3. (iii)

There exists a ${\textbf{v}}\in[0,\infty)^{d}$ such that

(29)
${\int_{\Theta}\bigvee_{j=1}^{d}w^{(j)}v^{(j)}\Phi(d{\textbf{w}})=\sum_{g=1}^{G}\int_{\Theta^{(O_{g})}}\bigvee_{j\in
O_{g}}w^{(j)}v^{(j)}\Phi^{(O_{g})}(d{\textbf{w}}^{(O_{g})}).}$

######  Proof D.7 (Proof of Proposition D.6).

The equivalence between (i) and (ii) falls down from definitions. The
implication (ii) $\implies$ 29 is trivial. We show now 29 $\implies$ (ii)
Notice that for every Borel set $B$ of $S_{d}$, we have

${\Phi(B)=\sum_{g=1}^{G}\Phi(B\cap\Theta^{(O_{g})})+\Phi\left(B\cap(\Theta\setminus\cup_{g=1}^{G}\Theta^{(O_{g})})\right)\geq\sum_{g=1}^{G}\Phi(B\cap\Theta^{(O_{g})})=\Phi_{\Pi}(B).}$

The identity in Equation (29) can be rewritten as

${\int_{\Theta}\bigvee_{j=1}^{d}w^{(j)}v^{(j)}(\Phi-\Phi_{\Pi})(d\textbf{w})=0.}$

From above, we know that $(\Phi-\Phi_{\Pi})$ defined a positive measure. For
every Borel set $B$ of $S_{d}$, we have

${\int_{B}\bigvee_{j=1}^{d}w^{(j)}v^{(j)}(\Phi-\Phi_{\Pi})(d\textbf{w})\leq\int_{\Theta}\bigvee_{j=1}^{d}w^{(j)}v^{(j)}(\Phi-\Phi_{\Pi})(d\textbf{w})=0.}$

Since the function $\textbf{w}\mapsto\bigvee_{j=1}^{d}w^{(j)}v^{(j)}$ is
strictly positive, continuous and defined on a compact set, we have that
$\bigvee_{j=1}^{d}w^{(j)}v^{(j)}\geq c$ for a certain constant $c$ strictly
positive and we obtain

${c(\Phi-\Phi)(B)\leq\int_{B}\bigvee_{j=1}^{d}w^{(j)}v^{(j)}(\Phi-\Phi_{\Pi})(d\textbf{w})=0.}$

The following identity is obtained

${\Phi(B)=\Phi_{\Pi}(B),}$

since $B$ is taken arbitrary from the Borelian of $\Theta$, we conclude.

One can notice that the integrals defined in (29) can be rewritten with the
help of stable tail dependence function, that is

${L\left(v^{(1)},\dots,v^{(d)}\right)=\sum_{g=1}^{G}L^{(O_{g})}\left(\textbf{v}^{(O_{g})}\right),\quad\textbf{v}\in[0,\infty)^{d},}$

since for every $\textbf{x}\in[0,\infty)^{d}$

${L(\textbf{v})=\int_{\Theta}\bigvee_{j=1}^{d}w^{(j)}v^{(j)}\Phi(d\textbf{w}).}$

###  D.2 Additional results of Section 3

To establish the strong consistency of the estimator $\hat{\nu}_{n,m}$ in (7),
certain conditions on the mixing coefficients must be satisfied. $\mathcal{C}$
Let $m_{n}=o(n)$. The series $\sum_{n\geq 1}\beta(m_{n})$ is convergent, where
$\beta$ is defined in (18). For the sake of notational simplicity, we will
write $m=m_{n}$, $k=k_{n}$. The convergence of the series of $\beta$-mixing
coefficients in Condition D.2 is necessary to obtain the strong consistency of
$\hat{\nu}_{n,m}$, and it can be achieved through the sufficiency condition of
the Glivencko-Cantelli lemma for almost sure convergence.

######  Proposition D.8.

Let $({\textbf{Z}_{t}},t\in\mathbb{Z})$ be a stationary multivariate random
process. Under Conditions 2.1 and D.2, the madogram estimator in (7) is
strongly consistent, i.e.,

$\left|\hat{\nu}_{n,m}-\nu\right|\overunderset{a.s.}{n\rightarrow\infty}{\longrightarrow}0,$

with $\nu$ the theoretical madogram of the extreme value random vector X given
in (5).

Let $C_{n,m}^{o}$ be the empirical estimator of the copula $C_{m}$ based on
the (unobservable) sample $(U_{m,1}^{(j)},\dots,U_{m,k}^{(j)})$ for
$j\in\\{1,\dots,d\\}$. The proof of Proposition D.8 will use twice Lemma E.1,
which shows that $||C_{n,m}^{o}-C||_{\infty}$ converges almost surely to $0$.
The proof of this lemma is postponed to E.1 of supplementary results.

######  Proof D.9 (Proof of Proposition D.8).

We aim to show the following convergence

$|\hat{\nu}_{n,m}-\nu|\overunderset{\textrm{a.s.}}{n\rightarrow\infty}{\longrightarrow}0,$

where $\nu$ is the theoretical madogram of the extreme value random vector X
given in (5) and $\hat{\nu}_{n,m}$ the madogram estimator in (7). Let us
define the following quantity

(30)
$\hat{\nu}_{n,m}^{o}=\frac{1}{k}\sum_{i=1}^{k}\left[\bigvee_{j=1}^{d}U_{m,i}^{(j)}-\frac{1}{d}\sum_{j=1}^{d}U_{m,i}^{(j)}\right],$

that is the madogram estimated through the sample
$\textbf{U}_{m,1},\dots,\textbf{U}_{m,k}$. Following Lemma A.1 of
[MARCON20171], we can show that

$\hat{\nu}_{n,m}^{o}-\nu=\phi(C_{n,m}^{o}-C),$

with $\phi:\ell^{\infty}([0,1]^{d})\rightarrow\ell^{\infty}(\Delta_{d-1})$,
$f\mapsto\phi(f)$ defined by

$\phi(f)=\frac{1}{d}\sum_{j=1}^{d}\int_{[0,1]}f(1,\dots,1,\underbrace{u}_{\textrm{$j$-th
component}},1,\dots,1)du-\int_{[0,1]}f(u,\dots,u)du.$

Using Conditions 2.1 and D.2, by Lemma E.1 in E.1, as
$||C_{n,m}^{o}-C||_{\infty}$ converges almost surely to $0$, we obtain that

(31)
$\left|\hat{\nu}_{n,m}^{o}-\nu\right|\overunderset{\textrm{a.s.}}{n\rightarrow\infty}{\longrightarrow}0.$

Furthermore, using the chain of inequalities and again Lemma E.1 in E.1,

$\displaystyle\left|\hat{\nu}_{n,m}-\hat{\nu}^{o}_{n,m}\right|$
$\displaystyle\leq
2\underset{j\in\\{1,\dots,d\\}}{\sup}\underset{x\in\mathbb{R}}{\sup}\left|\hat{F}_{n,m}^{(j)}(x)-F_{m}^{(j)}(x)\right|$
$\displaystyle\leq
2\underset{j\in\\{1,\dots,d\\}}{\sup}\underset{u\in[0,1]}{\sup}\left|\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{U_{m,i}^{(j)}\leq
u\\}}-u\right|.$

Then we obtain that

(32)
$\left|\hat{\nu}_{n,m}-\hat{\nu}^{o}_{n,m}\right|\overunderset{\textrm{a.s.}}{n\rightarrow\infty}{\longrightarrow}0.$

Now, write

$\left|\hat{\nu}_{n,m}-\nu\right|\leq\left|\hat{\nu}_{n,m}-\nu_{n,m}^{o}\right|+\left|\hat{\nu}_{n,m}^{o}-\nu\right|,$

and use Equations (31) and (32) to obtain the statement.

We present here the strong consistency of our procedure when the dimension $d$
is fixed the sample size $n$ grows at infinity. The main technicality of the
proof has already been tackled in Proposition D.8 and we state the precise
formulation of this theorem below.

######  Theorem D.10.

Consider the AI-block model as defined in Definition 2.2 under Condition 2.3
and $({\textbf{Z}_{t}},t\in\mathbb{Z})$ be a stationary multivariate random
process. For a given $\mathcal{X}$ and its corresponding estimator
$\hat{\mathcal{X}}$, if Conditions 2.1, D.2 holds, then taking $\tau=0$

$\mathbb{P}\left\\{\hat{O}=\bar{O}\right\\}=1,\quad\textrm{as
}n\rightarrow\infty.$

######  Proof D.11 (Proof of Theorem D.10).

If $a$ and $b$ are not in the same cluster according to $\bar{O}$, i.e.
$a\overset{\bar{O}}{\not\sim}b$, then $\chi(a,b)=0$. Therefore, using
Proposition D.8 along with Conditions 2.1 and D.2, we can conclude that almost
surely

$\underset{n\rightarrow\infty}{\lim}\,\hat{\chi}_{n,m}(a,b)=0\leq\tau.$

Now, if $a\overset{\bar{O}}{\sim}b$, then $\chi(a,b)>0$ and again by
Propositions D.8 and Conditions 2.1, D.2, we obtain

$\underset{n\rightarrow\infty}{\lim}\,\hat{\chi}_{n,m}(a,b)=\chi(a,b)>0,$

where the the strict positiveness is obtain through Condition 2.3, hence

$a\overset{\bar{O}}{\sim}b\iff\underset{n\rightarrow\infty}{\lim}\,\hat{\chi}_{n,m}(a,b)>\tau.$

Let us prove Theorem D.10 by induction on the algorithm step $l$. We consider
the algorithm at some step $l-1$ and assume that the algorithm was consistent
up to this step, i.e. $\hat{O}_{j}=\bar{O}_{j}$ for $j=1,\dots,l-1$.

If $\underset{n\rightarrow\infty}{\lim}\hat{\chi}_{n,m}(a_{l},b_{l})=0$, then
no $b\in S$ is in the same group of $a_{l}$. Since the algorithm has been
consistent up to this step $l$, it means that $a_{l}$ is a singleton and
$\hat{O}_{l}=\\{a_{l}\\}$.

If $\underset{n\rightarrow\infty}{\lim}\hat{\chi}_{n,m}(a_{l},b_{l})>\tau$,
then $a_{l}\overset{\bar{O}}{\sim}b$. The equivalence above implies that
$\hat{O}_{l}=S\cap\bar{O}_{l}$. Since the algorithm has been consistent up
until this step, we know that $\hat{O}_{l}=\bar{O}_{l}$. Therefore, the
algorithm remains consistent at step $l$ with probability tending to one as
$n\rightarrow\infty$, and Theorem D.10 follows by induction.

##  Appendix E Further results

###  E.1 A usefull Glivenko-Cantelli result for the copula with known margins
in a weakly dependent setting

In this section, we will prove an important auxiliary result: the empirical
copula estimator $\hat{C}_{n,m}^{o}$ based on the weakly dependent sample
$\textbf{U}_{m,1},\dots,\textbf{U}_{m,k}$ is uniformly strongly consistent
towards the extreme value copula $C$. This result is a main tool to obtain
important results in the paper such as Proposition D.8, Theorem D.10 and
Proposition A.1. For that purpose, the Berbee’s coupling lemma is of prime
interest (see, e.g., [rio2017asymptotic, Chapter 5]) which gives an
approximation of the original process by conveniently defined independent
random variables.

######  Lemma E.1.

Under conditions of Proposition D.8, we have

$||C_{n,m}^{o}-C||_{\infty}\overunderset{\textrm{a.s.}}{n\rightarrow\infty}{\longrightarrow}0.$

######  Proof E.2 (Lemma E.1).

Using triangle inequality, one obtain the following bound

(33)
$||C_{n,m}^{o}-C||_{\infty}\leq||C_{n,m}^{o}-C_{m}||_{\infty}+||C_{m}-C||_{\infty}.$

As $\\{C_{m},n\in\mathbb{N}\\}$ is an equicontinuous class of functions (for
every $m$, $C_{m}$ is a copula hence a 1-Lipschitz function), defined on the
compact set $[0,1]^{d}$ (by Tychonov’s theorem) which converges pointwise to
$C$ by Condition 2.1. Then the convergence is uniform over $[0,1]^{d}$. Thus
the second term of the RHS of Equation (33) converges to $0$ almost surely.

Now, let us prove that $||C_{n,m}^{o}-C_{m}||_{\infty}$ converges almost
surely to $0$. By Berbee’s coupling lemma (see [rio2017asymptotic, Theorem
6.1] or [bucher2014extreme, Theorem 3.1] for similar applications), we can
construct inductively a sequence
$(\bar{\textbf{Z}}_{im+1},\dots,\bar{\textbf{Z}}_{im+m})_{i\geq 0}$ such that
the following three properties hold:

  1. (i)

$(\bar{\textbf{Z}}_{im+1},\dots,\bar{\textbf{Z}}_{im+m})\overset{d}{=}(\textbf{Z}_{im+1},\dots,\textbf{Z}_{im+m})$
for any $i\geq 0$;

  2. (ii)

both $(\bar{\textbf{Z}}_{2im+1},\dots,\bar{\textbf{Z}}_{2im+m})_{i\geq 0}$ and
$(\bar{\textbf{Z}}_{(2i+1)m+1},\dots,\bar{\textbf{Z}}_{(2i+1)m+m})_{i\geq 0}$
sequences are independent and identically distributed;

  3. (iii)

$\mathbb{P}\\{(\bar{\textbf{Z}}_{im+1},\dots,\bar{\textbf{Z}}_{im+m})\neq(\textbf{Z}_{im+1},\dots,\textbf{Z}_{im+m})\\}\leq\beta(m)$.

Let $\bar{C}_{n,m}^{o}$ and $\bar{\textbf{U}}_{m,i}$ be defined analogously to
$C_{n,m}^{o}$ and $\textbf{U}_{m,i}$ respectively but with
$\textbf{Z}_{1},\dots,\textbf{Z}_{n}$ replaced with
$\bar{\textbf{Z}}_{1},\dots,\bar{\textbf{Z}}_{n}$. Now write

(34)
$C_{n,m}^{o}(\textbf{u})=\bar{C}_{n,m}^{o}(\textbf{u})+\left\\{C_{n,m}^{o}(\textbf{u})-\bar{C}_{n,m}^{o}(\textbf{u})\right\\}.$

We will show below that the term under brackets converges uniformly to $0$
almost surely. Write
$\bar{C}_{n,m}^{o}(\textbf{u})=\bar{C}_{n,m}^{o,\textrm{odd}}(\textbf{u})+\bar{C}_{n,m}^{o,\textrm{even}}(\textbf{u})$
where $\bar{C}_{n,m}^{o,\textrm{odd}}(\textbf{u})$ and
$\bar{C}_{n,m}^{o,\textrm{even}}(\textbf{u})$ are defined as sums over the odd
and even summands of $\bar{C}_{n,m}^{o}(\textbf{u})$, respectively. Since both
of these sums are based on i.i.d. summands by properties (i) and (ii), we have
$||\bar{C}^{o}_{n,m}-C_{m}||_{\infty}\overunderset{\textrm{a.s.}}{n\rightarrow\infty}{\longrightarrow}0$
using Glivenko-Cantelli (see [bestbook, Chapter 2.5]).

It remains to control the term under brackets on the right hand side of
Equation (34), we have that

$\displaystyle\left|C_{n,m}^{o}(\textbf{u})-\bar{C}_{n,m}^{o}(\textbf{u})\right|$
$\displaystyle\leq\frac{1}{k}\sum_{i=1}^{k}\left|\mathds{1}_{\\{\bar{\textbf{U}}_{m,i}\leq\textbf{u}\\}}-\mathds{1}_{\\{\textbf{U}_{m,i}\leq\textbf{u}\\}}\right|$
$\displaystyle\leq\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{(\bar{Z}_{im+1},\dots,\bar{Z}_{im+m})\neq(Z_{im+1},\dots,Z_{im+m})\\}}.$

Hence, using Markov’s inequality and property (iii), we have

$\mathbb{P}\left\\{\underset{u\in[0,1]^{d}}{\sup}\left|\bar{C}_{n,m}^{o}(\textbf{u})-C_{n,m}^{o}(\textbf{u})\right|>\epsilon\right\\}\leq\frac{\beta(m)}{\epsilon}.$

Thus by Condition D.2,

$\sum_{n\geq
1}\mathbb{P}\left\\{\underset{u\in[0,1]^{d}}{\sup}\left|\bar{C}_{n,m}^{o}(\textbf{u})-C_{n,m}^{o}(\textbf{u})\right|>\epsilon\right\\}<\infty.$

Applying Borel-Cantelli gives the desired convergence to $0$ almost surely of
the term under bracket in Equation (34). Gathering all results gives that the
term $||C_{n,m}^{o}-C_{m}||_{\infty}$ converges almost surely to 0. Hence the
statement using Equation (33).

###  E.2 Proof of Proposition A.1 in A.1

Proofs of consistency theorems for $k$-means clustering oftenly needs a
uniform strong law of large numbers (SLLN) stated as

(35) $\underset{g\in\mathcal{G}}{\sup}\left|\int
gd(\mathbb{P}_{n}-\mathbb{P})\right|\overunderset{a.s.}{n\rightarrow\infty}{\longrightarrow}0,$

(see Section 4 of [pollard1981strong]), where $\mathbb{P}$ is a probability
measure on $\mathcal{B}(\mathbb{R}^{d})$, $\mathbb{P}_{n}$ the empirical
measure and $\mathcal{G}$ a class of functions. Equation (35) is also stated
as the class of functions $\mathcal{G}$ is Glivenko-Cantelli (see Chapter 2 of
[bestbook]). In Proposition 3.3 of [janssen2020k], where $\mathbb{P}$ and
$\mathbb{P}_{n}$ are replaced respectively by the angular measure $S$ and its
empirical counterpart $S_{n}$, it is shown that condition (35) holds. In our
framework, we consider the extreme value copula $C$, the copula of block
maxima $C_{m}$, its empirical counterpart $\hat{C}_{n,m}$ where the second is
in the copula domain of attraction of $C$. The consistency of $k$-means
clustering using madogram directly comes down from argments given in
[janssen2020k] if we are able to state Equation (35) for a specific class of
function $\mathcal{G}$ where the madogram belongs to.

For this purpose, the notion of bounded variation of functions and in
particular the integration by part formula for Lebesgue-Stieltjes integral are
of prime interest (see, for example, [fermanian2004weak, Theorem 6]). We will
say that a function $f$ is of bounded variation in the sense of Hardy-Krause
if $V_{HK}(f)<\infty$ (see [radulovic2017weak, Section 2] for a definition).
Let us consider $\mathcal{G}$ as the class of functions which are continuous
and $V_{HK}(g)<\infty$.

######  Proof E.3 (Proof of Proposition A.1).

We want to prove that for every $g\in\mathcal{G}$

(36) $\int
gd\hat{C}_{n,m}\overunderset{a.s.}{n\rightarrow\infty}{\longrightarrow}\int
gdC.$

Using integration by parts, we have that

$\int gd\hat{C}_{n,m}=\Gamma(\hat{C}_{n,m},g),$

where $\Gamma(\cdot,g)$ is a linear and Lipschitz function, hence continuous.
Now, if we can state that

(37)
$||\hat{C}_{n,m}-C||_{\infty}\overunderset{a.s.}{n\rightarrow\infty}{\longrightarrow}0,$

we obtain that
$\Gamma(\hat{C}_{n,m},g)\overunderset{a.s.}{n\rightarrow\infty}{\longrightarrow}\Gamma(C,g)$,
using continuity, hence (36). Let us prove Equation (37). Using triangle
inequality, we have

$||\hat{C}_{n,m}-C||_{\infty}\leq||\hat{C}_{n,m}-C^{o}_{n,m}||_{\infty}+||C^{o}_{n,m}-C||_{\infty}.$

The second term in the right hand side converges almost surely to $0$ by Lemma
E.1 (see E.1) with Conditions 2.1 and D.2. It remains to work on the first
term. Now, we have

$\displaystyle||\hat{C}_{n,m}-C_{n,m}^{o}||_{\infty}$
$\displaystyle\leq\frac{1}{k}\sum_{i=1}^{k}|\mathds{1}_{\\{\hat{U}_{n,m,i}^{(j)}\leq
u^{(j)},1\leq j\leq d\\}}-\mathds{1}_{\\{U_{m,i}^{(j)}\leq u^{(j)},1\leq j\leq
d\\}}|$
$\displaystyle\leq\frac{1}{k}\sum_{i=1}^{k}{{{\Pi}}}_{j=1}^{d}\mathds{1}_{\\{\hat{U}_{n,m,i}^{(j)}\neq
U_{m,i}^{(j)}\\}}.$

Notice that, for every $i\in\\{1,\dots,k\\}$ and $j\in\\{1,\dots,d\\}$

$\displaystyle\mathds{1}_{\\{\hat{U}_{n,m,i}^{(j)}\neq U_{m,i}^{(j)}\\}}$
$\displaystyle=\mathds{1}_{\\{|\hat{U}_{n,m,i}^{(j)}-U_{m,i}^{(j)}|>0\\}}$
$\displaystyle\leq\mathds{1}_{\\{\underset{x\in\mathbb{R}}{\sup}|\hat{F}_{n,m}^{(j)}(x)-F_{m}^{(j)}(x)|>0\\}}=\mathds{1}_{\\{\underset{u\in[0,1]}{\sup}|\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{U_{m,i}^{(j)}\leq
u\\}}-u|>0\\}}.$

Thus

$||\hat{C}_{n,m}-C_{n,m}^{o}||_{\infty}\leq\Pi_{j=1}^{d}\mathds{1}_{\\{\underset{u\in[0,1]}{\sup}|\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{U_{m,i}^{(j)}\leq
u\\}}-u|>0\\}}.$

Denote by

$\displaystyle
A_{n}=\left\\{\omega\in\Omega:\underset{u\in[0,1]}{\sup}\left|\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{U_{m,i}^{(j)}\leq
u\\}}-u\right|(\omega)>0\right\\},$ $\displaystyle
A_{n,\epsilon}=\left\\{\omega\in\Omega:\underset{u\in[0,1]}{\sup}\left|\frac{1}{k}\sum_{i=1}^{k}\mathds{1}_{\\{U_{m,i}^{(j)}\leq
u\\}}-u\right|(\omega)>\epsilon\right\\},$

where $\epsilon>0$. Using Lemma E.1 in E.1, we have that

$\mathbb{P}\left\\{\underset{n\rightarrow\infty}{\mathrm{lim\,sup}}\,A_{n,\epsilon}\right\\}=0.$

Now, remark that $A_{n}=\bigcup_{N\geq 1}A_{n,1/N}$. We thus have

$\mathbb{P}\left\\{\underset{n\rightarrow\infty}{\mathrm{\lim\,sup}}\,A_{n}\right\\}=\mathbb{P}\left\\{\underset{n\rightarrow\infty}{\mathrm{\lim\,sup}}\,\bigcup_{N\geq
1}A_{n,1/N}\right\\}=\mathbb{P}\left\\{\bigcup_{N\geq
1}\underset{n\rightarrow\infty}{\mathrm{\lim\,sup}}\,A_{n,1/N}\right\\}.$

Using $\sigma$-subadditivity of measures, we have

$\mathbb{P}\left\\{\underset{n\rightarrow\infty}{\mathrm{\lim\,sup}}\,A_{n}\right\\}\leq\sum_{N\geq
1}\mathbb{P}\left\\{\underset{n\rightarrow\infty}{\mathrm{\lim\,sup}}\,A_{n,1/N}\right\\}=0.$

Hence $\mathds{1}_{A_{n}}=0$ almost surely as $n\rightarrow\infty$ which
results on $||\hat{C}_{n,m}-C_{n,m}^{o}||_{\infty}$ converges almost surely to
$0$. We thus obtain (37) hence (36).

In order to prove Proposition A.1, we prove in the following that the function

$g(\textbf{u})=\underset{g\in\\{1,\dots,G\\}}{\min}\frac{1}{2}\sum_{j=1}^{d}|u^{(j)}-\psi_{g}^{(j)}|,\quad\psi_{g}\in[0,1]^{d},\quad
g\in\\{1,\dots,G\\}.$

is of bounded variations in the sens of Hardy Krause (BHKV). Indeed, for every
$j\in\\{1,\dots,d\\}$, the functions $(\textbf{u},\textbf{v})\mapsto u^{(j)}$
and $(\textbf{u},\textbf{v})\mapsto v^{(j)}$ are BHKV on
$[0,1]^{d}\times[0,1]^{d}$ since it depends only on one variable and is
monotone in this variable. As the difference between two BHKV functions are
BHKV, it follows $\forall j\in\\{1,\dots,d\\}$ the function
$(\textbf{u},\textbf{v})\mapsto u^{(j)}-v^{(j)}$ is BHKV on
$[0,1]^{d}\times[0,1]^{d}$. Taking the absolute value preserves the BHKV
property on $[0,1]^{d}$. It follows that
$(\textbf{u},\textbf{v})\mapsto\sum_{j=1}^{d}|u^{(j)}-v^{(j)}|$ is BHKV as a
sum of functions that are BHKV. Multiplying by a constant preserves the
bounded variation property of the function. Finally, taking the $\min$ over a
finite set of values in $[0,1]^{d}$ maintains the BHKV property as it holds
for every $(\textbf{u},\textbf{v})\in[0,1]^{d}\times[0,1]^{d}$. Hence $g$ is
BHKV and clearly a continuous function. Thus using Equation (36) along with
arguments from [janssen2020k, Theorem 3.1] we obtain the consistency the
result.

###  E.3 Weak convergence of an estimator of $A^{(O)}-A$

We now state conditions on the block size $m$ and the number of blocks $k$, as
in [bucher2014extreme], to demonstrate the weak convergence of the empirical
copula process based on the (unobservable) sample
$(U_{n,m,1}^{(j)},\dots,U_{n,m,k}^{(j)})$ for every $j\in\\{1,\dots,d\\}$
under mixing conditions. An additional condition will be required within the
theorem to establish the weak convergence of the rank-based copula estimator
under the same mixing conditions.

$\mathcal{F}$ There exists a positive integer sequence $\ell_{n}$ such that
the following statement holds:

  1. (i)

$m_{n}\rightarrow\infty$ and $m_{n}=o(n)$

  2. (ii)

$\ell_{n}\rightarrow\infty$ and $\ell_{n}=o(m_{n})$

  3. (iii)

$k_{n}\alpha(\ell_{n})=o(1)$ and $(m_{n}/\ell_{n})\alpha(\ell_{n})=o(1)$

  4. (iv)

$\sqrt{k_{n}}\beta(m_{n})=o(1)$

For notational conveniency, we will write in the following $m_{n}=m$,
$k_{n}=k$, $\ell_{n}=\ell$. Note that Condition E.3 (iii) guarantees that the
limit $C$ is an extreme value copula by [hsing1989extreme, Theorem 4.2]. As
usual, the weak convergence of the empirical copula process stems down from
the finite dimensional convergence and the asymptotic tightness of the process
which then hold from Condition E.3 (iii) and (iv) respectively. In order to
apply Hadamard’s differentiability to obtain the weak convergence of the
empirical copula based on the sample’s scaled ranks, we need a classical
condition over the derivatives of the limit copula stated as follows.

$\mathcal{G}$ For any $j\in\\{1,\dots,d\\}$, the $j$th first order partial
derivative $\dot{C}^{(j)}=\partial C/\partial u^{(j)}$ exists and is
continuous on $\\{{\textbf{u}}\in[0,1]^{d},u^{(j)}\in(0,1)\\}$.

The estimator of the Pickands dependence function that we present is based on
the madogram concept ([cooley2006variograms, MARCON20171]), a notion borrowed
from geostatistics in order to capture the spatial dependence structure. Our
estimator is defined as

$\hat{A}_{n,m}(\textbf{t})=\frac{\hat{\nu}_{n,m}(\textbf{t})+c(\textbf{t})}{1-\hat{\nu}_{n,m}(\textbf{t})-c(\textbf{t})},$

where

$\displaystyle\hat{\nu}_{n,m}(\textbf{t})=\frac{1}{k}\sum_{i=1}^{k}\left[\bigvee_{j=1}^{d}\left\\{\hat{U}_{n,m,j}^{(j)}\right\\}^{1/t^{(j)}}-\frac{1}{d}\sum_{j=1}^{d}\left\\{\hat{U}_{n,m,i}^{(j)}\right\\}^{1/t^{(j)}}\right],\quad
c(\textbf{t})=\frac{1}{d}\sum_{j=1}^{d}\frac{t^{(j)}}{1+t^{(j)}},$

and $\hat{U}_{n,m,i}^{(j)}=\hat{F}_{n,m}^{(j)}(M_{m,i}^{(j)})$ corresponds to
ranks scaled by $k^{-1}$. By convention, here $u^{1/0}=0$ for $u\in(0,1)$. Let
$g\in\\{1,\dots,G\\}$ and define

$\hat{A}_{n,m}^{(O_{g})}\left(\textbf{t}^{(O_{g})}\right)=\hat{A}_{n,m}\left(\textbf{0},\textbf{t}^{(O_{g})},\textbf{0}\right)$

the empirical Pickands dependence function associated to the $k$-th subvector
of Xp. We consider the empirical process of the difference between estimates
of the Pickands dependence functions of subvectors
$\textbf{X}^{(O_{g})},g\in\\{1,\dots,G\\},$ and the estimator of the Pickands
dependence function of X:

$\mathcal{E}_{nG}(\textbf{t})=\sqrt{k}\left(\hat{A}^{(O)}_{n,m}(\textbf{t})-\hat{A}_{n,m}(\textbf{t})\right),$

where
$\hat{A}_{n,m}^{(O)}(\textbf{t})=\sum_{g=1}^{G}w^{(O_{g})}(\textbf{t})\hat{A}_{n,m}^{(O_{g})}(\textbf{t}^{(O_{g})})$.
Noticing that multiplying the above process by $d$ and taking
$\textbf{t}=(d^{-1},\dots,d^{-1})$ gives

$\sqrt{k}\widehat{SECO}(O)=\sqrt{k}\left(\sum_{g=1}^{G}\hat{\theta}_{n,m}^{(O_{g})}-\hat{\theta}_{n,m}\right).$

Hence, the weak convergence of the above empirical process will immediately
comes down from the one of the empirical process in $\mathcal{E}_{nG}$, as
stated in the theorem below.

######  Theorem E.4.

Consider the AI-block model in Definition 2.2 with a given partition $O$,
i.e., $A=A^{(O)}$. Under Conditions 2.1, E.3, E.3 and
$\sqrt{k}(C_{m}-C)\rightsquigarrow\Gamma$, the empirical process
$\mathcal{E}_{nG}$ converges weakly in $\ell^{\infty}(\Delta_{d-1})$ to a
tight Gaussian process having representation

$\displaystyle\mathcal{E}_{G}({\textbf{t}})$
$\displaystyle=\left(1+A({\textbf{t}})\right)^{2}\int_{[0,1]}(N_{C}+\Gamma)(u^{t^{(1)}},\dots,u^{t^{(d)}})du$
$\displaystyle-\sum_{g=1}^{G}w^{(O_{g})}({\textbf{t}})\left(1+A^{(O_{g})}({\textbf{t}}^{(O_{g})})\right)^{2}\int_{[0,1]}(N_{C}+\Gamma)({\textbf{1}},u^{t^{(i_{g,1})}},\dots,u^{t^{(i_{g,d_{g}})}},{\textbf{1}})du,$

where $N_{C}$ is a continuous tight Gaussian process with representation

$N_{C}(u^{(1)},\dots,u^{(d)})=B_{C}(u^{(1)},\dots,u^{(d)})-\sum_{j=1}^{d}\dot{C}_{j}(u^{(1)},\dots,u^{(d)})B_{C}({\textbf{1}},u^{(j)},{\textbf{1}}),$

and $B_{C}$ is a continuous tight Gaussian process with covariance function

$\mathrm{Cov}(B_{C}({\textbf{u}}),B_{C}({\textbf{v}}))=C({\textbf{u}\wedge\textbf{v}})-C({\textbf{u}})C({\textbf{v}})\overset{\mathcal{H}_{0}}{=}C_{\Pi}({\textbf{u}\wedge\textbf{v}})-C_{\Pi}({\textbf{u}})C_{\Pi}({\textbf{v}}).$

######  Proof E.5 (Theorem E.4).

The proof is straightforward, notice that by the triangle diagram in Figure 9

$\mathcal{E}_{nG}=\psi\circ\phi\left(\sqrt{k}(\hat{A}_{n,m}-A)\right),$

where $\phi$ is detailed as

$\begin{array}[]{ccccc}\phi&:&\ell^{\infty}(\Delta_{d-1})&\to&\ell^{\infty}(\Delta_{d-1})\otimes(\ell^{\infty}(\Delta_{d-1}),\dots,\ell^{\infty}(\Delta_{d-1}))\\\
&&x&\mapsto&(x,\phi_{1}(x),\dots,\phi_{G}(x)),\\\ \end{array}$

with for every $g\in\\{1,\dots,G\\}$

$\begin{array}[]{ccccc}\phi_{g}&:&\ell^{\infty}(\Delta_{d-1})&\to&\ell^{\infty}(S_{d})\\\
&&x&\mapsto&w^{(O_{g})}(t^{(1)},\dots,t^{(G)})x(\textbf{0},t^{(i_{g,1})},\dots,t^{(i_{g,d_{g}})},\textbf{0}),\\\
\end{array}$

and also

$\begin{array}[]{ccccc}\psi&:&\ell^{\infty}(\Delta_{d-1})\otimes(\ell^{\infty}(\Delta_{d-1}),\dots,\ell^{\infty}(\Delta_{d-1}))&\to&\ell^{\infty}(\Delta_{d-1})\\\
&&(x,\phi_{1}(x),\dots,\phi_{G}(x))&\mapsto&\sum_{g=1}^{G}\phi_{g}(x)-x.\\\
\end{array}$

$\sqrt{k}\left(\hat{A}_{n,m}-A\right)$$\mathcal{E}_{nG}$$\left(\sqrt{k}\left(\hat{A}_{n,m}-A\right);w^{(O_{1})}\sqrt{k}\left(\hat{A}_{n,m}^{(O_{1})}-A^{(O_{1})}\right),\dots,w^{(O_{G})}\sqrt{k}\left(\hat{A}_{n,m}^{(O_{G})}-A^{(O_{G})}\right)\right)$$\phi$$\psi$
Figure 9: Commutative diagram of composition of function.

The function $\phi_{g}$ is a linear and bounded function hence continuous for
every $g$, it follows that $\phi$ is continuous since each coordinate
functions is continuous. As a linear and bounded function, $\psi$ is also a
continuous function. Noticing that,

$(C_{m}-C)(\textbf{1},u,\textbf{1})=0,\quad\forall n\in\mathbb{N},$

where $m=m_{n}$ is the block length for a sample size $n$. We thus have

$\sqrt{k}(C_{m}-C)(\textbf{1},u,\textbf{1})\underset{n\rightarrow\infty}{\longrightarrow}0,$

with $k=k_{n}$ the number of blocks. Therefore
$\Gamma(\textbf{1},u,\textbf{1})=0$. Combining this equality with
[bucher2014extreme, Corollary 3.6] and the same techniques as in the proof of
[MARCON20171, Theorem 2.4], we obtain along with Conditions 2.1, E.3, E.3

$\sqrt{k}(\hat{A}_{n,m}(\textbf{t})-A(\textbf{t}))\rightsquigarrow-\left(1+A(\textbf{t})\right)^{2}\int_{[0,1]}(N_{C}+\Gamma)(u^{t^{(1)}},\dots,u^{t^{(d)}})du.$

Applying the continuous mapping theorem for the weak convergence in
$\ell^{\infty}(\Delta_{d-1})$ ([bestbook, Theorem 1.3.6]) leads the result.


# Long time behaviour of solutions to non-local and non-linear dispersion
problems

Maciej Tadej
<EMAIL_ADDRESS>Insitute of Mathematics, University of Wrocław,
pl. Grunwaldzki 2, 50-384 Wrocław

###### Abstract

This paper explores a non-linear, non-local model describing the evolution of
a single species. We investigate scenarios where the spatial domain is either
an arbitrary bounded and open subset of the $n$-dimensional Euclidean space or
a periodic environment modeled by $n$-dimensional torus. The analysis includes
the study of spectrum of the linear, bounded operator in the considered
equation, which is a scaled, non-local analogue of classical Laplacian with
Neumann boundaries. In particular we show the explicit formulas for
eigenvalues and eigenfunctions. Moreover we show the asymptotic behaviour of
eigenvalues. Within the context of the non-linear evolution problem, we
establish the existence of an invariant region, give a criterion for
convergence to the mean mass, and construct spatially heterogeneous steady
states.

Keywords : non-local dispersion, spectral analysis, asymptotics, equilibria,
stability

##  1 Introduction

###  1.1 Statement and main results

In this paper we study properties of solutions $u:=u(x,t)$ of the following
non-linear, non-local model describing the evolution of a density of single
species.

$\displaystyle u_{t}(x,t)=\mathcal{L}_{\varepsilon}u(x,t)+f(u(x,t))$
$(x,t)\in\Omega\times(0,\infty),$ (1.1) $\displaystyle u(x,0)=u_{0}(x)$
$x\in\Omega,$ (1.2)

where $\Omega\subset\mathbb{R}^{n}$ is a bounded and open domain. We assume
that an initial datum $u_{0}\in L^{2}\left(\Omega\right)$, a scaling parameter
$\varepsilon>0$ and the force term $f\in C^{1}\left(\mathbb{R}\right)$. The
dispersion operator
$\mathcal{L}_{\varepsilon}:L^{2}\left(\Omega\right)\rightarrow
L^{2}\left(\Omega\right)$ is given by the formula

$\displaystyle\mathcal{L}_{\varepsilon}u(x,t)=\int_{\Omega}J_{\varepsilon}(x-y)\left(u(y,t)-u(x,t)\right)\>dy,$
(1.3)

where $J_{\varepsilon}:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ is a
properly scaled version of the integral kernel
$J:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$. Here, we distinguishh two
frameworks stated in the following assumptions.

######  Assumption 1.1 (General case).

Denote by $\Omega\subset\mathbb{R}^{n}$ an open and bounded region. We
consider the integral kernels $J\in C(\mathbb{R}^{n})$ satisfying the
following properties

  1. (J1)

$J(z)\geq 0$ for $z\in\mathbb{R}^{n}$ and $J(0)>0$,

  2. (J2)

$J(z)=J(-z)$ for $z\in\mathbb{R}^{n}$,

  3. (J3)

$J(z)\rightarrow 0$ as $|z|\rightarrow\infty$,

  4. (J4)

$\int_{\mathbb{R}^{n}}J(z)|z|^{2}dz<\infty$.

  5. (J5)

$\int_{\mathbb{R}^{n}}J(z)dz=1$.

We introduce $\varepsilon>0$ and $m\in[0,2]$ to define the re-scaled kernel
$J_{\varepsilon}\in C(\mathbb{R}^{n})$

$J_{\varepsilon}(z)=\frac{C_{norm}}{\varepsilon^{m+n}}J\left(\frac{z}{\varepsilon}\right),\quad
C_{norm}=\left(\frac{1}{2}\int_{\mathbb{R}^{n}}J(z)|z|^{2}\>dz\right)^{-1}.$
(1.4)

Now we define the setting where dispersion outside the domain is identified by
the reflection of boundaries.

######  Assumption 1.2 (Periodic case).

Denote by $\mathbb{T}^{n}=(-\pi,\pi)^{n}$ an $n$-dimensional torus. We
consider the integral kernels $J\in C(\mathbb{R}^{n})$ satisfying claims (J1)
\- (J3) of Assumption 1.4. Next we introduce $\varepsilon>0$ and $m\in[0,2]$
to define the re-scaled, truncated and periodized kernel $J_{\varepsilon}\in
C(\mathbb{T}^{n})$ given by

$J_{\varepsilon}(z)=\frac{C_{norm}^{\varepsilon}}{\varepsilon^{m+n}}J_{per}\left(\frac{z}{\varepsilon}\right),\quad
C_{norm}^{\varepsilon}=\left(\frac{1}{2}\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}(z)|z|^{2}\>dz\right)^{-1}.$
(1.5)

We extended the kernel periodically in such a way that
$J_{per}\left(\frac{z}{\varepsilon}\right)$ is a $2\pi$-periodic function and
so $J_{per}(z)$ is $\frac{2\pi}{\varepsilon}$-periodic kernel.

The study of non-local dispersion, initiated by Hutson et al. [1, 2] and Fife
[3] in ecological contexts, characterizes population density $u:=u(x,t)$ in
the spatial domain $\Omega\subset\mathbb{R}^{n}$ over infinite time. The
dispersion operator $\mathcal{L}_{\varepsilon}$ captures population influx and
efflux at location $x$. Assumption 1.4 outlines confined dispersal without
boundary flux, posing technical challenges. In [1] a simpler, periodic
framework was introduced as in Assumption 1.2. This considers movements
outside the boundary as reflections back into the domain $\Omega$. Further
comments on related results follow after a brief overview of the main results.

We underscore the significance of analyzing the spectrum
$\sigma(\mathcal{L}_{\varepsilon})$ of the dispersion operator
$\mathcal{L}_{\varepsilon}$ in this work. In addition to addressing the
continuous part of this set, we aim to solve the corresponding eigenvalue
problem

$\mathcal{L}_{\varepsilon}\varphi_{k}=\beta_{k}\varphi_{k}.$ (1.6)

In Theorem 2.1 below, we characterize $\sigma(\mathcal{L}_{\varepsilon})$ when
$\Omega$ and $J_{\varepsilon}$ satisfy Assumption 1.4. In the periodic setting
defined by Assumption 1.2, Theorem 2.2 provides explicit formulas for
$(\beta_{k},\varphi_{k})$. Additionally, we explore the asymptotic behavior of
the obtained eigenvalues as $|k|\rightarrow\infty$ and $\varepsilon\rightarrow
0$.

The spatially homogeneous asymptotic behavior of solutions to the Cauchy
problem (1.1)-(1.2) is the focus of our next investigation. We seek an
invariant region $\Gamma\subset\mathbb{R}$. Specifically, we demonstrate that
every solution $u$ satisfies the following relation almost everywhere in
$\Omega$ for all $t>0.0$.

$u_{0}(x)\in\Gamma\implies u(x,t)\in\Gamma.$ (1.7)

The existence of an invariant region $\Gamma\subset\mathbb{R}$ is confirmed in
Theorem 3.2 when the force term $f$ has two distinct zeros. Let $\beta_{1}$
denote the largest non-zero eigenvalue of $\mathcal{L}_{\varepsilon}$. We
assume that the following condition holds true:

$\sigma=2\left(\beta_{1}+\max_{u\in\Gamma}|f^{\prime}(u)|\right)<0.$ (1.8)

Given this, we conclude in Theorem 3.2 the convergence to the average mass

$u(x,t)\xrightarrow{t\rightarrow\infty}\frac{1}{|\Omega|}\int_{\Omega}u(x,t)\>dx\quad\text{
in $L^{2}(\Omega)$}.$ (1.9)

Finally, we investigate the existence and stability of spatially heterogeneous
equilibria. In particular we solve the following steady state problem

$\displaystyle\mathcal{L}_{\varepsilon}U+f(U)=0.$ (1.10)

The continuous steady states are constructed in Theorem 4.1 through
bifurcation techniques. Additionally, in Theorem 4.2, we establish the
existence of discontinuous yet stable solutions, under the assumption that the
force term $f$ has two distinct zeros $u_{1}<u_{2}$ with $f^{\prime}(u_{1})<0$
and $f^{\prime}(u_{2})<0$.

###  1.2 Related results

The local counterpart of the Cauchy problem (1.1)-(1.2) is given by the
following reaction diffusion problem with no flux through boundaries

$\displaystyle u_{t}(x,t)=\Delta u(x,t)+f(u(x,t))$
$(x,t)\in\Omega\times(0,\infty),$ (1.11) $\displaystyle\frac{\partial
u}{\partial n}=0$ $x\in\partial\Omega$, (1.12) $\displaystyle u(x,0)=u_{0}(x)$
$x\in\Omega,$ (1.13)

where in this contexts, $n$ represents the outward normal vector to $\Omega$.
This model is already well known. We refer readers interested in general
properties of such problems to [31, 32, 33]. In [9] the invariant regions were
discussed with their applications to establishing spatially homogeneous
asymptotics. We emphasize the classical works [10, 11, 12] where the existence
and instability of non-constant steady states is explored.

The link between models (1.1)-(1.2) and (1.11)-(1.13) was established when
$f\equiv 0$. Namely the convergence of solutions $u^{\varepsilon}$ from the
non-local problem to solutions $u$ of the local problem was demonstrated in
[8]. We also direct the reader to a more general setting, considering the
convergence of non-local dispersion problems to the local parabolic
counterparts, discussed in [30].

We underscore the well-posed nature of problem (1.1)-(1.2) in the Hadamard
sense. In the linear case, existence and uniqueness were established in [8],
while the non-linear case was addressed in [5].

The eigenvalue problem for the operator (1.3), was studied for instance in
[4]. Authors demonstrated that $\beta_{1}$ is strictly negative. However, the
question of the existence of a corresponding eigenfunction remains unresolved.
Additional insights into this problem are presented in [19, 29]. More general
approach, outlined in [6, 7, 24], yields only the trivial eigenpair when
applied to our framework.

In the linear non-local case, the work of [22] explores spatially homogeneous
asymptotic behavior of solutions $u^{\varepsilon}$ of problem (1.1)-(1.2). The
non-local steady problem (1.10) was investigated in [22] and [6].

Other related works encompass diverse applications to real world, including
image processing [13, 14], epidemiology [21], ecology [25], and various
mathematical articles [28, 20, 26, 5].

##  2 Spectral analysis

###  2.1 Main results of the section

Here we perform the spectral analysis of the operator
$\mathcal{L}_{\varepsilon}:L^{2}(\Omega)\rightarrow L^{2}(\Omega)$ given by
formula (1.3). For that we decompose the dispersion operator

$\mathcal{L}_{\varepsilon}u(x)=\int_{\Omega}J_{\varepsilon}(x-y)u(y)\>dy+b_{\varepsilon}(x)u(x),$
(2.1)

where

$b_{\varepsilon}(x)=-\int_{\Omega}J_{\varepsilon}(x-y)\>dy.$ (2.2)

We characterise the spectrum $\sigma(\mathcal{L}_{\varepsilon})$ on an
arbitrary bounded domain $\Omega$.

######  Theorem 2.1 (General case).

Let $J_{\varepsilon}$ and $\Omega$ satisfy Assumption 1.4. Denote
$\mathcal{L}_{\varepsilon}:L^{2}\left(\Omega\right)\rightarrow
L^{2}\left(\Omega\right)$ as in a formula (2.1). Then the spectrum of
$\mathcal{L}_{\varepsilon}$ is given by

$\sigma(\mathcal{L}_{\varepsilon})=b_{\varepsilon}\left[\>\overline{\Omega}\>\right]\cup\left\\{\beta_{k}\right\\}_{k=0}^{\infty},$
(2.3)

where

$b_{\varepsilon}\left[\>\overline{\Omega}\>\right]=\left\\{-\int_{\Omega}J_{\varepsilon}(x-y)\>dy\>\Big{|}\>x\in\overline{\Omega}\right\\}$

and the sequence $\left\\{\beta_{k}\right\\}_{k=0}^{\infty}$ is a set of
isolated eigenvalues of $\mathcal{L}_{\varepsilon}$.

Next we consider the spatial domain $\Omega=\mathbb{T}^{n}$ being an
$n$-dimensional torus. In this approach we conclude the following

######  Remark 2.1.

Let $J_{\varepsilon}$ and $\mathbb{T}^{n}$ satisfy Assumption 1.2. Denote the
set $b_{\varepsilon}\left[\>\overline{\Omega}\>\right]$ as in Theorem 2.1.
Then this set is a single point. In particular we have

$b_{\varepsilon}\left[\>\overline{\Omega}\>\right]=\left\\{-\hat{J}_{\varepsilon}(0)\right\\},$

where
$\hat{J}_{\varepsilon}(k)=\int_{\mathbb{T}^{n}}J_{\varepsilon}(z)e^{ikz}\>dz$
is a $k$-th Fourier coefficient of the kernel $J_{\varepsilon}$.

In this setting we will also derive the explicit formulas for the eigenvalues
$\left\\{\beta_{k}\right\\}_{k=0}^{\infty}$ and their corresponding
eigenfunctions $\left\\{\varphi_{k}\right\\}_{k=0}^{\infty}$ from the Theorem
2.1.

######  Theorem 2.2 (Periodic case).

Let $J_{\varepsilon}$ and $\mathbb{T}^{n}$ satisfy Assumption 1.2. Denote
$\mathcal{L}_{\varepsilon}:L^{2}\left(\mathbb{T}^{n}\right)\rightarrow
L^{2}\left(\mathbb{T}^{n}\right)$ as in a formula (2.1). Then the spectrum of
$\mathcal{L}_{\varepsilon}$ is given by

$\sigma(\mathcal{L}_{\varepsilon})=\left\\{-\hat{J}_{\varepsilon}(0)\right\\}\cup\left\\{\beta\right\\}_{k=0}^{\infty}.$
(2.4)

Eigenvalues and the corresponding eigenfunctions are given explicitly as
follows

$\beta_{k}(\varepsilon):=\hat{J}_{\varepsilon}(k)-\hat{J}_{\varepsilon}(0)=\int_{\mathbb{T}^{n}}J_{\varepsilon}(z)(\cos(k\cdot
z)-1)\>dz,$ (2.5) $\varphi_{k}^{1}(x)=\cos(k\cdot
x),\quad\varphi_{k}^{2}(x)=\sin(k\cdot x),$ (2.6)

where
$\hat{J}_{\varepsilon}(k)=\int_{\mathbb{T}^{n}}J_{\varepsilon}(z)e^{ikz}\>dz$
is a $k$-th Fourier coefficient of the kernel $J_{\varepsilon}$. Moreover the
following properties hold

  1. (i)

The eigenvalues $\\{\beta_{k}\\}_{k=0}^{\infty}$ are real, bounded and non-
positive for $k\neq 0$.

  2. (ii)

The limit eigenvalue is an element of the spectrum. In particular

$\beta_{k}\xrightarrow{|k|\rightarrow\infty}\beta_{\infty}(\varepsilon)=-\hat{J}_{\varepsilon}(0)=b_{\varepsilon}(0).$
(2.7)

When $\varepsilon\rightarrow 0$ then the limit eigenvalue is bounded in a
following way

$-\varepsilon^{-m}\leq\beta_{\infty}(\varepsilon)\leq-\frac{\varepsilon^{2-m}}{\pi^{2}}.$
(2.8)

  3. (iii)

When $\varepsilon\rightarrow 0$ and $m=2$ then the eigenvalues
$\\{\beta_{k}\\}_{k=0}^{\infty}$ are given explicitly by

$\lim_{\varepsilon\rightarrow 0}\beta_{k}(\varepsilon)=-\frac{|k|^{2}}{n}$
(2.9)

If $m\in[0,2)$ then the limit becomes zero

The following subsections contain the proofs of theorems 2.1 and 2.2.

###  2.2 Proof of Theorem 2.1

Firstly we show that the set
$b_{\varepsilon}\left[\>\overline{\Omega}\>\right]$ belongs to the spectrum of
$\mathcal{L}_{\varepsilon}$.

######  Lemma 2.1.

Let $J_{\varepsilon}$ and $\Omega$ satisfy Assumption 1.4. Then for
$b_{\varepsilon}$ defined as in formula (2.2) we obtain the inclusion

$b_{\varepsilon}\left[\>\overline{\Omega}\>\right]\subseteq\sigma(\mathcal{L}_{\varepsilon}).$

###### Proof.

Let $\lambda\in b_{\varepsilon}\left[\>\Omega\>\right]$. Suppose that
$\lambda\notin\sigma(\mathcal{L}_{\varepsilon})$. Then for all $v\in
L^{2}\left(\Omega\right)$ there is a solution $u\in L^{2}\left(\Omega\right)$
to the equation

$\mathcal{L}_{\varepsilon}u(x)-\lambda u(x)=v(x).$

As a consequence of bounded inverse theorem, there exists $C>0$ such that

$C\|u\|_{2}\leq\|\mathcal{L}_{\varepsilon}u-\lambda u\|_{2}.$ (2.10)

By the continuity of $J$, there exists $x_{\lambda}\in\Omega$ such that
$b_{\varepsilon}(x_{\lambda})=\lambda$. Hence we take
$u_{R}^{\lambda}(x)=R^{-n/2}\mathds{1}_{B(x_{\lambda},R)}$ and easily verify
that it is a function of unit norm and it satisfies the upper bound of a
following form

$\displaystyle\|\mathcal{L}_{\varepsilon}u_{R}^{\lambda}-\lambda
u_{R}^{\lambda}\|_{2}$
$\displaystyle\leq\left\|\left(\int_{\Omega}J_{\varepsilon}(\cdot-y)u_{R}^{\lambda}(y)\>dy\right)\right\|_{2}+\|\left(b_{\varepsilon}-\lambda\right)u_{R}^{\lambda}\|_{2}$
$\displaystyle=\left(\int_{\Omega}\left(\int_{B(x_{\lambda},R)}J_{\varepsilon}(x-y)R^{-n/2}\>dy\right)^{2}\>dx\right)^{1/2}$
$\displaystyle+\left(\int_{B(x_{\lambda},R)}\left(b_{\varepsilon}(x)-\lambda\right)^{2}dx\right)^{1/2}$
$\displaystyle\leq\|J_{\varepsilon}\|_{\infty}|\Omega|^{1/2}R^{n/2}+\left(\int_{B(x_{\lambda},R)}\left(b_{\varepsilon}(x)-\lambda\right)^{2}\>dx\right)^{1/2}\xrightarrow{R\rightarrow
0}0.$

But it contradicts (2.10), which yields
$\lambda\in\sigma(\mathcal{L}_{\varepsilon})$. We conclude saying that
$\sigma(\mathcal{L}_{\varepsilon})$ must be a closed set since it is a
spectrum of bounded linear operator. ∎

Next, we show that every other element of the spectrum is an eigenvalue of
$\mathcal{L}_{\varepsilon}$.

######  Lemma 2.2.

Let $J_{\varepsilon}$ and $\Omega$ satisfy Assumption 1.4, $\varepsilon>0$,
$m\in\mathbb{N}$. Then every
$\lambda\in\sigma(\mathcal{L}_{\varepsilon})\setminus
b_{\varepsilon}\left[\>\overline{\Omega}\>\right]$ is an eigenvalue of the
operator $\mathcal{L}_{\varepsilon}$ given in (1.3).

###### Proof.

Denote a set $D=\mathbb{C}\setminus
b_{\varepsilon}\left[\>\overline{\Omega}\>\right]$. This set is open and
connected. Now we re-write equation (1.6) using formula (2.1) to obtain

$\int_{\Omega}J_{\varepsilon}(x-y)u(y)\>dy=\left(\beta-
b_{\varepsilon}(x)\right)u(x).$

We divide both sides of the equation to get the fixed point problem and denote
the analytic, operator valued function
$F:D\rightarrow\mathcal{L}(L^{2}(\Omega))$ as follows

$F(\beta)[u](x):=\int_{\Omega}\frac{J_{\varepsilon}(x-y)}{\beta-
b_{\varepsilon}(x)}u(y)\>dy=u(x).$

Now, let us verify the convergence of the difference quotient

$\displaystyle\frac{F(\beta)-F(\beta_{0})}{\beta-\beta_{0}}[u](x)$
$\displaystyle=\frac{1}{\beta-\beta_{0}}\left(\int_{\Omega}\frac{J_{\varepsilon}(x-y)}{\beta-
b_{\varepsilon}(x)}u(y)\>dy-\int_{\Omega}\frac{J_{\varepsilon}(x-y)}{\beta_{0}-b_{\varepsilon}(x)}u(y)\>dy\right)$
$\displaystyle=-\int_{\Omega}\frac{J_{\varepsilon}(x-y)}{(\beta-
b_{\varepsilon}(x))(\beta_{0}-b_{\varepsilon}(x))}u(y)\>dy$
$\displaystyle\xrightarrow{\beta_{0}\rightarrow\beta}-\int_{\Omega}\frac{J_{\varepsilon}(x-y)}{(\beta-
b_{\varepsilon}(x))^{2}}u(y)\>dy,$

which proves that $F:D\rightarrow\mathcal{L}(L^{2}\left(\Omega\right))$ is an
analytic operator, as claimed.

We emphasize that since $\beta$ never intersects the range of
$b_{\varepsilon}$ then the integral kernel
$K(x,y)=\frac{J_{\varepsilon}(x-y)}{\beta-b_{\varepsilon}(x)}$ has a finite
$L^{2}(\Omega\times\Omega)$ norm and so it is a Hilbert-Schmidt operator. In
particular it is a compact operator.

Now, we verify that $F$ is a contraction for some $\beta\in D$. For instance
we can take $\beta_{0}\in D$ such that

$\Re(\beta_{0})=-\inf_{x\in\Omega}-b_{\varepsilon}(x)+2\|J_{\varepsilon}\|_{\infty}|\Omega|^{\frac{1}{2}},\quad\Im(\beta_{0})=0.$

Having that we can bound the operator norm of $F(\beta_{0})$ to obtain the
following

$\displaystyle\|F(\beta_{0})\|$
$\displaystyle=\sup_{\|u\|_{2}=1}\|F(\beta_{0})[u]\|_{2}\leq\sup_{\|u\|_{2}=1}\left(\int_{\Omega}\int_{\Omega}\frac{\|J_{\varepsilon}\|^{2}u^{2}(y)}{(\beta_{0}-b_{\varepsilon}(x))^{2}}\>dy\>dx\right)^{\frac{1}{2}}\leq\frac{1}{2}$

Hence $(I-F(\beta_{0}))^{-1}$ exists. But now, by the Analytical Fredholm
Theorem [15], we know that $(I-F(\beta))^{-1}$ exists for all $\beta\in
D\setminus\left\\{\beta_{k}\right\\}_{k=0}^{\infty}$. On the other hand, for
any element $\beta_{k}\in\left\\{\beta_{k}\right\\}_{k=0}^{\infty}$ there
exists a corresponding function $\varphi_{k}\in L^{2}\left(\Omega\right)$ such
that $(\beta_{k},\varphi_{k})$ solves the equation (1.6). Thus $\beta_{k}$ is
an eigenvalue. ∎

Combining the results obtained in the Lemma 2.1 and 2.2 we finish the proof of
Theorem 2.1.

###  2.3 Proof of Theorem 2.2

Firstly we observe that by Theorem 2.1 and Remark 2.1 the spectrum is given by
formula (2.4) as claimed. To find the latter part of the spectrum we denote
the following complex valued function defined for
$x\in\mathbb{T}^{n},k\in\mathbb{Z}^{n}$

$\psi_{k}(x)=e^{ik\cdot x}.$

We plug $\psi_{k}$ into the equation (1.6) and use the periodicity of
$J_{\varepsilon}$ to obtain

$\displaystyle\mathcal{L}_{\varepsilon}\psi_{k}(x)$
$\displaystyle=\int_{\mathbb{T}^{n}}J_{\varepsilon}(x-y)e^{ik\cdot
y}\>dy-\int_{\mathbb{T}^{n}}J_{\varepsilon}(x-y)\>dy\>\psi_{k}(x)$
$\displaystyle=\left(\int_{\mathbb{T}^{n}}J_{\varepsilon}(x-y)e^{ik\cdot(y-x)}\>dy-\int_{\mathbb{T}^{n}}J_{\varepsilon}(x-y)\>dy\right)\psi_{k}(x)$
$\displaystyle=\left(\int_{\mathbb{T}^{n}+x}J_{\varepsilon}(z)e^{ik\cdot
z}\>dz-\int_{\mathbb{T}^{n}+x}J_{\varepsilon}(z)\>dz\right)\psi_{k}(x)$
$\displaystyle=\left(\int_{\mathbb{T}^{n}}J_{\varepsilon}(z)e^{ik\cdot
z}\>dz-\int_{\mathbb{T}^{n}}J_{\varepsilon}(z)\>dz\right)\psi_{k}(x)$
$\displaystyle=\left(\hat{J}_{\varepsilon}(k)-\hat{J}_{\varepsilon}(0)\right)\psi_{k}(x).$

Using the symmetry of $J$, we verify formula (2.5). Indeed we have that

$\beta_{k}(\varepsilon):=\hat{J}_{\varepsilon}(k)-\hat{J}_{\varepsilon}(0)=\int_{\mathbb{T}^{n}}J_{\varepsilon}(z)(\cos(k\cdot
z)-1)\>dz,$

as asserted in the Theorem. We conclude that both real and imaginary part of
$\psi_{k}$ solves the equation (1.6) and so

$\varphi_{k}^{1}(x):=\Re(\psi_{k})=\cos(k\cdot x),$
$\varphi_{k}^{2}(x):=\Im(\psi_{k})=\sin(k\cdot x),$

which is exactly formula (2.6) as we claimed.

(i) It follows immediately from formula (2.5).

(ii) The Riemann-Lebesgue Lemma applied to formula (2.5) yields

$\beta_{\infty}(\varepsilon):=\lim_{|k|\rightarrow\infty}\beta_{k}(\varepsilon)=-\hat{J}_{\varepsilon}(0)=b_{\varepsilon}(0).$

Next we compute

$\displaystyle\beta_{\infty}(\varepsilon)$
$\displaystyle=-\int_{\mathbb{T}^{n}}J_{\varepsilon}(x)\>dx$
$\displaystyle=-\frac{C_{norm}^{\varepsilon}}{\varepsilon^{m+n}}\int_{\left(-\pi,\pi\right)^{n}}J_{per}\left(\frac{x}{\varepsilon}\right)\>dx$
$\displaystyle=-\frac{C_{norm}^{\varepsilon}}{\varepsilon^{m}}\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}\left(z\right)\>dz$
$\displaystyle=-\frac{1}{\varepsilon^{m}}\frac{\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}\left(z\right)\>dz}{\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}\left(z\right)|z|^{2}\>dz}=:-\frac{\rho(\varepsilon)}{\varepsilon^{m}},$

where we denoted

$\rho(\varepsilon)=\frac{\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}\left(z\right)\>dz}{\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}\left(z\right)|z|^{2}\>dz}.$

We bound this quantity from below in the following way

$\displaystyle\rho(\varepsilon)$
$\displaystyle\geq\frac{\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}\left(z\right)\>dz}{\frac{\pi^{2}}{\varepsilon^{2}}\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}(z)\>dz}=\frac{\varepsilon^{2}}{\pi^{2}}.$

Reversely we see that for $\varepsilon>0$ sufficiently small we have

$\displaystyle\rho(\varepsilon)=\frac{\int_{|z|<1}J_{per}\left(z\right)\>dz+\int_{\begin{subarray}{c}\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}\\\
|z|>1\end{subarray}}J_{per}\left(z\right)\>dz}{\int_{|z|<1}J_{per}\left(z\right)|z|^{2}\>dz+\int_{\begin{subarray}{c}\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}\\\
|z|>1\end{subarray}}J_{per}\left(z\right)|z|^{2}\>dz}\leq 1.$

Hence we obtained the bound for a limit eigenvalue

$-\varepsilon^{-m}\leq\beta_{\infty}(\varepsilon)\leq-\frac{\varepsilon^{2-m}}{\pi^{2}}.$

(iii) We use the Taylor series expansion to re-write $\beta_{k}$ with $k\neq
0$.

$\displaystyle\beta_{k}(\varepsilon)$
$\displaystyle=\int_{(-\pi,\pi)^{n}}J_{\varepsilon}(x)\cos(k\cdot
x)\>dx-\int_{(-\pi,\pi)^{n}}J_{\varepsilon}(x)\>dx$
$\displaystyle=-\int_{(-\pi,\pi)^{n}}J_{\varepsilon}(x)\frac{(k\cdot
x)^{2}}{2}\>dx+\sum_{l=2}^{\infty}(-1)^{l}\int_{(-\pi,\pi)^{n}}J_{\varepsilon}(x)\frac{(k\cdot
x)^{2l}}{(2l)!}\>dx$

Substitution $z=x/\varepsilon$ gives a simplified formula in the asymptotic
form.

$\displaystyle\beta_{k}(\varepsilon)$
$\displaystyle=-\frac{\varepsilon^{2-m}C_{norm}^{\varepsilon}}{2}\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}(z)(k\cdot
z)^{2}\>dz+o(\varepsilon^{4-m})$
$\displaystyle=-\frac{\varepsilon^{2-m}C_{norm}^{\varepsilon}}{2}\sum_{i=1}^{n}\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}(z)k_{i}^{2}z_{i}^{2}\>dz$
$\displaystyle-\frac{\varepsilon^{2-m}C_{norm}^{\varepsilon}}{2}\sum_{i\neq
j}^{n}\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}(z)k_{i}k_{j}z_{i}z_{j}\>dz+o(\varepsilon^{4-m})$
$\displaystyle=-\varepsilon^{2-m}\sum_{i=1}^{n}k_{i}^{2}\frac{\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}z_{i}^{2}\>dz}{\int_{\left(-\frac{\pi}{\varepsilon},\frac{\pi}{\varepsilon}\right)^{n}}J_{per}|z|^{2}\>dz}+o(\varepsilon^{4-m})$
$\displaystyle=-\varepsilon^{2-m}\frac{|k|^{2}}{n}+o(\varepsilon^{4-m})$

This ends the proof of Theorem 2.2.

##  3 Existence, invariant regions and spatially homogeneous asymptotics

###  3.1 Main results of the section

We begin by stating results on asymptotic properties of solutions to the
initial value problem (1.1)-(1.2).

######  Theorem 3.1 (Existence of solutions).

Consider the integral kernel $J_{\varepsilon}$ and domain $\Omega$ satisfying
either Assumptions 1.4 or 1.2. Assume that $f\in C^{1}(\mathbb{R})$ such that
$f^{\prime}\in L^{\infty}(\Omega)$. The following statements are true

  1. (i)

For every $u_{0}\in L^{2}\left(\Omega\right)$ there exists a unique solution
$u\in C\left([0,\infty),L^{2}\left(\Omega\right)\right)$ to the Cauchy problem
(1.1)-(1.2).

  2. (ii)

Solutions $u,v$ to the Cauchy problem (1.1)-(1.2) depend continuously on the
initial data $u_{0},v_{0}$. In particular for all $T>0$ there exists $C(T)>0$
such that

$\|u-v\|_{2}\leq C(T)\|u_{0}-v_{0}\|_{2}.$ (3.1)

  3. (iii)

If $u_{0}\in C(\overline{\Omega})$ then both $u\in
C\left(\overline{\Omega}\times[0,T]\right)$ and $u_{t}\in
C\left(\overline{\Omega}\times[0,T]\right)$ for each $T>0$

Next we study existence of invariant regions in the considered evolution
problem.

######  Theorem 3.2 (Invariant region).

Let the assumptions of the Theorem 3.1 hold true. Additionally assume there
exists $u_{1}<u_{2}$ with $f(u_{1})=f(u_{2})=0$. Let $u\in
C([0,\infty),L^{2}\left(\Omega\right))$ be a solution of initial value problem
(1.1)-(1.2) with an initial datum $u_{0}\in L^{2}\left(\Omega\right)$. Denote
the set $\Gamma=[u_{1},u_{2}]$. If $u_{0}(x)\in\Gamma$ a.e. then
$u(x,t)\in\Gamma$ a.e. for all $t>0$.

And finally we derive a criterion for the spatially homogeneous asymptotics.

######  Theorem 3.3 (Convergence to the mean).

Let the assumptions of the Theorem 3.2 hold true. Let $u\in
C\left([0,\infty),L^{2}\left(\Omega\right)\right)$ be a solution of the Cauchy
problem (1.1) - (1.2) with an initial datum $u_{0}\in
L^{2}\left(\Omega\right)$. Denote the mean mass

$u_{\Omega}(t)=\frac{1}{|\Omega|}\int_{\Omega}u(x,t)\>dx.$

There are positive numbers $a_{1}>0,a_{2}>0$ and a critical value given by
formula

$\sigma=2(\beta_{1}+\max_{u\in\Gamma}|f^{\prime}(u)|),$ (3.2)

where $\beta_{1}$ is largest non-trivial eigenvalue. For these values the
following statements hold true for all $t>0$.

  1. (i)

The difference between the solution $u$ and a mean mass $u_{\Omega}$ is
estimated by

$\|u(t)-u_{\Omega}(t)\|_{2}\leq a_{1}e^{\sigma t}.$ (3.3)

  2. (ii)

The mean mass $u_{\Omega}$ satisfies the following ordinary differential
equation

$\begin{cases}u_{\Omega}^{\prime}(t)&=f(u_{\Omega}(t))+O(e^{\sigma t}),\\\
u_{\Omega}(0)&=\frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)\>dx.\end{cases}$ (3.4)

  3. (iii)

The Dirichlet form associated with $\mathcal{L}_{\varepsilon}$ is estimated by

$|\left\langle\mathcal{L}_{\varepsilon}u(t),u(t)\right\rangle|\leq
a_{2}e^{2\sigma t}.$ (3.5)

######  Remark 3.1.

Observe that once the value of $\sigma$ given by formula (3.2) is negative
then Theorem 3.3 yields the convergence to the mean mass in the sense of
$L^{2}$ norm. What is more, we see by Theorem 2.2 that this criterion becomes

$\max_{u\in\Gamma}|f^{\prime}(u)|<\varepsilon^{2-m}\frac{|k|^{2}}{n}+o(\varepsilon^{4-m}).$

######  Remark 3.2.

The constants $a_{1},a_{2}$ in Theorem 3.3 are given explicitly by

$a_{1}=\|u_{0}-u_{\Omega}(0)\|_{2},\quad
a_{2}=\|\mathcal{L}_{\varepsilon}\|_{L^{2}\rightarrow
L^{2}}\|u_{0}-u_{\Omega}(0)\|_{2}^{2}.$

Reminder of this section is focused on the proofs of the above theorems.

###  3.2 Proof of Theorem 3.1

Observe that (i) is a consequence of a more general result by Sun [5], who
constructed a unique solution $u\in C([0,\infty),L^{1}(\Omega))$. This
solution is obtained as a fixed point of the following equation

$u(x,t)=u_{0}(x)+\int_{0}^{t}\mathcal{L}_{\varepsilon}u(x,s)\>ds+\int_{0}^{t}f(u(x,s))\>ds.$
(3.6)

Obviously $L^{2}(\Omega)\subset L^{1}(\Omega)$ for a bounded domain $\Omega$
and hence (i) follows. Denote by $w=u-v$ the difference of solutions $u,v$ to
initial value problem (1.1)-(1.2) and proceed for $t\in[0,T]$ as follows

$\displaystyle\|w(t)\|_{2}$
$\displaystyle\leq\|w_{0}\|_{2}+\int_{0}^{t}\|\mathcal{L}_{\varepsilon}w(s)\|_{2}+\|f(u(s))-f(v(s))\|_{2}\>ds$
$\displaystyle\leq\|w_{0}\|_{2}+\int_{0}^{t}\left(\int_{\Omega}\int_{\Omega}J_{\varepsilon}^{2}(x-y)\left(w(y,s)-w(x,s)\right)^{2}\>dy\>dx\right)^{1/2}\>ds$
$\displaystyle+\int_{0}^{t}\left(\int_{\Omega}\left(f(u(x,s))-f(v(x,s))\right)^{2}\>dx\right)^{1/2}\>ds$
$\displaystyle\leq\|w_{0}\|_{2}+\left(\sqrt{2}\|J_{\varepsilon}\|_{\infty}|\Omega|^{1/2}+\|f^{\prime}\|_{\infty}\right)\int_{0}^{t}\|w(s)\|_{2}\>ds.$

The Gronwall inequality yields

$\|w(t)\|_{2}\leq
e^{T\left(\sqrt{2}\|J_{\varepsilon}\|_{\infty}|\Omega|^{1/2}+\|f^{\prime}\|_{\infty}\right)}\|w_{0}\|_{2}.$

This proves (ii) with
$C(T)=e^{T\left(\sqrt{2}\|J_{\varepsilon}\|_{\infty}|\Omega|^{1/2}+\|f^{\prime}\|_{\infty}\right)}$.
Now, let $u_{0}\in C(\overline{\Omega})$. Denote by
$\mathbb{X}=C\left(\overline{\Omega}\times[0,T_{0}]\right)$ the Banach space
equipped with the norm

$|||u|||=\max_{t\in[0,T_{0}]}\max_{x\in\overline{\Omega}}|u(x,t)|.$

Now apply the Banach Fixed Point Theorem and the usual procedure. We deduce
that there exists unique solution $u\in
C\left(\overline{\Omega}\times[0,T]\right)$. But now observe that second and
third terms in formula (3.6) become continuously differentiable in time, and
so in fact $u_{t}\in C\left(\overline{\Omega}\times[0,\infty)\right)$ which is
(iii). Now the proof of Theorem 3.1 is finished.

###  3.3 Proof of Theorem 3.2

Our strategy to show an existence of an invariant region is to differentiate
maximum and minimum of the solution to (1.1)-(1.2). The first step is to solve
the problem when $u_{0}\in C(\overline{\Omega})$. Indeed we have

######  Lemma 3.1.

For $T>0$ and bounded domain $\Omega\subset\mathbb{R}^{n}$ let $u\in
C\left(\overline{\Omega},[0,T]\right)$ such that $u_{t}\in
C\left(\overline{\Omega},[0,T]\right)$. Denote the functions
$\xi:[0,T]\rightarrow\overline{\Omega}$,
$\zeta:[0,T]\rightarrow\overline{\Omega}$ and
$\mu:[0,T]\rightarrow\mathbb{R}$, $\nu:[0,T]\rightarrow\mathbb{R}$ given by
the following formulas

$\mu(t):=u(\xi(t),t)=\max_{x\in\Omega}u(x,t),\quad\nu(t):=u(\zeta(t),t)=\min_{x\in\Omega}u(x,t)$
(3.7)

Then both $\mu$ and $\nu$ are differentiable a.e. and their derivatives are
given by

$\mu^{\prime}(t)=u_{t}(\xi(t),t),\quad\nu^{\prime}(t)=u_{t}(\zeta(t),t).$
(3.8)

###### Proof.

Let $t>0,s>0$. Assume that $\mu(t)>\mu(s)$. formula (3.7) yields

$0<\mu(t)-\mu(s)\leq
u(\xi(t),t)-u(\xi(t),s)\leq\max_{(x,t)\in\overline{\Omega}\times[0,T]}|u_{t}(x,t)||t-s|$

which yields Lipschitz continouity of $\mu$. By the Radamacher theorem [16] we
conclude that $\mu$ is differentiable almost everywhere. Now, we estimate the
difference quotient as follows

$\displaystyle\frac{\mu(t+h)-\mu(t)}{h}$
$\displaystyle=\frac{u(\xi(t+h),t+h)-u(\xi(t),t)}{h}$
$\displaystyle\leq\frac{u(\xi(t+h),t+h)-u(\xi(t+h),t)}{h}$
$\displaystyle=u_{t}(\xi(t+h),c(t,h)),$

where by the Mean Value Theorem $c(t,h)\in[t-h,t+h]$. Passing to the limit as
$h\rightarrow 0$ we get

$\mu^{\prime}(t)\leq u_{t}(\xi(t),t).$ (3.9)

The reverse estimate can be written down as follows

$\displaystyle\frac{\mu(t+h)-\mu(t)}{h}$
$\displaystyle\geq\frac{u(\xi(t),t+h)-u(\xi(t),t)}{h}$
$\displaystyle=u_{t}(\xi(t),c(t,h)).$

Similarly as before, we pass to the limit as $h\rightarrow 0$ and obtain

$\mu^{\prime}(t)\geq u_{t}(\xi(t),t).$ (3.10)

Combining inequalities (3.9) and (3.10) we get formula (3.8) for
$\mu^{\prime}$. The proof for $\nu^{\prime}$ is analogous. That completes the
proof of the Lemma 3.8. ∎

Next we show that $\mu$ and $\nu$ are bounded by the minimum and maximum of
$u_{0}$.

######  Lemma 3.2.

Let the assumptions of the Theorem 3.2 hold true. Let $u\in
C\left(\overline{\Omega},[0,T]\right)$ be a solution to the Cauchy problem
(1.1)-(1.2) with an initial datum $u_{0}\in C(\overline{\Omega})$. Denote the
functions $\xi,\zeta,\mu,\nu$ as in the Lemma 3.2. Then for all $t\geq 0$ the
following estimates hold true

$u_{1}\leq\nu(t)\leq\mu(t)\leq u_{2}$

###### Proof.

Recall the definition of the dispersion operator $\mathcal{L}_{\varepsilon}$
as in formula (1.3). In maximum point $\xi$ we have

$\mathcal{L}_{\varepsilon}u(\xi(t),t)=\int_{\Omega}J_{\varepsilon}(x-y)\left(u(y,t)-u(\xi(t),t)\right)\>dy\leq
0.$

Moreover in minimum point $\zeta$ we have

$\mathcal{L}_{\varepsilon}u(\zeta(t),t)=\int_{\Omega}J_{\varepsilon}(x-y)\left(u(y,t)-u(\zeta(t),t)\right)\>dy\geq
0.$

By Lemma 3.8 both $\mu$ and $\nu$ are differentiable almost everywhere in time
$t>0$. Formula (3.8) yields

$\mu^{\prime}(t)=u_{t}(\xi(t),t)=\mathcal{L}_{\varepsilon}u(\xi(t),t)+f(\mu(t))\leq
f(\mu(t)).$

Similarly we have

$\nu^{\prime}(t)=u_{t}(\zeta(t),t)=\mathcal{L}_{\varepsilon}u(\zeta(t),t)+f(\nu(t))\geq
f(\nu(t)).$

The proof is completed by the usual comparison principle for the differential
inequalities [17]. ∎

In order to proceed with the proof of Theorem 3.2 observe that every $u_{0}\in
L^{2}(\Omega)$ can be approximated by the sequence $u^{n}_{0}\in
C(\overline{\Omega})$. Now let the functions $u\in
C([0,T],L^{2}\left(\Omega\right))$ and $u^{n}\in C(\overline{\Omega},[0,T])$
be the solutions of the Cauchy problem (1.1)-(1.2) equipped respectively with
the initial data $u_{0},u^{n}_{0}$. Assume that $u_{0}(x)\in\Gamma$ a.e., then
$u_{0}^{n}\in\Gamma$ a.e.. By the Theorem 3.1 (ii) we have the following
estimate

$\|u-u^{n}\|_{2}\leq C(T)\|u_{0}-u_{0}^{n}\|_{2}.$

Passing to the limit as $n\rightarrow\infty$ we get that $u^{n}\rightarrow u$
in the sense of $L^{2}$ norm. But now one can take the subsequence $u^{n_{k}}$
such that $u^{n_{k}}\rightarrow u$ a.e. as $k\rightarrow\infty$. By the Lemma
3.2 $u^{n_{k}}(x,t)\in\Gamma$ a.e. for all $t\in[0,T]$. Hence
$u(x,t)\in\Gamma$ a.e. for all $t\in[0,T]$. Thus the proof of Theorem 3.2 is
completed.

###  3.4 Proof of Theorem 3.3

We introduce the notion of energy $E_{v}$ of the distance function $v$ by the
formula

$E_{v}(t)=\frac{1}{2}\int_{\Omega}|v(x,t)|^{2}\>dx,\quad
v(x,t)=u(x,t)-u_{\Omega}(t),$

where $u$ is a solution to Cauchy problem (1.1)-(1.2) and $u_{\Omega}(t)$ is
the mean mass.

We differentiate the energy to get

$\displaystyle E_{v}^{\prime}(t)$ $\displaystyle=\int_{\Omega}v_{t}v\>dx$
$\displaystyle=\int_{\Omega}\left(u_{t}-u_{\Omega}^{\prime}\right)v\>dx$
$\displaystyle=\int_{\Omega}\left(\mathcal{L}_{\varepsilon}u+f(u)-\frac{1}{|\Omega|}\int_{\Omega}f(u)\>dy\right)v\>dx$
$\displaystyle=\left\langle\mathcal{L}_{\varepsilon}u,v\right\rangle+\int_{\Omega}\left(f(u)-f(u_{\Omega})-\frac{1}{|\Omega|}\int_{\Omega}\left(f(u)-f(u_{\Omega})\right)\>dy\>\right)v\>dx$
$\displaystyle\leq\left\langle\mathcal{L}_{\varepsilon}v,v\right\rangle+\max_{u\in\Gamma}|f^{\prime}(u)|E_{v}(t)-\frac{1}{|\Omega|}\int_{\Omega}\left(f(u)-f(u_{\Omega})\right)\>dy\int_{\Omega}v\>dx$
$\displaystyle=\langle\mathcal{L}_{\varepsilon}v,v\rangle+\max_{u\in\Gamma}|f^{\prime}(u)|E_{v}(t)$

Now we divide both sides by $E_{v}(t)$ and obtain the final inequality

$\displaystyle\frac{E_{v}^{\prime}(t)}{E_{v}(t)}$ $\displaystyle\leq
2\left(\sup_{v\in
L^{2}(\Omega),\int_{\Omega}v=0}\frac{\langle\mathcal{L}_{\varepsilon}v,v\rangle}{\|v\|_{2}^{2}}+\max_{u\in\Gamma}|f^{\prime}(u)|\right)$
$\displaystyle=2\left(\beta_{1}+\max_{u\in\Gamma}|f^{\prime}(u)|\right)=:\sigma.$

Solving this differential inequality we get that

$\|u(t)-u_{\Omega}(t)\|_{2}\leq\|u_{0}-u_{\Omega}(0)\|_{2}e^{\sigma
t}=:a_{1}e^{\sigma t}.$

Next, we compute the derivative of the mean mass $u_{\Omega}$, apply the Mean
Value Theorem and Cauchy-Schwarz inequality to obtain

$\displaystyle u_{\Omega}^{\prime}(t)$
$\displaystyle=\frac{1}{|\Omega|}\int_{\Omega}f(u)\>dx$ $\displaystyle\leq
f(u_{\Omega})+\frac{1}{|\Omega|}\int_{\Omega}\left|f(u)-f(u_{\Omega})\right|\>dx$
$\displaystyle\leq
f(u_{\Omega})+\frac{\max_{u\in\Gamma}|f^{\prime}(u)|}{|\Omega|}\int_{\Omega}|u-u_{\Omega}|\>dx$
$\displaystyle\leq
f(u_{\Omega})+\max_{u\in\Gamma}|f^{\prime}(u)|\|v\|_{2}^{2}$

Finally, using claim (i) we get that mean mass satisfies the following
differential inequality

$\displaystyle\begin{cases}u_{\Omega}^{\prime}(t)=f(u_{\Omega}(t))+O(e^{\sigma
t})\\\ u_{\Omega}(0)=\frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)\>dx\end{cases}$

Lastly, it suffices to use (i) once more to get

$\displaystyle\left|\langle\mathcal{L}_{\varepsilon}u,u\rangle\right|$
$\displaystyle=\left|\langle\mathcal{L}_{\varepsilon}v,u\rangle\right|$
$\displaystyle=\left|\langle\mathcal{L}_{\varepsilon}v,v\rangle+u_{\Omega}(t)\int_{\Omega}\mathcal{L}_{\varepsilon}v\>dx\right|$
$\displaystyle=\left|\langle\mathcal{L}_{\varepsilon}v,v\rangle\right|$
$\displaystyle\leq\|\mathcal{L}_{\varepsilon}v\|_{2}\|v\|_{2}$
$\displaystyle\leq\|\mathcal{L}_{\varepsilon}\|\|v\|_{2}^{2}$
$\displaystyle\leq\|\mathcal{L}_{\varepsilon}\|u_{0}-u_{0,\Omega}\|_{2}^{2}e^{2\sigma
t}=a_{2}e^{2\sigma t},$

The proof of Theorem 3.2 is now completed.

##  4 Spatially heterogeneous steady states

###  4.1 Main results of the section

In this section we construct non-constant solution $U$ of steady equation
(1.10). The first class of continuous stationary states is given by the
following theorem.

######  Theorem 4.1 (Continuous equilibrium).

Let $J_{\varepsilon}$ and $\mathbb{T}^{n}$ satisfy Assumption 1.2. Assume that
$f\in C^{1}(\mathbb{R})$ is such that there exist $u_{k}\in\mathbb{R}$
satisfying $f(u_{k})=0$, $f^{\prime}(u_{k})=\beta_{k}$, where $\beta_{k}$ is a
$k$-th eigenvalue of dispersion operator $\mathcal{L}_{\varepsilon}$. Consider
the equation

$\mathcal{L}_{\varepsilon}U+f(U)=\lambda U.$ (4.1)

Then for $\ell>\frac{n}{2}$ there exists a sequence of non-constant solutions
$\left\\{U_{i}\right\\}_{i=0}^{\infty}\subset W^{\ell,2}(\mathbb{T}^{n})$ and
corresponding numbers
$\left\\{\lambda_{i}\right\\}_{i=0}^{\infty}\subset\mathbb{R}$ to equation
(4.1) such that

$(U_{i},\lambda_{i})\xrightarrow{i\rightarrow\infty}(u_{k},0).$

Now we derive the class of discontinuous equilibria in the following theorem.

######  Theorem 4.2 (Discontinuous equilibrium).

Consider the integral kernel $J_{\varepsilon}$ and domain $\Omega$ satisfying
either Assumption 1.4 or 1.2. Assume that $f\in C^{2}_{b}(\mathbb{R})$ is such
that there exists $u_{1},u_{2}$ satisfying $f(u_{1})=f(u_{2})=0$ and
$f^{\prime}(u_{1})\neq 0$, $f^{\prime}(u_{2})\neq 0$. For arbitrary open, non-
empty $\Omega_{1}\subset\Omega$ and $\Omega_{2}\subset\Omega$ such that

$\Omega_{1}\cap\Omega_{2}=\emptyset\quad\text{ and
}\Omega=\left(\overline{\Omega_{1}\cup\Omega_{2}}\right)^{o}.$ (4.2)

Suppose that $\varepsilon>0$ is sufficiently large. Then the steady problem
(1.10) admits a solution of form $U=\hat{U}+\varphi$. The jump function
$\hat{U}$ is defined by formula

$\tilde{U}(x)=\begin{cases}u_{1}\quad x\in\Omega_{1},\\\ u_{2}\quad
x\in\Omega_{2},\end{cases}$ (4.3)

and the perturbation $\varphi\in\overline{B(0,R)}\subset\L^{\infty}(\Omega)$
for a small radius $R>0$. If we additionally assume that $f^{\prime}(u_{1})<0$
and $f^{\prime}(u_{2})<0$ and $R$ is sufficiently small, then $U$ is linearly,
exponentially stable in $L^{2}(\Omega)$.

The remaining part of this section is dedicated to the proofs of both
theorems.

###  4.2 Proof of Theorem 4.1

We start with an abstract equation with linear operator $L$ and nonlinear $H$

$Lv+H(v)=\lambda v,$ (4.4)

where $H$ is such that $H(v)=o(\|v\|)$ when $\|v\|\rightarrow 0$. Obviously
there is a trivial solution $(0,\lambda)$ for all $\lambda\in\mathbb{R}$. If
equation (4.4) has a variational formulation then bifurcation techniques allow
us to construct a family of its non-trivial solutions.

######  Definition 4.1 (Bifurcation point).

We call $(0,\lambda)$ a bifurcation point for equation (4.4) if every
neighbourhood of $(0,\lambda)$ contains a non-zero solution.

Now, for convenience of the reader we supply this subsection with the
classical result by Rabinowitz [18].

######  Theorem 4.3 (Rabinowitz Bifurcation Theorem).

Let $\mathbb{H}$ be a Hilbert space, $V$ a neighbourhood of $0$ in
$\mathbb{H}$, and $I\in C^{2}\left(V,\mathbb{R}\right)$ with $DI(u)=Lu+H(u)$
with $L:\mathbb{H}\rightarrow\mathbb{H}$ being linear and potential
$H(u)=o(\|u\|)$ when $\|u\|\rightarrow 0$. If $\lambda$ is an isolated
eigenvalue of $L$ of finite multiplicity, then $(0,\lambda)$ is a bifurcation
point of the equation (4.4). Moreover, at least one of the following
alternatives occur.

  * (i)

$(0,\lambda)$ is not an isolated solution in $\mathbb{H}\times\\{\lambda\\}$,

  * (ii)

There is a one-sided neighbourhood, $\Lambda$, of $\lambda$ such that for all
$\mu\in\Lambda\setminus\\{\lambda\\}$ there are at least two distinct
nontrivial solutions,

  * (iii)

There is a neighbourhood, $\Lambda$, of $\lambda$ such that for all
$\mu\in\Lambda\setminus\\{\lambda\\}$ there is at least one nontrivial
solution.

We show commutation of $\mathcal{L}_{\varepsilon}$ with weak derivative when
environment is periodic.

######  Lemma 4.1.

Let $\mathbb{T}^{n},J_{\varepsilon}$ satisfy the Assumption 1.2. Denoting by
$D^{\alpha}$ the weak derivative with multi-index $\alpha\in\mathbb{Z}_{\geq
0}^{n}$ the following equation holds true

$D^{\alpha}\mathcal{L}_{\varepsilon}u=\mathcal{L}_{\varepsilon}D^{\alpha}u.$
(4.5)

We skip a direct proof of Lemma 4.5. Next, we proceed with properties of the
non-linear functional $I$ defined by equation (4.6).

######  Lemma 4.2.

Let $\mathbb{T}^{n},J_{\varepsilon}$ satisfy either Assumption 1.4 or 1.2.
Assume that for $f\in C^{1}(\mathbb{R})$ there exists a primitive
$F^{\prime}=f$. Define a nonlinear functional $I:\Theta\rightarrow\mathbb{R}$,
where $\Theta\subset W^{k,2}(\mathbb{T}^{n})$ and $k>\frac{n}{2}$ is an open
neighbourhood of $0$, as follows

$\displaystyle\begin{split}I(u)&=\frac{1}{2}\int_{\mathbb{T}^{n}\times\mathbb{T}^{n}}J_{\varepsilon}(x-y)\left(u(y)-u(x)\right)^{2}\>dy\>dx+\int_{\mathbb{T}^{n}}F(u(x))\>dx.\end{split}$
(4.6)

Then $I\in C^{2}(\Theta,\mathbb{R})$. Moreover, the Frechete derivative is
given by formula

$DI(u)=\mathcal{L}_{\varepsilon}u+f(u).$

We skip a standard proof of Lemma 4.2. Next we leverage the bifurcation theory
to obtain the following result

######  Lemma 4.3.

Let the assumptions of Theorem 4.1 hold true. Without loss of generality
suppose that $f(0)=0$ and $f^{\prime}(0)=-\beta_{k}$, where $\beta_{k}$ is a
$k$-th eigenvalue of dispersion operator $\mathcal{L}_{\varepsilon}$. Take
$\ell\geq\frac{n}{2}$. Then every neighbourhood of $(0,0)\in
W^{\ell,2}(\mathbb{T}^{n})\times\mathbb{R}$ possesses a non-trivial solution
to equation (4.1).

###### Proof.

See that Lemma 4.5 yields that $\mathcal{L}_{\varepsilon}$ is a bounded,
linear mapping of $W^{\ell,2}(\mathbb{T}^{n})$ onto itself. Re-write equation
(4.1) using Taylor’s Theorem to obtain

$\mathcal{L}_{\varepsilon}u+f^{\prime}(0)u+R_{1}(u)=0,$

and finally we consider the equation perturbed by the factor $\lambda u$ for
some $\lambda\in\mathbb{R}$

$\mathcal{L}_{\varepsilon}u+R_{1}(u)=(\lambda+\beta_{k})u.$ (4.7)

The Taylor’s Theorem yields that $R_{1}(u)=o(\|u\|_{\infty})$ as
$\|u\|_{\infty}\rightarrow 0$. Thus the embedding
$W^{\ell,2}(\mathbb{T}^{n})\subset L^{\infty}(\mathbb{T}^{n})$ yields that
$R_{1}$ is a desired potential operator. By Lemma 4.2 considered problem has
the variational formulation. Since by Theorem 2.1 $\beta_{k}$ is an isolated
eigenvalue of $\mathcal{L}_{\varepsilon}$ then by the Rabinowitz Bifurcation
Theorem 4.3 there exists a sequence of numbers $\lambda_{i}\rightarrow 0$ and
non-constant functions $U_{i}\rightarrow 0$ which solves equation (4.7). ∎

This ends the proof of Theorem 4.1.

###  4.3 Proof of Theorem 4.2

We study the existence of a discontinuous solutions to problem (1.10). In fact
we start with a following Lemma.

######  Lemma 4.4.

Let $\Omega,J_{\varepsilon}$ satisfy either Assumption 1.4 or 1.2. Assume that
$f\in C^{2}_{b}(\mathbb{R})$ is such that there exists
$u_{1},u_{2}\in\mathbb{R}$ satisfying $f(u_{1})=f(u_{2})=0$ and
$f^{\prime}(u_{1})\neq 0$, $f^{\prime}(u_{2})\neq 0$. Then there exists unique
perturbation $\varphi$ such that $U=\bar{U}+\varphi$ is a solution to steady
problem (1.10) with $\hat{U}$ defined as in Theorem 4.2.

###### Proof.

Plug $U=\tilde{U}+\varphi$ into equation (1.10) and re-write the obtained
formula to get fixed point problem

$\mathcal{F}_{\varepsilon}\varphi=\frac{1}{f^{\prime}(\tilde{U})}\left(\mathcal{L}_{\varepsilon}[\tilde{U}+\varphi]+f(\tilde{U}+\varphi)-f(\tilde{U})-f^{\prime}(\tilde{U})\varphi\right)=\varphi.$

We show that the mapping $\mathcal{F}_{\varepsilon}$ is a Banach contraction
for some large value of $\varepsilon>0$. For the Banach space we choose the
closed ball $\overline{B(0,R)}\subset L^{\infty}(\Omega)$ with a small radius
$R>0$. Let $\varphi\in\overline{B(0,R)}$, then the norm is bounded as follows

$\displaystyle\|\mathcal{F}_{\varepsilon}\varphi\|_{\infty}$
$\displaystyle\leq\left\|\frac{1}{f^{\prime}(\tilde{U})}\right\|_{\infty}\left(C(\mathcal{L}_{\varepsilon})(\|\tilde{U}\|_{\infty}+R)+\|f^{\prime\prime}\|_{\infty}R^{2}\right).$

Here, the quadratic term is estimated using the Mean Value Theorem and the
constant $C(\mathcal{L}_{\varepsilon})$ is given by the formula

$C(\mathcal{L}_{\varepsilon}):=2\sup_{x\in\Omega}\int_{\Omega}J_{\varepsilon}(x-y)\>dy=O(\frac{1}{\varepsilon^{m}}).$

Thus in order to have
$\mathcal{F}_{\varepsilon}:\overline{B(0,R)}\rightarrow\overline{B(0,R)}$ the
following must be fulfilled

$\left\|\frac{1}{f^{\prime}(\tilde{U})}\right\|_{\infty}\left(\frac{1}{\varepsilon^{m}}(\|\tilde{U}\|_{\infty}+R)+\|f^{\prime\prime}\|_{\infty}R^{2}\right)\leq
R.$ (4.8)

Now we derive the contractivity criterion. Take
$\varphi,\psi\in\overline{B(0,R)}$ and estimate

$\displaystyle\|\mathcal{F}_{\varepsilon}\varphi-\mathcal{F}_{\varepsilon}\psi\|_{\infty}$
$\displaystyle\leq\left\|\frac{1}{f^{\prime}(\tilde{U})}\right\|_{\infty}\left(C(\mathcal{L}_{\varepsilon})+\|f^{\prime}(\zeta)-f^{\prime}(\tilde{U})\|_{\infty}\right)\|\varphi-\psi\|_{\infty},$
$\displaystyle\leq\left\|\frac{1}{f^{\prime}(\tilde{U})}\right\|_{\infty}\left(\frac{1}{\varepsilon^{m}}+2\|f^{\prime}\|_{\infty}R\right)\|\varphi-\psi\|_{\infty}.$

The criterion of the following form has to hold true to get contractivity

$\left\|\frac{1}{f^{\prime}(\tilde{U})}\right\|_{\infty}\left(\frac{1}{\varepsilon^{m}}+2\|f^{\prime}\|_{\infty}R\right)<1.$
(4.9)

Observe that conditions (4.8) and (4.9) are satisfied for a sufficiently large
values of dispersion parameter $\varepsilon>0$ and small radius $R>0$. Now the
proof proceeds by the standard Banach fixed point argument. ∎

Next, we emphasize that the obtained discontinuous solution is stable.

######  Lemma 4.5.

Let $\Omega,J_{\varepsilon}$ satisfy either Assumption 1.4 or 1.2. Assume that
$f\in C^{1}(\mathbb{R})$ is such that there exists $u_{1},u_{2}\in\mathbb{R}$
satisfying $f(u_{1})=f(u_{2})=0$ and $f^{\prime}(u_{1})<0$,
$f^{\prime}(u_{2})<0$. Then solution $U$ given by Lemma 4.4 is linearly,
exponentially stable in $L^{2}(\Omega)$.

###### Proof.

By Lemma 4.4 the function $U=\tilde{U}+\varphi$ is a steady state, where the
jump function $\hat{U}$ and small perturbation $\psi$ are defined in the
lemma. We linearise problem (1.1)-(1.2) around $U$ by introducing the new
function $\psi=u-U$. This functions satisfies the following problem

$\displaystyle\begin{cases}\psi_{t}(x,t)&=\mathcal{L}_{\varepsilon}\psi(x,t)+f^{\prime}(U(x))\psi(x,t),\\\
\psi(x,0)&=\psi_{0}(x),\end{cases}$

In order to proceed we define the energy of a linearized system as follows

$E_{\phi}(t):=\frac{1}{2}\int_{\Omega}|\phi(x,t)|^{2}\>dx.$

We verify that the eigenvalue of linearization operator is strictly negative

$\displaystyle\begin{split}\gamma_{0}&=\sup_{\varphi\in
L^{2}\left(\Omega\right),\varphi\neq
0}\frac{\langle\mathcal{L}_{\varepsilon}\varphi+f^{\prime}(U)\varphi,\varphi\rangle}{\|\varphi\|_{2}^{2}}\\\
&\leq\sup_{\varphi\in L^{2}\left(\Omega\right),\|\varphi\|_{2}=1}\langle
f^{\prime}(U)\varphi,\varphi\rangle\\\ &=\sup_{\varphi\in
L^{2}\left(\Omega\right),\|\varphi\|_{2}=1}\int_{\Omega}f^{\prime}(U)\varphi^{2}(x)\>dx<0,\end{split}$
(4.10)

where last strict inequality holds true by continuity of $f^{\prime}$ and
perturbation $\varphi$ being defined in sufficiently small ball. Next we
differentiate the energy $E_{\varphi}$.

$\displaystyle E_{\varphi}^{\prime}(t)$
$\displaystyle=\int_{\Omega}\varphi_{t}(x,t)\varphi(x,t)\>dx$
$\displaystyle=\int_{\Omega}\left(\mathcal{L}_{\varepsilon}\varphi(x,t)+f^{\prime}(U)\varphi(x,t)\right)\varphi(x,t)\>dx$
$\displaystyle=\frac{\langle\mathcal{L}_{\varepsilon}\varphi+f^{\prime}(U)\varphi,\varphi\rangle}{\|\varphi\|_{2}^{2}}\|\varphi\|_{2}^{2}$
$\displaystyle\leq\gamma_{0}E_{\varphi}(t).$

When solved, this inequality yields the following limit

$\|\varphi(\cdot,t)-0\|_{2}^{2}=E_{\varphi}(t)\leq\|\psi_{0}\|_{2}^{2}e^{\gamma_{0}t}.$
(4.11)

Inequality (4.10) and (4.11) yields the stability of zero in a linearized
problem. ∎

Combining lemmas 4.4 and 4.5 completes the proof of Theorem 4.2.

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