ds$
(5.1)
$\displaystyle\leq(e^{\kappa_{j}(T-\tau)\Delta}v_{j}(\tau))(x)+C\int_{\tau}^{T}e^{(T-s)\kappa_{j}\Delta}e^{Z_{j}\cdot
V}(s)ds.$

We used the exponential bound (1.6) above.

We first treat the second term in the right side of (5.1). Let us set

$h(t,x)=g_{\kappa_{j}t}(x),$

with the convention $g_{t}(x)=0$ for $t<0$ and similarly extend $e^{Z_{j}\cdot
V}(t,x)=0$ for $t<\tau$. This allows us to represent the integral in (5.1) as

$\int_{\tau}^{T}e^{(T-s)\kappa_{j}\Delta}e^{Z_{j}\cdot
V}(s)ds=h*_{t,x}[e^{Z_{j}\cdot V}](T,x).$ (5.2)

Next, we use the decomposition

$h(t,x)=\sum_{m=1}^{\infty}h_{m}(t,x),~{}~{}h_{m}(t,x)=h(t,x)\mathbbm{1}_{\\{m-1\leq|x|<m\\}}(x),$
(5.3)

so that

$h*_{t,x}[e^{Z_{j}\cdot V}](T,x)=\sum_{m=1}^{\infty}h_{m}*_{t,x}(e^{Z_{j}\cdot
V})(T,x).$ (5.4)

For each $m\geq 1$ and $p>1$, an application of Hölder’s inequality gives

$\displaystyle h_{m}*(e^{Z_{j}\cdot
V})(T,x)=\int_{\tau}^{T}\int\mathbbm{1}_{\\{m-1\leq|x-y|<m\\}}(y)\frac{e^{-{|x-y|^{2}}/{(4\kappa_{j}(T-s))}}}{(4\pi\kappa_{j}(T-s))^{{n}/{2}}}[e^{Z_{j}\cdot
V}](s,y)dyds$ (5.5)
$\displaystyle\leq\Big{(}\int_{\tau}^{T}\int_{m-1\leq|x-y|<m}\Big{(}\frac{e^{-{{(m-1)}^{2}}/{(4\kappa_{j}(T-s))}}}{(4\pi\kappa_{j}(T-s))^{{n}/{2}}}\Big{)}^{p^{\prime}}dyds\Big{)}^{{1}/{p^{\prime}}}\Big{(}\int_{\tau}^{T}\int_{m-1\leq|x-y|<m}e^{p(Z_{j}\cdot
V)}dyds\Big{)}^{1/{p}}$ $\displaystyle\leq
Cm^{(n-1)/p^{\prime}}\Big{(}\int_{\tau}^{T}\frac{e^{-{{(m-1)}^{2}p^{\prime}}/{(4\kappa_{j}(T-s))}}}{(4\pi\kappa_{j}(T-s))^{{n}{p^{\prime}}/{2}}}ds\Big{)}^{{1}/{p^{\prime}}}\Big{(}\int_{(B_{m}(x)\setminus
B_{m-1}({x}))\times[\tau,T]}e^{p(Z_{j}\cdot V)}dyds\Big{)}^{{1}/{p}}.$

Here, $p^{\prime}\in(1,\infty)$ is the conjugate exponent to $p$:

$\frac{1}{p}+\frac{1}{p^{\prime}}=1,$ (5.6)

which satisfies $np^{\prime}<2$.

For $m>1$, the first integral in the right side of (5.5) can be estimated as

$\displaystyle\int_{\tau}^{T}\frac{e^{-{{(m-1)}^{2}p^{\prime}}/{(4\kappa_{j}(T-s))}}}{(4\pi\kappa_{j}(T-s))^{{n}p^{\prime}/{2}}}ds$
$\displaystyle=\int_{0}^{T-\tau}\frac{e^{-{{(m-1)}^{2}p^{\prime}}/{(4\kappa_{j}s)}}}{(4\pi\kappa_{j}s)^{{n}p^{\prime}/{2}}}ds\leq
C(m-1)^{2-np^{\prime}}\int_{0}^{2/(m-1)^{2}}\frac{e^{-{p^{\prime}}/{(4\kappa_{j}s)}}}{s^{{n}p^{\prime}/{2}}}ds$
(5.7)
$\displaystyle=C(m-1)^{2-np^{\prime}}\zeta\Big{(}\frac{2}{(m-1)^{2}}\Big{)},$

with

$\zeta(t)=\int_{0}^{t}\frac{e^{-{p^{\prime}}/{(4\kappa_{j}s)}}}{s^{{n}p^{\prime}/{2}}}ds.$
(5.8)

Note that $\zeta(t)\to 0$ as $t\to 0$ faster than any power of $t$: for any
$r>0$ there exists $C_{r}$ so that

$\zeta(t)\leq C_{r}t^{r}.$ (5.9)

For $m=1$, the first integral is bounded since $np^{\prime}<2$.

The second integral in the right side of (5.5) can be bounded using Lemma 4.1
and the fact that the domain of integration

$(B_{m}(x)\setminus B_{m-1}(x))\times[\tau,T]\subset(B_{m}(x)\setminus
B_{m-1}(x))\times[T-2,T]$

can be covered by $O(m^{n-1})$ parabolic cylinders of size $1$. This leads to
the bound

$\int_{(B_{m}({x})\setminus B_{m-1}({x}))\times[0,T]}e^{p(Z_{j}\cdot
V)}dyds\leq Cm^{n-1}.$ (5.10)

Going back to (5.5) and using (5.7) and (5.10), as well as (5.6), we obtain

$\displaystyle h_{m}*(e^{Z_{j}\cdot V})(T,x)\leq
Cm^{-1+\frac{2}{p^{\prime}}}\Big{(}\zeta\Big{(}\frac{2}{(m-1)^{2}}\Big{)}\Big{)}^{1/p^{\prime}}.$
(5.11)

Inserting this bound into (5.1), with the help of (5.9), with $r>1$ slightly
larger than $1$, gives a bound

$\displaystyle\int_{\tau}^{T}e^{(T-s)\kappa_{j}\Delta}e^{Z_{j}\cdot V}(s)ds$
$\displaystyle\leq
C\sum_{m=1}^{\infty}m^{-1+\frac{2}{p^{\prime}}}\left(\zeta\Big{(}\frac{2}{(m-1)^{2}}\Big{)}\right)^{1/p^{\prime}}\leq
C\sum_{m=1}^{\infty}m^{-1+\frac{2}{p^{\prime}}-\frac{2r}{p^{\prime}}}\leq C,$
(5.12)

which is uniform in $\tau\in[T-2,T-1]$. Therefore, we have

$\int_{T-2}^{T-1}\int_{\tau}^{T}e^{(T-s)\kappa_{j}\Delta}e^{Z_{j}\cdot
V}(s)dsd\tau\leq C,$ (5.13)

as well.

Now, averaging (5.1) over $\tau\in[T-2,T-1]$ and using (5.13) gives

$v_{j}(t,x)\leq\int_{T-2}^{T-1}(e^{\kappa_{j}(T-\tau)\Delta}v_{j}(\tau))(x)d\tau+C.$
(5.14)

We recall that by Lemma 3.2, we have $(v_{j})_{Q}\leq C$ for any unit
parabolic cylinder $Q$, with a constant $C$ that is independent of $T$.
Therefore, we have

$\displaystyle\int_{T-2}^{T-1}(e^{\kappa_{j}(T-\tau)\Delta}v_{j}(\tau))(x)d\tau=\sum_{m=1}^{\infty}\int_{(B_{m}(0)\setminus
B_{m-1}(0))\times[T-2,T-1]}g_{\kappa_{j}(T-\tau)}(y)v_{j}(x-y,\tau)dyd\tau$
(5.15) $\displaystyle\leq\sum_{m=1}^{\infty}\int_{(B_{m}(0)\setminus
B_{m-1}(0))\times[T-2,T-1]}\frac{e^{-(m-1)^{2}/(8\kappa_{j})}}{(4\pi\kappa_{j})^{n/2}}v_{j}(x-y,\tau)dyd\tau$
$\displaystyle\leq C\sum_{m=1}^{\infty}e^{-(m-1)^{2}/(8\kappa_{j})}m^{n-1}\leq
C.$

Using this bound in (5.14) gives

$v_{j}(t,x)\leq C,$ (5.16)

finishing the proof of Theorem 1.1. $\Box$

##  6 Generalization to fractional diffusions

In this section, we prove Theorem 1.2. We consider solutions to (1.10):

$\begin{gathered}\partial_{t}u=-\eta(-\Delta)^{s}u-f_{1}(u,v),\\\
\partial_{t}v=-\kappa(-\Delta)^{s}v+f_{2}(u,v),\\\ \end{gathered}$ (6.1)

with bounded non-negative conditions $u_{0}(x)$, $v_{0}(x)$ such that

$0\leq u_{0}(x)\leq K_{1},~{}~{}0\leq v_{0}(x)\leq K_{2},~{}~{}\hbox{ for all
$x\in{\mathbb{R}}^{n}$.}$ (6.2)

To simplify the notation, we will assume throughout the proof that $\eta=1$,
without loss of generality. The nonlinearities $f_{1,2}(u,v)$ are assumed to
satisfy assumptions (1.11), (1.12) and (1.13). We will use the notation
$\mathcal{P}(t,x)$ for the heat kernel for $(-\Delta)^{s}$: the solution to

$\displaystyle\partial_{t}\varphi=-(-\Delta)^{s}\varphi,~{}~{}t>0,~{}x\in{\mathbb{R}}^{n},$
(6.3) $\displaystyle\varphi(0,x)=\phi_{0}(x),$

is given by

$\varphi(t,x)=\int\mathcal{P}(t,x-y)\phi_{0}(y)dy.$ (6.4)

This kernel is none-negative: $\mathcal{P}(t,x)>0$ for all $t>0$ and
$x\in{\mathbb{R}}^{n}$ and has mass equal to one:

$\int_{{\mathbb{R}}^{n}}\mathcal{P}(t,x)dx=1,~{}~{}\hbox{ for all $t>0$.}$
(6.5)

It also satisfies the estimate

$|\mathcal{P}(t,x)|\leq\frac{C}{t^{{n}/{(2s)}}}\Big{(}1+\frac{|x|^{2}}{t^{{1}/{s}}}\Big{)}^{-{(n+2s)}/{2}}.$
(6.6)

The basic idea is as in the proof of Theorem 1.1: the maximum principle for
the fractional heat equation implies that $u(t,x)$ obeys a uniform bound

$0<u(t,x)\leq K_{1},~{}~{}\hbox{ for all $t>0$ and $x\in{\mathbb{R}}^{n}$.}$
(6.7)

Let us write

$u(t,x)=e^{-t(-\Delta)^{s}}u_{0}(x)-F(t,x).$ (6.8)

The function $F(t,x)$ is the solution to

$\displaystyle\partial_{t}F=-(-\Delta)^{s}F+f_{1}(u,v),~{}~{}t>0,~{}x\in{\mathbb{R}}^{n},$
(6.9) $\displaystyle F(0,x)=0,~{}x\in{\mathbb{R}}^{n}.$

It follows from (6.7) that $F(t,x)$ is globally bounded:

$0\leq F(t,x)\leq K_{1}.$ (6.10)

On the other hand, it follows from (1.12) that the function $v(t,x)$ obeys an
upper bound

$v(t,x)\leq e^{-\kappa t(-\Delta)^{s}}v_{0}(x)+CH(t,x).$ (6.11)

Here, the function $H(t,x)$ is the solution to

$\displaystyle\partial_{t}H=-\kappa(-\Delta)^{s}H+f_{1}(u,v),~{}~{}t>0,~{}x\in{\mathbb{R}}^{n},$
(6.12) $\displaystyle H(0,x)=0,~{}x\in{\mathbb{R}}^{n}.$

Thus, as in the proof of Theorem 1.1, the question is what kind of a bound on
the solution to (6.12) is implied by the uniform bound (6.10) on the solution
to (6.9), given that the forcing terms in the two equations are identical.

Following the aforementioned proof, we first show that the function $H(t,x)$
is bounded in a parabolic-BMO space, with a modified space-time length ratio,
to account for the fractional Laplacian. Then, we prove that its local space-
time averages are also bounded, as in Lemma 3.2. Combining these two results
with a version of the parabolic John-Nirenberg inequality, we prove that the
local space-times averages of subexponentials $\exp\big{(}Cv^{\rho}\big{)}$,
with $0<\rho<1$, are bounded. This is an analogue of Proposition 4.1. The
conclusion of Theorem 1.2 will follow from this estimate and (1.13).

###  6.1 A parabolic BMO bound

Let us recall the definition of the $s$-parabolic BMO semi-norm. An
$s$-parabolic cylinder of size $R>0$ centered at a point
$(t,x)\in{\mathbb{R}}\times{\mathbb{R}}^{n}$ is a set of the form

$Q=Q_{R}(t,x)=\\{(s,y):~{}|x-y|<R,~{}|t-s|<R^{2s}\\}\subset{\mathbb{R}}\times{\mathbb{R}}^{n}.$
(6.13)

The $s$-parabolic BMO semi-norm is defined by

$\|f\|_{\operatorname{s-p-
BMO}}=\sup\Big{\\{}\frac{1}{|Q|}\int_{Q}|f-(f)_{Q}|dxdt\Big{\\}},$

with the supremum taken over all $s$-parabolic cylinders
$Q\subset\mathbb{R}_{+}\times\mathbb{R}^{n}.$

In order to obtain an $s$-$\operatorname{p-BMO}$ bound on the function
$H(t,x)$, let us write, similarly to (2.15):

$H=F+(\kappa-1)\mathcal{T}_{\kappa}^{(s)}F.$ (6.14)

Here, we have set

$\mathcal{T}_{\kappa}^{(s)}F:=-(-\Delta)^{s}(\partial_{t}+\kappa(-\Delta)^{s})^{-1}F.$
(6.15)

In other words, $\Phi=\mathcal{T}_{\kappa}^{(s)}F$ is the solution to

$\displaystyle\partial_{t}\Phi+\kappa(-\Delta)^{s}\Phi=-(-\Delta)^{s}F,~{}~{}t>0,~{}x\in{\mathbb{R}}^{n},$
(6.16) $\displaystyle\Phi(0,x)=0,~{}x\in{\mathbb{R}}^{n}.$

Here is the analogue of Lemma 2.2.

######  Lemma 6.1.

There exists $C>0$ so that for all $F\in
L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}^{n})$ we have

$\|\mathcal{T}_{\kappa}^{(s)}F\|_{\operatorname{s-p-BMO}}\leq
C\|F\|_{L^{\infty}(\mathbb{R}_{+}\times\mathbb{R}^{n})}.$ (6.17)

An immediate consequence of Lemma 6.1 and (6.10), (6.11) and (6.14), is an
$\operatorname{s-p-BMO}$ bound on the solution to (6.1).

######  Corollary 6.2.

Let $u(t,x)$, $v(t,x)$ be the solution to the system (6.1) under the
assumptions (1.11) and (1.12). There exists a constant $C_{1}$ that depends on
the constants $K_{1}$ and $K_{2}$ in (6.2), and also on the dimension $n$,
diffusivities $\kappa$ and $\eta$, and the constant $C$ that appears in (1.12)
such that

$\|v\|_{\operatorname{s-p-BMO}({\mathbb{R}}_{+}\times{\mathbb{R}}^{n})}\leq
C_{1}.$ (6.18)

As in the case of Proposition 2.1, the conclusion of Corollary 6.2 does depend
on the assumption (1.13) on the growth of the function $f_{2}(u,v)$.

### The proof of Lemma 6.1

Extending $F$ by 0 to negative times, the action of
$\mathcal{T}_{\kappa}^{(s)}$ can be written as a singular integral operator

$\mathcal{T}_{\kappa}^{(s)}F(t,x)=\int_{\mathbb{R}^{n}\times\mathbb{R}}A(t-s,x-y)\mathbbm{1}(t>s)F(s,y)dyds.$
(6.19)

Fortunately, some of the properties of the kernel $A(t,x)$ that we will need
in the present proof have been established in [43], where they was used to
study the Hölder regularity of the solutions to linear fractional heat
equations. First, it is self-similar: there is a function $\Psi(z)$ so that

$A(t,x)=\frac{1}{t^{1+n/(2s)}}\Psi\Big{(}\frac{x}{t^{1/(2s)}}\Big{)},$ (6.20)

and is mean-zero:

$\int A(t,x)dx=0~{}~{}\hbox{ for al $t>0$.}$ (6.21)

Furthermore, there exists a constant $C>0$ such that $A$ is bounded by

$|A(t,x)|\leq\frac{C}{\left(|x|+t^{{1}/{(2s)}}\right)^{n+2s}},$ (6.22)

and its derivatives are bounded as

$|\partial_{t}A(t,x)|\leq\frac{C}{\left(|x|+t^{{1}/{(2s)}}\right)^{n+4s}},~{}~{}|\nabla_{x}A(t,x)|\leq\frac{C}{\left(|x|+t^{{1}/{(2s)}}\right)^{n+2s+1}}.$
(6.23)

To get the $\operatorname{s-p-BMO}$ bound, we need to show that there exists a
constant $C_{\mathcal{T}}$ such that for any $s$-parabolic cylinder

$Q=Q_{R}(x_{0},t_{0})=\\{(t,x)\in{\mathbb{R}}\times\mathbb{R}^{n}:~{}|x-x_{0}|<R,|t-t_{0}|<R^{2s}\\},$
(6.24)

and any function $F\in L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})$, we
have

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F-(\mathcal{T}_{\kappa}^{(s)}F)_{Q}|dxdt\leq
C_{\mathcal{T}}\|F\|_{L^{\infty}}.$ (6.25)

As in the proof of Proposition 2.1, it suffices to show that there exists a
constant $K_{\mathcal{T}}$, such that for any $s$-parabolic cylinder $Q$ and
function $F\in L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})$ we can find a
constant $a_{Q}(F)\in{\mathbb{R}}$ that may depend on $Q$ and $F$ so that

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}F-a_{Q}(F)|dxdt\leq
K_{\mathcal{T}}\|F\|_{L^{\infty}}.$ (6.26)

Given a cylinder $Q=Q_{R}(t_{0},x_{0})$, we use the notation
$2Q=Q_{2R}(t_{0},x_{0})$ and decompose

$F=(F-(F)_{Q})\chi_{2Q}+(F-(F)_{Q})\chi_{2Q^{c}}+(F)_{Q}=:F_{1}+F_{2}+(F)_{Q}.$
(6.27)

As in (2.26), we write

$\mathcal{T}_{\kappa}^{(s)}F=\mathcal{T}_{\kappa}^{(s)}F_{1}+\mathcal{T}_{\kappa}^{(s)}F_{2},$
(6.28)

with

$F_{1}=(F-(F)_{Q})\mathbbm{1}_{2Q},~{}~{}F_{2}=(F-(F)_{Q})\mathbbm{1}_{2Q^{c}}.$
(6.29)

#### A bound on $\mathcal{T}_{\kappa}^{(s)}F_{1}$

For the first term in (6.28), we write

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{1}|dxdt\leq\frac{1}{|Q|}\Big{(}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{1}|^{2}dxdt\Big{)}^{{1}/{2}}|Q|^{{1}/{2}}\leq\Big{(}\frac{1}{|Q|}\int_{\mathbb{R}^{n}\times\mathbb{R}}|\mathcal{T}_{\kappa}^{(s)}F_{1}|^{2}dxdt\Big{)}^{{1}/{2}}.$
(6.30)

To bound the right side of (6.30), let us note that the operator
$\mathcal{T}_{\kappa}^{(s)}$ is bounded on
$L^{2}({\mathbb{R}}\times{\mathbb{R}}^{n})$:

$\|\mathcal{T}_{\kappa}^{(s)}F\|_{L^{2}(\mathbb{R}^{n}\times\mathbb{R})}\leq\frac{1}{\kappa}\|F\|_{L^{2}(\mathbb{R}^{n}\times\mathbb{R})}.$
(6.31)

This is proved exactly as the corresponding estimate (2.32) for classical
diffusion. Using this estimate in (6.30), we obtain

$\displaystyle\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{1}|dxdt$
$\displaystyle\leq\Big{(}\frac{1}{|Q|}\int_{\mathbb{R}^{n}\times\mathbb{R}}|\mathcal{T}_{\kappa}^{(s)}F_{1}|^{2}dxdt\Big{)}^{{1}/{2}}\leq\frac{C}{|Q|^{{1}/{2}}}\Big{(}\int_{\mathbb{R}^{n}\times\mathbb{R}}|F_{1}|^{2}dxdt\Big{)}^{{1}/{2}}$
(6.32)
$\displaystyle=C\Big{(}\frac{1}{|Q|}\int_{2Q}|F-(F)_{Q}|^{2}dxdt\Big{)}^{{1}/{2}}\leq
2C\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}.$

It follows that

$\|\mathcal{T}_{\kappa}^{(s)}F_{1}\|_{\operatorname{s-p-BMO}}\leq
C\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{+})}.$ (6.33)

#### A bound on $\mathcal{T}_{\kappa}^{(s)}F_{2}$

Next, we show that if we take

$b=\int A(t_{0}-\tau,x-y)\mathbbm{1}(t_{0}-\tau>0)F_{2}(y,\tau)dyd\tau,$
(6.34)

then

$\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{2}-b|dxdt\leq
C\|F\|_{L^{\infty}},$ (6.35)

with a constant $C$ that does not depend on $Q$ or $F$.

The first step in the proof of (6.35) is an analog of (2.38):

$|b|\leq
C_{1}(\kappa,n)\frac{t_{0}}{R^{2s}}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}.$
(6.36)

To prove (6.36), let use use (6.22) to write

$\displaystyle|b|$
$\displaystyle\leq\int|A(t_{0}-\tau,x_{0}-y)\mathbbm{1}(t_{0}-\tau>0)||F_{2}(\tau,y)|dyd\tau\leq
I\cdot\|F_{2}\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})},$ (6.37)

where

$\displaystyle I$ $\displaystyle=\int_{0}^{t_{0}}\int_{|x_{0}-y|\geq
2R}|A(t_{0}-\tau,x_{0}-y)|dyd\tau+\int_{0}^{t_{0}-(2R)^{2s}}\int_{|x_{0}-y|<2R}|A(t_{0}-\tau,x_{0}-y)|dyd\tau$
(6.38) $\displaystyle\leq\int_{0}^{t_{0}}\int_{|x_{0}-y|\geq
2R}\frac{C}{(|x_{0}-y|+(t_{0}-\tau)^{{1}/{(2s)}})^{n+2s}}dyd\tau$
$\displaystyle+\int_{0}^{t_{0}-(2R)^{2s}}\int_{|x_{0}-y|<2R}\frac{C}{(|x_{0}-y|+(t_{0}-\tau)^{{1}/{(2s)}})^{n+2s}}dyd\tau$
$\displaystyle\leq Ct_{0}\int_{|x_{0}-y|\geq
2R}\frac{1}{|x_{0}-y|^{n+2s}}dy+C(2R)^{n}\int_{0}^{t_{0}-(2R)^{2s}}\frac{1}{(t_{0}-\tau)^{1+{n}/{(2s)}}}d\tau\leq
C_{s}t_{0}R^{-2s}$

We have assumed in the last step, without loss of generality that
$t_{0}>R^{2s}$. This gives (6.37), and (6.36) follows.

Next, we prove (6.35). Using the mean-zero property (6.21) of $A(t,x)$, we
write

$\displaystyle\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{2}-b|dxdt$ (6.39)
$\displaystyle=\int_{Q}\Big{|}\int_{\mathbb{R}^{n}\times\mathbb{R}}(A(t-\tau,x-y)\mathbbm{1}(t>\tau)-A(t_{0}-\tau,x_{0}-y)\mathbbm{1}(t_{0}>\tau))F_{2}(\tau,y)dyd\tau\Big{|}dxdt$
$\displaystyle=\int_{Q}\Big{|}\int_{\mathbb{R}^{n}\times\mathbb{R}}\\!\\!(A(t-\tau,x-y)\mathbbm{1}(t>\tau)-A(t_{0}-s,x_{0}-y)\mathbbm{1}(t_{0}>\tau))(F(\tau,y)-(F)_{Q})\mathbbm{1}_{(2Q)^{c}}(\tau,y)dyd\tau\Big{|}dxdt$
$\displaystyle\leq\int_{Q}\Big{(}\int_{\mathbb{R}^{n}\times\mathbb{R}\setminus
2Q}|F(\tau,y)-(F)_{Q}||A(t-\tau,x-y)\mathbbm{1}(t>\tau)-A(t_{0}-\tau,x_{0}-y)\mathbbm{1}(t_{0}>\tau)|dyd\tau\Big{)}dxdt$
$\displaystyle=\int_{\mathbb{R}^{n}\times\mathbb{R}\setminus
2Q}|F(\tau,y)-(F)_{Q}|\Big{(}\int_{Q}|A(t-\tau,x-y)\mathbbm{1}(t>\tau)-A(t_{0}-\tau,x_{0}-y)\mathbbm{1}(t_{0}>\tau)|dxdt\Big{)}dyd\tau$
$\displaystyle\leq
2\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}\sum_{j=1}^{\infty}\int_{2^{j+1}Q\setminus
2^{j}Q}\Big{(}\int_{Q}|A(t-\tau,x-y)-A(t_{0}-\tau,x_{0}-y)|dxdt\Big{)}dyd\tau.$

The gradient and time derivative estimates (6.23) give

$\displaystyle|A(x_{1},t_{1})-A(x_{2},t_{2})|\leq|A(x_{1},t_{1})-A(x_{2},t_{1})|+|A(x_{2},t_{1})-A(x_{2},t_{2})|$
(6.40) $\displaystyle\leq|x_{1}-x_{2}||(\nabla_{x}A)(\lambda
x_{1}+(1-\lambda)x_{2},t_{1})|$
$\displaystyle+(|A(x_{2},t_{1})|+|A(x_{2},t_{2})|)^{{(2s-1)}/{2s}}|t_{2}-t_{1}|^{{1}/{2s}}|\partial_{t}A(x_{2},\lambda^{\prime}t_{1}+(1-\lambda^{\prime})t_{2})|^{{1}/{2s}}$
$\displaystyle\leq\frac{C|x_{1}-x_{2}|}{({t_{1}}^{1/2s}+|\lambda
x_{1}+(1-\lambda)x_{2}|)^{n+2s+1}}$
$\displaystyle+\Big{(}\frac{1}{({t_{1}}^{1/2s}+|x_{2}|)^{\frac{(n+2s)(2s-1)}{2s}}}+\frac{1}{({t_{2}}^{1/2s}+|x_{2}|)^{\frac{(n+2s)(2s-1)}{2s}}}\Big{)}\frac{C|t_{2}-t_{1}|^{1/2s}}{((\lambda^{\prime}t_{1}+(1-\lambda^{\prime})t_{2})^{{1}/{2s}}+|x_{2}|)^{\frac{(n+4s)}{2s}}},$

with some $\lambda,\lambda^{\prime}\in[0,1]$.

We will use (6.40) with

$t_{1}=t-\tau$, $t_{2}=t_{0}-\tau$, $x_{1}=x-y$, $x_{2}=x_{0}-y$, (6.41)

as in the integral in the right side of (6.39). As in (2.45), since the point
$(t_{0},x_{0})$ is the center of the parabolic cylinder $Q$, the point $(t,x)$
lies in $Q$, and the point $(s,y)$ lies outside of $2Q$, there exists a
constant $B_{3}(n)$ such that

$\frac{1}{B_{3}}\leq\frac{|\lambda
x+(1-\lambda)x_{0}-y|}{|x_{0}-y|},\Big{(}\frac{|t-s|}{|t_{0}-s|}\Big{)}^{{1}/{2s}},\Big{(}\frac{|\lambda^{\prime}t+(1-\lambda^{\prime})t_{0}-s|}{|t_{0}-s|}\Big{)}^{{1}/{2s}}\leq
B_{3}.$ (6.42)

Using (6.40) for the points given by (6.41), we get, with the help of (6.42),

$\displaystyle|A(x-y,t-\tau)-A(x_{0}-y,t_{0}-\tau)|\leq
B_{4}(n,\kappa,s)\frac{|x-x_{0}|+|t-t_{0}|^{{1}/{2s}}}{\left(|t_{0}-\tau|^{{1}/{2s}}+|x_{0}-y|\right)^{n+2s+1}}.$
(6.43)

Thus, since $(t,x)\in Q_{R}(t_{0},x_{0})$, we get, for each $(\tau,y)\in
2^{j+1}Q\setminus 2^{j}Q$:

$\displaystyle\int_{Q}|A(x-y,t-\tau)-A(x_{0}-y,t_{0}-\tau)|dxdt\leq\frac{2B_{4}R|Q|}{\left(|t_{0}-\tau|^{{1}/{2s}}+|x_{0}-y|\right)^{n+2s+1}}\leq\frac{B_{5}}{2^{(n+2s+1)j}},$
(6.44)

with the constant $B_{5}$ that depends on $n$ and $\kappa$. Using (6.44) in
(6.39) gives

$\displaystyle\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{2}-b|dxdt\leq
2B_{5}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}\sum_{j=1}^{\infty}\frac{|2^{j+1}Q\setminus
2^{j}Q|}{2^{(n+2s+1)j}}\leq
2B_{5}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}\sum_{j=1}^{\infty}\frac{2^{(j+1)(n+2s)}}{2^{(n+2s+1)j}}|Q|$
(6.45)
$\displaystyle=2^{n+2s}B_{5}\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}|Q|=B_{6}(n,\kappa,s)\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}|Q|.$

This gives (6.35).

Now that we have both (6.32) and (6.35), we get (6.26) with $a_{Q}=b$:

$\displaystyle\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F-b|dxdt$
$\displaystyle=\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{1}+\mathcal{T}_{\kappa}^{(s)}F_{2}-b|dxdt$
(6.46)
$\displaystyle\leq\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{1}|dxdt+\frac{1}{|Q|}\int_{Q}|\mathcal{T}_{\kappa}^{(s)}F_{2}-b|dxdt$
$\displaystyle\leq
C\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})}+B_{6}(n,\kappa,s)\|F\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R})},$

finishing the proof of Lemma 6.1. $\Box$

###  6.2 A bound on space-time averages

We next prove an analogue of Proposition 3.1, a bound on the space-time
averages of the solution.

######  Proposition 6.3.

Let $u(t,x)$, $v(t,x)$ be the solution to the system (6.1) under the
assumptions (1.11) and (1.12). There exists a constant $C_{1}$ that depends on
the constants $K_{1}$ and $K_{2}$ in (6.2), and also on the dimension $n$,
diffusivities $\kappa$ and $\eta$, and the constant $C$ that appears in (1.12)
such that for any parabolic cylinder $Q_{1}(t_{0},x_{0})$, with
$x_{0}\in{\mathbb{R}}^{n}$ and $t_{0}>1$, we have

$|(v)_{Q_{1}}|\leq C_{1}.$ (6.47)

Once again, the bound (6.47) does not rely on the sub-exponential bound (1.13)
on the reaction rate. The main step in the proof of Proposition 6.3 is the
following analogue of Lemma 3.2.

######  Lemma 6.4.

For any $\gamma\in(0,1)$, and any parabolic cylinder $Q_{R}(t_{0},x_{0})$, we
have

$|(\mathcal{T}_{\kappa}^{(s)}F)_{Q}|\leq\frac{C(n,\kappa,\gamma)}{R^{2s(1-\gamma)}}\|F\|_{L^{\infty}(\mathbb{R}\times\mathbb{R}^{n})}.$
(6.48)

###### Proof.

As in (3.5), the function $\Phi=\mathcal{T}_{\kappa}^{(s)}F$ can be
represented as

$\Phi=\frac{1}{\kappa}(\partial_{t}J-F).$ (6.49)

Here, $J(t,x)$ is the solution to

$\partial_{t}J+\kappa(-\Delta)^{s}J=F,$ (6.50)

with the initial condition $J(0,x)=0$.

Thus, to prove (6.48) we only need to bound the space-time averages
$(\partial_{t}J)_{Q}$ that can be written as

$\displaystyle(\partial_{t}J)_{Q}$
$\displaystyle=\frac{1}{|Q|}\int_{|x-x_{0}|<R}\int_{|t-t_{0}|<R^{2s}}\partial_{t}J(t,x)dtdx$
(6.51)
$\displaystyle=\frac{1}{R^{n+2s}|Q_{1}|}\int_{|x-x_{0}|<R}\left(J(t_{0}+R^{2s},x)-J(t_{0}-R^{2s},x)\right)dx.$

Let us now show that, as in the classical diffusion case, there exists a
constant $D_{\alpha}$ that depends only on $\kappa$ and $\alpha\in(0,1)$ so
that

$|J(t_{1},x)-J(t_{2},x)|\leq
D_{\alpha}\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}|t_{1}-t_{2}|^{\alpha},~{}~{}\hbox{
for all $t_{1},t_{2}>0$ and $x\in{\mathbb{R}}^{n}$.}$ (6.52)

Let us assume, without loss of generality, that $t_{2}\geq t_{1}$, and write

$\displaystyle J(t_{1},x)-J(t_{2},x)=$
$\displaystyle\int_{0}^{t_{1}}\int_{{\mathbb{R}}^{n}}\Big{(}\mathcal{P}(\kappa(t_{1}-\tau),x-y)-\mathcal{P}(\kappa(t_{2}-\tau),x-y)\Big{)}F(\tau,y)dyd\tau$
(6.53)
$\displaystyle-\int_{t_{1}}^{t_{2}}\int_{{\mathbb{R}}^{n}}\mathcal{P}(\kappa(t_{2}-\tau),x-y)F(\tau,y)dyd\tau=I_{1}(t_{1},t_{2},x)+I_{2}(t_{1},t_{2},x).$

The second term above can be bounded simply using (6.5) by

$|I_{2}(t_{1},t_{2},x)|\leq\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}|t_{2}-t_{1}|.$
(6.54)

To bound $I_{1}(t_{1},t_{2},x)$, we write, using the Newton-Leibniz formula in
the $t$ variable, for a fixed $\tau\in[0,t_{1}]$

$\mathcal{P}(\kappa(t_{1}-\tau),x-y)-\mathcal{P}(\kappa(t_{2}-\tau),x-y)=\kappa\int_{t_{1}}^{t_{2}}(\partial_{t}\mathcal{P})(\kappa(\tilde{\tau}-\tau),x-y)d\tilde{\tau}.$
(6.55)

The kernel $\mathcal{P}(t,x)$ obeys the estimate

$|\partial_{t}\mathcal{P}(t,x)|\leq\frac{C}{t}\mathcal{P}(t,x).$ (6.56)

Using (6.5) and (6.56) in (6.55) gives

$\displaystyle|I_{1}(t_{1},t_{2},x)|$ $\displaystyle\leq
C\|F\|_{L^{\infty}}\int_{0}^{t_{1}}\int_{t_{1}}^{t_{2}}\frac{d\tilde{\tau}}{\tau-\tilde{\tau}}d\tau=C\|F\|_{L^{\infty}}(t_{2}\log
t_{2}-t_{1}\log t_{1}-(t_{2}-t_{1})\log(t_{2}-t_{1})).$ (6.57)

We deduce that

$|I_{1}(t_{1},t_{2},x)|\leq
C(\kappa,\gamma)\|F\|_{L^{\infty}}|t_{2}-t_{1}|^{\gamma},$ (6.58)

for all $\gamma\in(0,1)$, and (6.52) follows.

Going back to (6.51) and using (6.52), we obtain, for any $\gamma\in(0,1)$:

$\displaystyle|(\partial_{t}J)_{Q}|$
$\displaystyle\leq\frac{1}{R^{n+2s}|Q_{1}|}\int_{|x-x_{0}|<R}|J(x,t_{0}+R^{2s})-J(x,t_{0}-R^{2s})|dx\leq\frac{C(n,\kappa,\gamma)}{R^{n+2s}}R^{n}R^{2s\gamma}\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}$
(6.59)
$\displaystyle=\frac{C(n,\kappa,\gamma)}{R^{2s(1-\gamma)}}\|F\|_{L^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n})}.$

This implies (6.48) and finishes the proof of Lemma 6.4. ∎

###  6.3 A sub-exponential moment bound

The next step is an analogue of Proposition 4.1.

######  Proposition 6.5.

For any $r>0$ and $\rho\in(0,1)$ there is $C_{r,\rho}>0$ so that for all
$t_{0}>1$ and $x_{0}\in{\mathbb{R}}^{n}$, we have

$\frac{1}{|Q_{1}(t_{0},x_{0})|}\int_{Q_{1}(t_{0},x_{0})}e^{rv^{\rho}}dxdt\leq
C_{r,\rho}.$ (6.60)

Note that, unlike in Proposition 4.1, the moment in the left side of (6.60) is
sub-exponential in $v$. This is the reason for the sub-exponential growth
assumption (1.13) on the nonlinearity $f_{2}(u,v)$. Similarly to the
exponential moment bound in in Proposition 4.1, the estimate (6.60) seems to
be optimal without using any assumptions on the reaction rates $f_{1}(u,v)$
and $f_{2}(u,v)$ as long as the production rate is controlled by the
consumption rate as in (1.12).

The parabolic John-Nirenberg inequality in Proposition 4.3 has the following
s-parabolic version [41].

######  Lemma 6.6.

(John-Nirenberg inequality) There exist constants $A,B>0$, such that for all
$s$-parabolic cylinders $Q=Q_{R}(t_{0},x_{0})$ and all $\lambda>0$, we have

$\big{|}\\{(t,x)\in Q:~{}|g(t,x)-(g)_{Q}|\geq\lambda\\}|\leq
B\exp\Big{(}-\frac{A\lambda}{\|g\|_{\operatorname{s-p-BMO}}}\Big{)}|Q|,$
(6.61)

for all $g\in\operatorname{s-p-BMO}$. Here, $|Q|$ is the $(n+1)$-dimensional
Lebesgue measure of $Q$.

#### The proof of Proposition 6.5

In order to bound the sub-exponential $e^{Cv^{\rho}}$, we recall (6.11)

$v(t,x)\leq e^{-\kappa t(-\Delta)^{s}}v_{0}(x)+CH(t,x),$ (6.62)

with the function $H(t,x)$ that is the solution to (6.12). As the first term
in the right side is bounded, to prove (6.60) it suffices to show that for any
$r>0$ and $\rho\in(0,1)$, we can find $C(r,\rho)$ such that for all $t_{0}>1$
and $x_{0}\in{\mathbb{R}}^{n}$, we have

$\displaystyle\frac{1}{|Q_{1}|}\int_{Q_{1}}e^{rH(t,x)^{\rho}}dxdt\leq
C(r,\rho),$ (6.63)

with $Q_{1}=Q_{1}(t_{0},x_{0})$. Let us use (6.14) to write

$\displaystyle\frac{1}{|Q_{1}|}\int_{Q_{1}}e^{rH(t,x)^{\rho}}dxdt\leq\frac{C}{|Q_{1}|}\int_{Q_{1}}\exp\Big{\\{}Cr\big{(}|(\mathcal{T}_{\kappa}^{(s)}F)_{Q}|^{\rho}+|(\mathcal{T}_{\kappa}^{(s)}F-(\mathcal{T}_{\kappa}^{(s)}F)_{Q})|^{\rho}\big{)}\Big{\\}}dxdt,$
(6.64)

We may now use Proposition 6.3 to bound the first term in the exponential,
leading to

$\displaystyle\frac{1}{|Q_{1}|}\int_{Q_{1}}e^{rH(t,x)^{\rho}}dxdt$
$\displaystyle\leq\frac{C}{|Q_{1}|}\int_{Q_{1}}\exp\Big{\\{}Cr|(\mathcal{T}_{\kappa}^{(s)}F-(\mathcal{T}_{\kappa}^{(s)}F)_{Q})|^{\rho}\Big{\\}}dxdt$
(6.65)
$\displaystyle\leq\frac{1}{|Q_{1}|}\sum_{m=1}^{\infty}\Big{|}Q_{1}\cap\big{\\{}m-1\leq|\mathcal{T}_{\beta}F-(\mathcal{T}_{\beta}F)_{Q}|<m\big{\\}}\Big{|}e^{Crm^{\rho}}.$

The John-Nirenberg inequality (6.61) applied to the right side of (6.65),
together with the $\operatorname{s-p-BMO}$ bound on $v(t,x)$ in Corollary 6.2
gives

$\displaystyle\frac{1}{|Q_{1}|}\int_{Q_{1}}e^{rH(t,x)^{\rho}}dxdt$
$\displaystyle\leq\frac{1}{|Q_{1}|}\sum_{m=1}^{\infty}\Big{|}Q_{1}\cap\big{\\{}m-1\leq|\mathcal{T}_{\beta}F-(\mathcal{T}_{\beta}F)_{Q}|<m\big{\\}}\Big{|}e^{Crm^{\rho}}$
(6.66) $\displaystyle\leq C\sum_{m=1}^{\infty}e^{Crm^{\rho}-C^{-1}m}\leq C,$

finishing the proof of Proposition 6.5. $\Box$

###  6.4 The proof of Theorem 1.2

We now finish the proof of Theorem 1.2. Once again, (6.11) shows that it
suffices to bound the function $H(t,x)$. Also, as before, the existence of
solution for an arbitrary $T>0$ comes from the same argument as the following
one, with changing initial time to $t=0$. Let $T>2$ be fixed, take some
$\tau\in[T-2,T-1]$ and write

$\displaystyle H(T,x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\mathcal{P}(\kappa(T-\tau),x-y)H(\tau,y)dy+\int_{\tau}^{T}\int_{\mathbb{R}^{n}}\mathcal{P}(\kappa(T-\tilde{\tau}),x-y)f_{1}(\tilde{\tau},y)dyd\tilde{\tau}$
(6.67)
$\displaystyle\leq\int_{\mathbb{R}^{n}}\mathcal{P}(\kappa(T-\tau),x-y)H(\tau,y)dy+\int_{\tau}^{T}\int_{\mathbb{R}^{n}}\mathcal{P}(\kappa(T-\tilde{\tau}),x-y)Ce^{CH^{\rho}(\tilde{\tau},y)}dyd\tilde{\tau}$
$\displaystyle=H_{1}(T,x)+H_{2}(T,x),$

with $\rho\in(0,1)$.

We first look at the second term in the right side of (6.67). Let us set

$h(t,x)=\mathcal{P}(\kappa t,x),$

with the convention $h(t,x):=0$ for $t<0$ and similarly extend
$f_{1}(t,x):=0=:e^{CH^{\rho}}(t,x)$ for $t<\tau$, so that

$H_{2}(T,x)=Ch*_{t,x}[e^{CH^{\rho}}](T,x)=\sum_{m=1}^{\infty}h_{m}*_{t,x}(e^{CH^{\rho}})(T,x),$
(6.68)

with

$h_{m}(t,x)=h(t,x)\mathbbm{1}_{\\{m-1\leq|x|<m\\}}(x).$ (6.69)

For each $p>1$, we use Hölder’s inequality to obtain

$\displaystyle
h_{m}*(e^{CH^{\rho}})(T,x)=\int_{\tau}^{T}\int\mathbbm{1}_{\\{m-1\leq|x-y|<m\\}}(y)\mathcal{P}(\kappa(T-\tilde{\tau}),x-y)[e^{CH^{\rho}}](\tilde{\tau},y)dyd\tilde{\tau}$
(6.70)
$\displaystyle\leq\Big{(}\int_{\tau}^{T}\int_{m-1\leq|x-y|<m}\Big{(}\mathcal{P}(\kappa(T-\tilde{\tau}),x-y)\Big{)}^{p^{\prime}}dyd\tilde{\tau}\Big{)}^{{1}/{p^{\prime}}}\Big{(}\int_{\tau}^{T}\int_{m-1\leq|x-y|<m}(e^{CH^{\rho}})^{p}(\tilde{\tau},y)dyd\tilde{\tau}\Big{)}^{1/{p}}$
$\displaystyle=I_{1}^{1/p^{\prime}}\cdot I_{2}^{1/p}.$

Here, $p^{\prime}\in(1,\infty)$ is the conjugate exponent to $p$:

$\frac{1}{p}+\frac{1}{p^{\prime}}=1.$

The integral $I_{1}$ in the right side of (6.70) can be estimated using the
decay bound (6.6) on the fractional heat kernel as

$\displaystyle I_{1}\leq
C\int_{\tau}^{T}\int_{m-1\leq|x-y|<m}\Big{(}{(\kappa(T-\tilde{\tau}))^{{n}/{(2s)}}}{\Big{[}1+\frac{|x-y|^{2}}{(\kappa(T-\tilde{\tau}))^{{1}/{s}}}\Big{]}^{\frac{n}{2}+s}}\Big{)}^{-p^{\prime}}dyd\tilde{\tau}.$
(6.71)

The standard change of variables $z=(x-y)/(\kappa(T-\tilde{\tau}))^{1/(2s)}$
gives

$\displaystyle I_{1}$ $\displaystyle\leq
C\int_{\tau}^{T}(T-\tilde{\tau})^{-\frac{n}{2s}(p^{\prime}-1)}\int_{\frac{m-1}{(\kappa(T-\tilde{\tau}))^{{1}/{(2s)}}}\leq|z|<\frac{m}{(\kappa(T-\tilde{\tau}))^{{1}/{(2s)}}}}\frac{1}{(1+|z|^{2})^{p^{\prime}(\frac{n}{2}+s)}}dzd\tilde{\tau}$
(6.72) $\displaystyle\leq
C\int_{\tau}^{T}(T-\tilde{\tau})^{-\frac{n}{2s}(p^{\prime}-1)}\int_{\frac{m-1}{(\kappa(T-\tilde{\tau}))^{{1}/{(2s)}}}}^{\frac{m}{(\kappa(T-\tilde{\tau}))^{{1}/{(2s)}}}}\frac{r^{n-1}}{(1+r^{2})^{p^{\prime}(\frac{n}{2}+s)}}drd\tilde{\tau}.$

Using the inequality

$\int_{a}^{b}\frac{r^{n-1}}{(1+r^{2})^{p(n/2+s)}}dr\leq\frac{b-a}{(1+a^{2})^{p(n/2+s)-(n-1)/2}}.$
(6.73)

in (6.72) gives

$\displaystyle I_{1}$ $\displaystyle\leq
C\int_{\tau}^{T}(T-\tilde{\tau})^{-\frac{n}{2s}(p^{\prime}-1)}\frac{1}{(T-\tilde{\tau})^{{1}/{2s}}}\Big{(}1+\frac{(m-1)^{2}}{(\kappa(T-\tilde{\tau}))^{{1}/{s}}}\Big{)}^{-(\frac{n}{2}(p^{\prime}-1)+p^{\prime}s+\frac{1}{2})}d\tilde{\tau}$
(6.74) $\displaystyle\leq
C(m-1)^{-(n(p^{\prime}-1)+2p^{\prime}s+1)}\int_{\tau}^{T}(T-\tilde{\tau})^{-\frac{1}{2s}(np^{\prime}-n+1)+\frac{1}{2s}(n(p^{\prime}-1)+2p^{\prime}s+1)}d\tilde{\tau}$
$\displaystyle=C(m-1)^{-(n(p^{\prime}-1)+2p^{\prime}s+1)}\int_{\tau}^{T}(T-\tilde{\tau})^{p^{\prime}}d\tilde{\tau}\leq
C(m-1)^{-(n(p^{\prime}-1)+2p^{\prime}s+1)},$

for $m>1$. We used the fact that $T-2\leq\tau\leq T-1$ in the last step. On
the other hand, for $m=1$, we use (6.72) to get the bound

$\displaystyle I_{1}^{m=1}\leq
C\int_{\tau}^{T}(T-\tilde{\tau})^{-\frac{n}{2s}(p^{\prime}-1)}\int_{0}^{\infty}\frac{r^{n-1}}{(1+r^{2})^{p^{\prime}(\frac{n}{2}+s)}}drd\tilde{\tau}\leq
C,$ (6.75)

as long as

$p^{\prime}\big{(}\frac{n}{2}+s\big{)}>\frac{1}{2},$ (6.76)

and

$\frac{n}{2s}(p^{\prime}-1)<1.$ (6.77)

Note that (6.76) holds simply because $p^{\prime}>1$, while for (6.77) to hold
we need

$1<p^{\prime}<1+\frac{2s}{n}.$ (6.78)

The integral $I_{2}$ in the right side of (6.70) can be bounded using Lemma
6.5 and the fact that the set

$(B_{m}(x)\setminus B_{m-1}(x))\times[\tau,T]\subset(B_{m}(x)\setminus
B_{m-1}(x))\times[T-2,T]$

can be covered by $O(m^{n-1})$ $s$-parabolic cylinders of size $1$. This gives

$\int_{(B_{m}({x})\setminus
B_{m-1}({x}))\times[0,T]}(e^{CH^{\rho}})^{p}dyds\leq Cm^{n-1}.$ (6.79)

Using (6.71) and (6.79) in (6.70) leads to

$\displaystyle h_{m}*(e^{cH^{\rho}})(T,x)\leq
Cm^{\frac{n-1}{p}-n(1-\frac{1}{p^{\prime}})-2s-\frac{1}{p^{\prime}}}=Cm^{-2s-1}.$
(6.80)

Inserting this bound into (6.68) gives

$\displaystyle H_{2}(T,x)$ $\displaystyle\leq\sum_{m=1}^{\infty}Cm^{-2s-1}\leq
C,$ (6.81)

which is uniform in $\tau$ and $T$.

Next, we look at the term $H_{1}(T,x)$ in (6.67). Averaging over
$\tau\in[T-2,T-1]$ we obtain

$H_{1}(T,x)\leq\int_{T-2}^{T-1}\int_{\mathbb{R}^{n}}\mathcal{P}(\kappa(T-\tau),x-y)H(\tau,y)dyd\tau.$
(6.82)

By Proposition 6.3, we have $(H)_{Q}\leq M$ for any unit parabolic cylinder
$Q$, independent of $T$ as well. Therefore, we have, with the help of (6.6)

$\displaystyle H_{1}(T,x)$
$\displaystyle=\sum_{m=1}^{\infty}\int_{(B_{m}(0)\setminus
B_{m-1}(0))\times[T-2,T-1]}\mathcal{P}(\kappa(T-\tau),y)H(x-y,\tau)dyd\tau$
(6.83) $\displaystyle\leq C\sum_{m=1}^{\infty}\int_{(B_{m}(0)\setminus
B_{m-1}(0))\times[T-2,T-1]}\frac{1}{(1+(m-1)^{2})^{\frac{n+2s}{2}}}H(x-y,\tau)dyd\tau$
$\displaystyle\leq C\sum_{m=1}^{\infty}m^{n-1}m^{-n-2s}\leq C.$

This finishes the proof of Theorem 1.2. $\Box$

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