constellation yielding existence of some eigenvalue $\lambda_{j}(k)$ of $M(k)$
with eventually constant real part.

Theorem 7.11 and (145) have massive consequences for the long-term behavior of
solutions of the transport-reaction system (122).

In the context of transport-driven instabilities, we assume

######  Assumption 2.

The spectrum $\sigma(B)$ is contained in the left half plane of $\mathbb{C},$
that is

$\sigma(B)\subseteq\mathbb{C}_{-}=\\{\lambda\in\mathbb{C}\,\colon\,\operatorname{Re}\lambda<0\\}.$

Let us recall the definition of Turing patterns. To this end, we again use the
notation

$\Sigma(k)=\max_{\lambda\in\sigma(M(k))}\operatorname{Re}\lambda.$ (147)

######  Definition 7.16 (Turing pattern formation).

The transport-reaction model (122) generates Turing patterns if $B$ fulfills 2
and if there exist finitely many $k_{1},\cdots,k_{n}\in\mathbb{Z}$ such that
the following holds:

  1. (i)

$\Sigma(k_{i})=\Sigma(k_{j})>0\qquad\text{for all }i,j=1,\cdots,n,$

  2. (ii)

$\sup_{\begin{subarray}{c}k\in\mathbb{Z}\\\ k\neq
k_{1},\cdots,k_{n}\end{subarray}}\Sigma(k)<\Sigma(k_{i})\qquad\text{for all
}i=1,\cdots,n.$

In addition to Turing patterns, there is a second possible qualitative
behavior for the solutions of (122).

######  Definition 7.17 (Hyperbolic instabilities).

The transport-reaction model (122) generates hyperbolic instabilities if $B$
fulfills 2 and if there exists $b>0$ and $K\in\mathbb{N}$ such that the
following holds:

  1. (i)

The sequence $(\Sigma(k))_{k\geq K}$ is strictly monotonically increasing with
limit $b$,

  2. (ii)

$\sup_{\begin{subarray}{c}k\in\mathbb{Z}\\\ |k|\leq
K\end{subarray}}\Sigma(k)<b.$

######  Remark 7.18.

The definition relies on the symmetry $\sigma(M(-k))=\overline{\sigma(M(k))}$
for all $k\in\mathbb{N},$ see Lemma 4.11.

Intuitively, hyperbolic instabilities describe chaotic and increasingly
oscillating behavior: for large $t$, high wave numbers $|k|$ are dominant and
for even larger $t$, even higher waves numbers are dominant due to the
eventual strict monotonicity of $(\Sigma(k))_{k\in\mathbb{Z}}$. The remarkable
thing is that every single high frequency is important. Given that the
spectrum of the generator has to be contained in a strip
$\mathbb{C}_{c_{1},c_{2}}=\\{\lambda\in\mathbb{C}\,\colon\,c_{1}<\operatorname{Re}\lambda<c_{2}\\}$
for some constants $c_{1},\,c_{2}\in\mathbb{R}$ by Lemma 3.14, hyperbolic
instabilities in some sense correspond to the most chaotic qualitative
behavior theoretically possible.
In general, if the initial function has an infinite Fourier series,
increasingly high frequencies are dominant. If the initial function has a
finite Fourier series, either the highest wave number or finitely many wave
numbers independent of the initial function ($\text{argmax}_{|k|\leq
K}\Sigma(k)$) are dominant. The number of peaks of the solution typically
increases over time.

Regarding transport-driven instabilities, there is the following dichotomy.

######  Theorem 7.19.

Let the transport-directions $v_{1},\cdots,v_{N}$ be pairwise different.
Moreover, assume that $B$ is a stable matrix, i.e. 1 and 2 are fulfilled. If
the transport-reaction model (122) generates an instability, the instability
is either a Turing pattern or a hyperbolic instability.

###### Proof.

Notice that Theorem 7.11 implies

$\displaystyle\Sigma(k)$
$\displaystyle=\max_{\lambda\in\sigma(M(k))}\operatorname{Re}\lambda=\max_{j=1,\cdots,N}\operatorname{Re}\lambda_{j}(k)=\max\big{\\{}\Sigma_{\text{incr}}(k),\,\Sigma_{\text{decr}}(k)\big{\\}},$

with

$\Sigma_{\text{incr}}(k)\coloneqq\max_{\begin{subarray}{c}j=1,\cdots,N\\\
\operatorname{Re}\lambda_{j}(k)\text{ ev. strictly }\\\ \text{ increasing
}\end{subarray}}\operatorname{Re}\lambda_{j}(k)$

and

$\Sigma_{\text{decr}}(k)\coloneqq\max_{\begin{subarray}{c}j=1,\cdots,N\\\
\operatorname{Re}\lambda_{j}(k)\text{ ev. strictly }\\\ \text{ decreasing
}\end{subarray}}\operatorname{Re}\lambda_{j}(k).$

Hence, $\Sigma(k)$ is eventually the maximum of a strictly monotonically
increasing and a strictly monotonically decreasing function. Moreover, (145)
implies

$\displaystyle\lim_{|k|\rightarrow\infty}\Sigma_{\text{incr}}(k)$
$\displaystyle=\max\\{b_{jj}\,\colon\,\operatorname{Re}\lambda_{j}(k)\text{
eventually strictly monotonically increasing}\\}\eqqcolon b_{1},$
$\displaystyle\lim_{|k|\rightarrow\infty}\Sigma_{\text{decr}}(k)$
$\displaystyle=\max\\{b_{jj}\,\colon\,\operatorname{Re}\lambda_{j}(k)\text{
eventually strictly monotonically decreasing}\\}\eqqcolon b_{2},$
$\displaystyle\lim_{|k|\rightarrow\infty}\Sigma(k)$
$\displaystyle=\max\\{b_{1},\,b_{2}\\}=\max_{j=1,\cdots,N}b_{jj}\eqqcolon b.$
(148)

A case distinction yields the dichotomy: if we have $b_{1}\leq b_{2}$, the
sequence $\Sigma(k)$ converges eventually strictly monotonically decreasing to
$b$ and Turing patterns emerge. If we have $b_{1}>b_{2},$ there exists some
$K\in\mathbb{N}$ such that the sequence $(\Sigma(k))_{k\geq K}$ converges
strictly monotonically increasing to $b$. Now, if there exists some $k$ with
absolute value smaller than $K$ and

$\Sigma(k)>b\qquad,$

the transport-reaction model generates Turing patterns. If not,

$\Sigma(k)<b$

holds for all $k\in\mathbb{Z}$ and the instability is hyperbolic. ∎

We also obtain a useful criteria, which guarantees the existence of transport-
driven instabilities.

######  Corollary 7.20.

Let the transport-directions $v_{1},\cdots,v_{N}$ be pairwise different.
Moreover, assume that $B$ is a stable matrix, i.e. 1 and 2 are fulfilled.
Assume that $B$ has a positive diagonal entry $b_{jj}>0$. Then the transport-
reaction model (122) is unstable and it either generates a Turing pattern or a
hyperbolic instability.

###### Proof.

By assumption and (11), we have

$\lim_{|k|\rightarrow\infty}\Sigma(k)=\max_{j=1,\cdots,N}b_{jj}>0$

and the transport-reaction model (122) generates an instability. It is either
a Turing pattern or a hyperbolic instability by the dichotomy Theorem 7.19. ∎

7.13 and (145) also reveal a useful criterion for the existence of Turing
patterns.

######  Corollary 7.21.

Let the transport-directions $v_{1},\cdots,v_{N}$ be pairwise different.
Moreover, assume that $B$ is a stable matrix, i.e. 1 and 2 are fulfilled. Let
$j^{*}$ be the index with $b_{j^{*}j^{*}}=\max_{j=1,\cdots,N}b_{jj}$ (wlog
unique). Assume that the model is unstable and assume

$\lambda_{j^{*}}^{(3)}=\sum_{\begin{subarray}{c}i,l=1\\\ i,l\neq
j^{*}\end{subarray}}^{N}(v_{l}-v_{j^{*}})^{-1}(v_{i}-v_{j^{*}})^{-1}b_{{j^{*}}l}b_{li}b_{i{j^{*}}}-\sum_{\begin{subarray}{c}i=1\\\
i\neq{j^{*}}\end{subarray}}^{N}(v_{i}-v_{j^{*}})^{-2}b_{i{j^{*}}}b_{{j^{*}}i}b_{{j^{*}}{j^{*}}}<0.$

Then the model generates Turing patterns.

###### Proof.

We eventually have

$\Sigma(k)=\operatorname{Re}\lambda_{j^{*}}(k),$

which is eventually strictly monotonically decreasing due to
$\lambda_{j^{*}}^{(3)}<0,$ see 7.13. ∎

####  11.1 Transport-Driven Instabilities for $N=2$

Until now, we almost always considered the general case of arbitrary many
components $N\in\mathbb{N}$. This subsection gives a very brief excursion to
the case $N=2.$ In this simple case, the transport-reaction model (122) reads

$\partial_{t}\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}+\begin{pmatrix}v_{1}u_{1}^{\prime}\\\
v_{2}u_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}u_{1}\\\ u_{2}\end{pmatrix},$ (149)

where we used the notation

$B=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

for the matrix $B\in\mathbb{R}^{2\times 2}.$ The key takeaway is that
basically the same parameter choices for $B$ which lead to Turing pattern
formation in reaction-diffusion equations, see Theorem 7.2, lead to hyperbolic
instabilities in transport-reaction equations. The difference is that the
transport speeds $v_{1}$ and $v_{2}$ as well as the length $L$ of the circle
$\mathbb{T}_{L}$ have no influence on the instability to occur.
In contrast to the general case with $N$ components, $N=2$ ensures that the
sequences of the real parts of the eigenvalues of

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&0\\\
0&v_{2}\end{pmatrix}+\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

are not only eventually monotone, but even monotone from $k=0$ on. That is,
$\big{(}\operatorname{Re}\lambda_{1}(k)\big{)}_{k\geq 0}$ and
$\big{(}\operatorname{Re}\lambda_{2}(k)\big{)}_{k\geq 0}$ are monotone
sequences. Also the assumption $v_{1}\neq v_{2}$ from Corollary 7.8 is no
mathematical restriction at all. However, the case $v_{1}=v_{2}$ is
uninteresting, because thereby the eigenvalues of $M(k)$ lie on straight
lines.

In any case, for a transport-driven instability to occur, $B$ has to be
stable, i.e.

$\displaystyle\operatorname{tr}(B)$ $\displaystyle=a+d<0,$
$\displaystyle\det(B)$ $\displaystyle=ad-bc>0.$

Additionally, the case of eventually constant real parts of the eigenvalues of
$M(k)$ in Theorem 7.11 can be ruled out by Lemma 7.14. If both eigenvalues are
eventually constant, the eigenvalues lie either on straight lines or
eventually have a real part equal to $a=d.$ Under the assumption that $B$ is
stable and given the immediate monotonicity of the real parts of the
eigenvalues, the whole spectrum of $-A+B$ lies in $\mathbb{C}_{-}$ and the
transport-reaction model does not generate instabilities.
This consideration, (145) and the immediate monotonicity of the real parts of
the eigenvalues can be summarized in the following theorem.

######  Theorem 7.22.

Let $N=2$. The transport-reaction model (149) generates instabilities if and
only if $v_{1}\neq v_{2}$ and the matrix $B$ fulfills

$\displaystyle\operatorname{tr}(B)$ $\displaystyle=a+d<0,$
$\displaystyle\det(B)$ $\displaystyle=ad-bc>0,$ $\displaystyle a>0$
$\displaystyle\text{ or }d>0.$

In this case, the instability is a hyperbolic instability in the sense of
Definition 7.17.

######  Remark 7.23.

Wlog, let $d<0.$ For hyperbolic instabilities to exist, $a$ has to be positive
and $b$ and $c$ need to have different signs.
The situation in which $b,\,d<0$ and $a,\,c>0$ fulfill all conditions is
remarkably similar to the activator-inhibitor scenario for reaction-diffusion
equations. The “hyperbolic interpretation” of these parameters is to see
$u_{1}$ as a prey and $u_{2}$ as a predator. The prey $u_{1}$ increases by
itself and its growth can be offset by the predator $u_{2},$ which increases
only in the presence of the prey.

###  12 Supplementary Figures and Simulations

Let $v_{1},\cdots,v_{N}\in\mathbb{R}$ and let $B\in\mathbb{R}^{N\times N}.$
Consider the transport-reaction equation

$\partial_{t}u+\begin{pmatrix}v_{1}u_{1}^{\prime}\\\ \vdots\\\
v_{N}u_{N}^{\prime}\end{pmatrix}=Bu$ (150)

Complementary to our abstract findings from Corollary 7.8, we add figures to
demonstrate how the spectrum

$\sigma(-A+B)=\overline{\bigcup_{k\in\mathbb{Z}}\sigma(M(k))}$

of generator of the transport-reaction semigroup $(R(t))_{t\geq 0}$ on
$L^{2}(\mathbb{T},\,\mathbb{C}^{N})$ typically looks like. As always, the
matrices $M(k)$ are given by

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&&\\\ &\ddots&\\\
&&v_{N}\end{pmatrix}+B=-2\pi ikV+B.$

In addition, we simulated the solution of (150) in one exemplary case of
Turing patterns and hyperbolic instabilities respectively. The simulations are
based on formula (55). We have no expertise in numerical simulations of
hyperbolic equations and used the exact formula in order to avoid difficulties
like numerical diffusion.

####  12.1 Figures of The Spectrum for $N=2$

In the case $N=2$, the transport-reaction model (150) reads

$\partial_{t}\begin{pmatrix}u_{1}\\\
u_{2}\end{pmatrix}+\begin{pmatrix}v_{1}u_{1}^{\prime}\\\
v_{2}u_{2}^{\prime}\end{pmatrix}=B\begin{pmatrix}u_{1}\\\ u_{2}\end{pmatrix},$

for a matrix $B\in\mathbb{R}^{2\times 2}.$ What follows are plots of

$\sigma(-A+B)=\overline{\bigcup_{k\in\mathbb{Z}}\sigma(M(k))},$

where the matrices $M(k)$ are given by

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&0\\\ 0&v_{2}\end{pmatrix}+B.$

Obviously, the plots only contain a part of the spectrum and there is a cutoff
in $|k|.$ Each dot in the pictures corresponds to one eigenvalue of
$M(k),\,k\in\mathbb{Z}$. The black crosses are the eigenvalues of $B$ and the
parameter choices are written down on the right of each plot.

Figure 2: A typical spectrum of $-A+B$ for $N=2.$

Parameter choices:

$B=\begin{pmatrix}-2&3\\\ -1&-1\end{pmatrix}$ $\displaystyle v_{1}$
$\displaystyle=0.5$ $\displaystyle v_{2}$ $\displaystyle=-0.1$

Figure 3: A typical spectrum of $-A+B$ for $N=2.$

Parameter choices:

$B=\begin{pmatrix}-5&2\\\ -4&-1\end{pmatrix}$ $\displaystyle v_{1}$
$\displaystyle=-0.5$ $\displaystyle v_{2}$ $\displaystyle=-0.1$

Figure 4: An example of the spectrum of $-A+B$ in the case of hyperbolic
instabilities for $N=2.$

Parameter choices:

$B=\begin{pmatrix}3&8\\\ -3&-7\end{pmatrix}$ $\displaystyle v_{1}$
$\displaystyle=0.1$ $\displaystyle v_{2}$ $\displaystyle=-0.1$

Figure 5: An example of the spectrum of $-A+B$ in the case of hyperbolic
instabilities for $N=2$ with only one moving component.

Parameter choices:

$B=\begin{pmatrix}-7&4\\\ -5&2\end{pmatrix}$ $\displaystyle v_{1}$
$\displaystyle=-0.1$ $\displaystyle v_{2}$ $\displaystyle=0$

Figure 6: An example of the spectrum of $-A+B$ in the case of eventually
constant real parts for $N=2,$ see Lemma 7.14.

Parameter choices:

$B=\begin{pmatrix}1&2\\\ 4&1\end{pmatrix}$ $\displaystyle v_{1}$
$\displaystyle=-0.1$ $\displaystyle v_{2}$ $\displaystyle=-0.2$

####  12.2 Figures of The Spectrum for $N=3$

In the case $N=3$, the transport-reaction model (150) reads

$\partial_{t}\begin{pmatrix}u_{1}\\\ u_{2}\\\
u_{3}\end{pmatrix}+\begin{pmatrix}v_{1}&0&0\\\ 0&v_{2}&0\\\
0&0&v_{3}\end{pmatrix}\begin{pmatrix}u_{1}^{\prime}\\\ u_{2}^{\prime}\\\
u_{3}^{\prime}\end{pmatrix}=B\begin{pmatrix}u_{1}\\\ u_{2}\\\
u_{3}\end{pmatrix}$

with $B\in\mathbb{R}^{3\times 3}.$ What follows are plots of

$\sigma(-A+B)=\overline{\bigcup_{k\in\mathbb{Z}}\sigma(M(k))},$

where the matrices $M(k)$ are given by

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&0&0\\\ 0&v_{2}&0\\\
0&0&v_{3}\end{pmatrix}+B.$ (151)

Again, the plots only contain a part of the spectrum and there is a cutoff in
$|k|.$ Each dot in the pictures corresponds to one eigenvalue of
$M(k),\,k\in\mathbb{Z}$. The black crosses are the eigenvalues of $B$ and the
parameter choices are written down on the right of each plot.

In contrast to $N=2,$ there are a variety of possibilities for the location of
the eigenvalues for small $|k|.$ In particular, the real parts of the
eigenvalues are generally no longer immediately monotone.

Figure 7: An example of the spectrum of $-A+B$ for $N=3.$

Parameter choices:

$B=\begin{pmatrix}1&2&-4\\\ -5&-2&2\\\ -4&-5&-1\end{pmatrix}$ $\displaystyle
v_{1}$ $\displaystyle=0.1$ $\displaystyle v_{2}$ $\displaystyle=-0.1$
$\displaystyle v_{3}$ $\displaystyle=0.2$

Figure 8: An example of the spectrum of $-A+B$ in the case of hyperbolic
instabilities for $N=3.$

Parameter choices:

$B=\begin{pmatrix}-6&2&-9\\\ 4&-10&-5\\\ 8&10&2\end{pmatrix}$ $\displaystyle
v_{1}$ $\displaystyle=0.1$ $\displaystyle v_{2}$ $\displaystyle=-0.1$
$\displaystyle v_{3}$ $\displaystyle=0.5$

Figure 9: An example of the spectrum of $-A+B$ in the case of Turing patterns
for $N=3.$

Parameter choices:

$B=\begin{pmatrix}-1&2&-4\\\ -2&-2&2\\\ -6&-7&-8\end{pmatrix}$ $\displaystyle
v_{1}$ $\displaystyle=0.1$ $\displaystyle v_{2}$ $\displaystyle=-0.1$
$\displaystyle v_{3}$ $\displaystyle=0.2$

Figure 10: An example of the spectrum of $-A+B$ in the case of Turing patterns
for $N=3.$

Parameter choices:

$B=\begin{pmatrix}-8&2&-9\\\ -5&-3&-10\\\ 9&-9&-1\end{pmatrix}$ $\displaystyle
v_{1}$ $\displaystyle=0.1$ $\displaystyle v_{2}$ $\displaystyle=-0.2$
$\displaystyle v_{3}$ $\displaystyle=0.2$

Figure 11: An example of the spectrum of $-A+B$ in the case of Turing patterns
due to Corollary 7.21.

Parameter choices:

$B=\begin{pmatrix}-3&2&-4\\\ -5&-5&2\\\ -5&-5&1\end{pmatrix}$ $\displaystyle
v_{1}$ $\displaystyle=-0.1$ $\displaystyle v_{2}$ $\displaystyle=-0.2$
$\displaystyle v_{3}$ $\displaystyle=0.2$

Figure 12: An example of the spectrum of $-A+B$ in the case of Turing patterns
for $N=3$ with only two moving components.

Parameter choices:

$B=\begin{pmatrix}-3&2&-4\\\ -5&-6&2\\\ -5&-5&1\end{pmatrix}$ $\displaystyle
v_{1}$ $\displaystyle=-0.1$ $\displaystyle v_{2}$ $\displaystyle=0.1$
$\displaystyle v_{3}$ $\displaystyle=0$

####  12.3 A Simulation for $N=3$

As already mentioned, the solution formula (55) was used for the simulations.
We cut off the frequencies at $K=100,$ i.e. we considered an initial function
$u_{0}$ with

$\hat{u}_{0}(k)=0$

for all $|k|>100.$ Then, the formula

$u(t,x)=\sum_{k=-100}^{100}e^{tM(k)}\hat{u}_{0}(k)e^{2\pi ikx}$

for the solution to (150) with initial function $u_{0}$ is exact. Regarding
the initial function, we perturbed each component $j=1,2,3$ of the constant
equilibrium by sampling the Fourier coefficients as random normal variables
with

$\hat{u}_{0,j}(k)\sim\mathcal{N}\left(0,10^{-4}(k+1)^{-2}\right)$

for $k=0,\cdots,100$ and setting
$\hat{u}_{0,j}(-k)=\overline{\hat{u}_{0,j}(k)}$. The computation of the matrix
exponential in the solution formula was implemented with scipy.linalg.expm
from the SciPy library.
Concerning the $3$D-plots below, an exponential scaling of the solution is
necessary in order to get a reasonable visualization of the time evolution. In
both our examples, we plotted the rescaled solution

$\widetilde{u}(t,x)\coloneqq e^{-\frac{3}{2}t}u(t,x).$ (152)

Given a specific model, the appropriate rate of the rescaling depends on the
maximum time $t_{\text{max}}$ of the simulation and on the growth rate of the
solution - the latter can be read off from a plot of the spectrum of the
generator.

The simulation of Turing patterns was performed in the exemplary case of

$\partial_{t}\begin{pmatrix}u_{1}\\\ u_{2}\\\
u_{3}\end{pmatrix}+\begin{pmatrix}0.1&0&0\\\ 0&-0.2&0\\\
0&0&0.2\end{pmatrix}\partial_{x}\begin{pmatrix}u_{1}\\\ u_{2}\\\
u_{3}\end{pmatrix}=\begin{pmatrix}-8&2&-9\\\ -5&-3&-10\\\
9&-9&-1\end{pmatrix}\begin{pmatrix}u_{1}\\\ u_{2}\\\ u_{3}\end{pmatrix}.$
(153)

This parameter choice coincides with the one from Figure 10. In particular,
the spectrum of the generator of the transport-reaction semigroup, see Figure
10, indicates that Turing patterns emerge. It is probably not easily apparent
from Figure 10 but the wave numbers $k=\pm 4$ maximize $\Sigma(k),$ with
$\Sigma(k)$ being defined in (147). The matrices $M(k)$ are accordingly given
by

$M(k)=-2\pi ik\begin{pmatrix}0.1&0&0\\\ 0&-0.2&0\\\
0&0&0.2\end{pmatrix}+\begin{pmatrix}-8&2&-9\\\ -5&-3&-10\\\
9&-9&-1\end{pmatrix},$

see (151). Therefore, the number of peaks of the solution of (153) for large
times should be equal to $4$. Our simulation verifies these theoretical
considerations.

Figure 13: A Simulation of Turing pattern formation in the exemplary case
(153). The graphs in the upper row are plots of the three components of the
initial function $u_{0}$. The lower row shows plots of each component of the
solution at the time $t=10.$

(a) The time evolution of $u_{1}$ (b) The time evolution of $u_{2}$

(c) The time evolution of $u_{3}$

Figure 14: The emerging of Turing patterns in the exemplary case (153). Notice
that the values of $u$ are scaled, see (152).

The simulation of hyperbolic instabilities was performed in the exemplary case
of

$\partial_{t}\begin{pmatrix}u_{1}\\\ u_{2}\\\
u_{3}\end{pmatrix}+\begin{pmatrix}0.1&0&0\\\ 0&-0.2&0\\\
0&0&0.5\end{pmatrix}\partial_{x}\begin{pmatrix}u_{1}\\\ u_{2}\\\
u_{3}\end{pmatrix}=\begin{pmatrix}-6&2&-9\\\ 4&-10&-5\\\
8&10&2\end{pmatrix}\begin{pmatrix}u_{1}\\\ u_{2}\\\ u_{3}\end{pmatrix}.$ (154)

This parameter choice coincides with the one from Figure 8. In particular, the
spectrum of the generator of the transport-reaction semigroup, see Figure 8,
indicates that hyperbolic instabilities emerge. Therefore, the number of peaks
of the solution to (154) should increase and chaotic, increasingly oscillating
behavior should emerge. Indeed, the simulation verifies our theoretical
considerations.

Figure 15: A Simulation of hyperbolic instabilities in the exemplary case
(154). The graphs in the upper row are plots of the three components of the
initial function $u_{0}$. The lower rows show plots of each component of the
solution at the times $t=5,10.$

(a) The time evolution of $u_{1}$ (b) The time evolution of $u_{2}$

(c) The time evolution of $u_{3}$

Figure 16: The emerging of hyperbolic instabilities in the exemplary case
(154). Notice that the values of $u$ are scaled, see (152).

The purpose of these detailed appendices is to strongly facilitate reading the
thesis and to write the paper as self-contained as possible.
We decided to present the most important results from semigroup and spectral
theory we applied, including standard references and comments for readers
unfamiliar with semigroup theory.

Our summaries of semigroup and spectral theory are based on the excellent
books from A. Pazy [30] and K.-J. Engel and R. Nagel [8].

##  Appendix A Postponed Proofs

######  Lemma 2.2.

Let $u\in L^{p}(\mathbb{T}^{d})$ and let $k=1,\cdots,d.$

  1. (i)

Let $D_{k}^{h}u\rightharpoonup v$ in $L^{p}(\mathbb{T}^{d})$. Then the $k$-th
weak derivative exists in $L^{p}(\mathbb{T}^{d})$ and $\partial_{k}u=v.$

  2. (ii)

Conversely, if $u\in W^{1,p}(\mathbb{T}^{d})$, it follows that
$D_{k}^{h}u\rightarrow\partial_{k}u$ in $L^{p}(\mathbb{T}^{d})$ for $p>1$ and
$D_{k}^{h}u\rightharpoonup\partial_{k}u$ for $p=1$.

All convergences are with respect to $h\rightarrow 0.$

###### Proof of Lemma 2.2.

  1. (i)

Let $\varphi\in C_{c}^{\infty}(\mathbb{T}^{d})$ be a test function. A
translation yields

$\int_{\mathbb{T}^{d}}D_{k}^{h}u(\mathbf{x})\varphi(\mathbf{x})\,d\mathbf{x}=-\int_{\mathbb{T}^{d}}u(\mathbf{x})D_{k}^{-h}\varphi(\mathbf{x})\,d\mathbf{x}$

and by assumption, the left-hand side converges to

$\int_{\mathbb{T}^{d}}v(\mathbf{x})\varphi(\mathbf{x})\,d\mathbf{x}.$

On the other hand $\varphi$ is smooth, so the difference quotient converges
and dominated convergence implies

$\int_{\mathbb{T}^{d}}v(\mathbf{x})\varphi(\mathbf{x})\,d\mathbf{x}=-\int_{\mathbb{T}^{d}}u(\mathbf{x})\partial_{k}\varphi(\mathbf{x})\,d\mathbf{x},$

i.e. the $k$-th weak derivative of $u$ exists and is given by $v$.

  2. (ii)

The converse direction is a little bit more involved and we first prove a
uniform $L^{p}$ bound of the difference quotients for $1\leq p<\infty$. To
this end, let us firstly consider $u\in C^{1}(\mathbb{T}^{d}).$ By the
fundamental theorem of calculus

$\left|\frac{u(\mathbf{x}+h\mathbf{e_{k}})-u(\mathbf{x})}{h}\right|\leq\int_{0}^{1}\left|\partial_{k}u(\mathbf{x}+sh\mathbf{e_{k}})\right|\,ds$

holds for arbitrary $\mathbf{x}\in\mathbb{T}^{d}.$ We now integrate over the
torus, apply Jensen’s inequality and use Fubini to obtain

$\displaystyle\int_{\mathbb{T}^{d}}|D_{k}^{h}u(\mathbf{x})|^{p}\,d\mathbf{x}$
$\displaystyle\leq\int_{\mathbb{T}^{d}}\left(\int_{0}^{1}|\partial_{k}u(\mathbf{x}+sh\mathbf{e_{k}})|\,ds\right)^{p}d\mathbf{x}\leq\int_{\mathbb{T}^{d}}\int_{0}^{1}|\partial_{k}u(\mathbf{x}+sh\mathbf{e_{k}})|^{p}\,dsd\mathbf{x}$
$\displaystyle\leq\int_{0}^{1}\int_{\mathbb{T}^{d}}|\partial_{k}u(\mathbf{x}+sh\mathbf{e_{k}})|^{p}\,d\mathbf{x}ds=\|\partial_{k}u\|_{L^{p}(\mathbb{T}^{d})}^{p}.$

For $u\in W^{1,p}(\mathbb{T}^{d}),$ the bound follows by the density of
$C^{1}(\mathbb{T}^{d})$ with respect to $\|\cdot\|_{W^{1,p}(\mathbb{T}^{d})}.$
Notice that the difference quotients of an approximation sequence
$(u^{(n)})_{n\in\mathbb{N}}$ converge for fixed $h$ as

$\|D_{k}^{h}u-D_{k}^{h}u^{(n)}\|_{L^{p}(\mathbb{T}^{d})}\leq
2|h|^{-1}\|u-u^{(n)}\|_{L^{p}(\mathbb{T}^{d})}\rightarrow 0\qquad\text{ as
}n\rightarrow\infty.$

This shows

$\|D_{k}^{h}u\|_{L^{p}(\mathbb{T}^{d})}\leq\|\partial_{k}u\|_{L^{p}(\mathbb{T}^{d})}$
(155)

for all $W^{1,p}(\mathbb{T}^{d}).$
In the following, let $1<p<\infty.$ Since $L^{p}(\mathbb{T}^{d})$ is reflexive
in this case, we can choose a subsequence (without relabeling) such that
$D_{k}^{h}u\rightharpoonup v$ in $L^{p}(\mathbb{T}^{d}).$ It follows by (i)
that $v=\partial_{k}u.$ Notice that starting with a subsequence of
$(D_{k}^{h}u)_{h}$ implies that every subsequence has a subsequence which
converges weakly to the same limit in $L^{p}(\mathbb{T}^{d}),$ namely
$\partial_{k}u$. This implies weak convergence for $p>1$. Since (155) implies

$\limsup_{h\rightarrow
0}\|D_{k}^{h}u\|_{L^{p}(\mathbb{T}^{d})}\leq\|\partial_{k}u\|_{L^{p}(\mathbb{T}^{d})},$

strong convergence follows from [3, Proposition 3.32, p. 78] as
$L^{p}(\mathbb{T}^{d})$ is uniformly convex for $p>1.$
Unfortunately, $L^{1}(\mathbb{T}^{d})$ is not reflexive. However, the Dunford-
Pettis theorem, see [3, Theorem 4.30, p. 115], characterizes weakly compact
sets and it suffices to show that $(D_{k}^{h}u)_{h}$ is equiintegrable. The
same subsubsequence argument as before then implies weak convergence of the
whole sequence. Let $A\in\mathcal{B}(\mathbb{T}^{d})$ and let $u\in
C^{1}(\mathbb{T}^{d}).$ Note that we also have

$\int_{A}|D_{k}^{h}u(\mathbf{x})|\,d\mathbf{x}\leq\int_{0}^{1}\int_{A}|\partial_{k}u(\mathbf{x}+sh\mathbf{e_{k}})|\,d\mathbf{x}ds=\int_{0}^{1}\int_{A_{s,h}}|\partial_{k}u(\mathbf{x})|\,d\mathbf{x}ds$

for arbitrary measurable $A$ and with $A_{s,h}$ being the translated set. Now,
consider an arbitrary function $u\in W^{1,1}(\mathbb{T}^{d})$ and take a
sequence $(u^{(n)})_{n\in\mathbb{N}}\subset C^{1}(\mathbb{T}^{d})$ such that
$u^{(n)}\rightarrow u$ in $W^{1,1}(\mathbb{T}^{d})$. In particular,
$(\partial_{k}u^{(n)})_{n\in\mathbb{N}}$ is uniformly integrable and
$D_{k}^{h}u^{(n)}\rightarrow D_{k}^{h}u$ in $L^{1}(\mathbb{T}^{d})$ for every
fixed $h$. Let $\varepsilon>0.$ By the uniform integrability of the
approximation sequence, there exists $\delta>0$ such that

$\mathcal{L}^{d}(B)\leq\delta\quad\implies\quad\sup_{n\in\mathbb{N}}\int_{B}|\partial_{k}u^{(n)}|\,d\mathbf{x}\leq\varepsilon$

for all measurable sets $B\in\mathcal{B}(\mathbb{T}^{d}).$ Now take
$A\in\mathcal{B}(\mathbb{T}^{d})$ with $\mathcal{L}^{d}(A)\leq\delta$ and fix
$h>0.$ It holds

$\int_{A}|D_{k}^{h}u(\mathbf{x})|\,d\mathbf{x}=\lim_{n\rightarrow\infty}\int_{A}|D_{k}^{h}u^{(n)}(\mathbf{x})|\,d\mathbf{x}\leq\limsup_{n\rightarrow\infty}\int_{0}^{1}\int_{A_{s,h}}|\partial_{k}u^{(n)}(\mathbf{x})|\,d\mathbf{x}ds\leq\varepsilon$

due to $\mathcal{L}^{d}(A_{s,h})=\mathcal{L}^{d}(A).$ This estimate is uniform
in $h$ and hence, it holds

$\sup_{h\neq 0}\int_{A}|D_{k}^{h}u|\,d\mathbf{x}\leq\varepsilon,$

i.e. $(D_{k}^{h}u)_{h}$ is equiintegrable.

∎

######  Lemma 2.15.

Let $\mathbf{v_{1}},\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ be non-vanishing,
i.e. $\mathbf{v_{1},}\cdots,\mathbf{v_{N}}\neq 0.$ Moreover, let
$j=1,\cdots,N.$ The following assertions are equivalent:

  1. (i)

Component $j$ is transport periodic.

  2. (ii)

$\mathbf{v_{j,k}}\in\mathbf{v_{j,k^{*}}}\mathbb{Q}$ for all $k=1,\cdots,d$ and
one (and hence all) $k^{*}$ with $\mathbf{v_{j,k^{*}}}\neq 0.$

  3. (iii)

$\displaystyle\bigcap^{d}_{\begin{subarray}{c}k=1\\\ \mathbf{v_{j,k}}\neq
0\end{subarray}}\mathbf{v_{j,k}}\mathbb{Z}\neq\\{0\\}.$

###### Proof of Lemma 2.15.

$(i)\implies(ii)$ If component $j$ is transport periodic, there exists
$t^{*}>0$ with
$t^{*}\mathbf{v_{j}}=(l_{1},\cdots,l_{d})^{T}\in\mathbb{Z}^{d}.$ Then

$\mathbf{v_{j,k}}=\mathbf{v_{j,k^{*}}}\frac{l_{k}}{l_{k^{*}}}\in\mathbf{v_{j,k^{*}}}\mathbb{Q}$

for all $k=1,\cdots,d$ and all $k^{*}$ with $\mathbf{v_{j,k^{*}}}\neq 0.$

$(ii)\implies(iii)$ By assumption, there exists $k^{*}$ with
$\mathbf{v_{j,k^{*}}}\neq 0$ and

$\mathbf{v_{j,k}}=\mathbf{v_{j,k^{*}}}\frac{p_{k}}{q_{k}}$

for all $k=1,\cdots,d$ and some $p_{k}\in\mathbb{Z},\,q_{k}\in\mathbb{N}.$
Take an arbitrary $k$ with $\mathbf{v_{j,k}}\neq 0.$ It follows that

$0\neq\mathbf{v_{j,k^{*}}}\prod_{\begin{subarray}{c}l=1\\\ p_{l}\neq
0\end{subarray}}^{d}p_{l}=\mathbf{v_{j,k}}\bigg{(}q_{k}\prod_{\begin{subarray}{c}l\neq
k\\\ p_{l}\neq 0\end{subarray}}p_{l}\bigg{)}\in\mathbf{v_{j,k}}\mathbb{Z}.$

$(iii)\implies(ii)\implies(i)$ Condition $(iii)$ implies that there exists
$\lambda\neq 0$ and $p_{k}\in\mathbb{Z}$ such that

$\lambda=\mathbf{v_{j,k}}p_{k}$

for all $k$ with $\mathbf{v_{j,k}}\neq 0.$ In this case

$\mathbf{v_{j,k}}=\frac{\lambda}{p_{k}}=\mathbf{v_{j,k^{*}}}\frac{p_{k^{*}}}{p_{k}}\in\mathbf{v_{j,k^{*}}}\mathbb{Q}$

for one (and hence all) $k^{*}$ with $\mathbf{v_{j,k^{*}}}\neq 0.$ For

$t\coloneqq\frac{1}{\mathbf{v_{j,k^{*}}}}\prod_{\begin{subarray}{c}l=1\\\
\mathbf{v_{j}^{l}\neq 0}\end{subarray}}^{d}p_{l}>0$

we obtain

$t\mathbf{v_{j,k}}=t\mathbf{v_{j,k^{*}}}\frac{p_{k^{*}}}{p_{k}}=p_{k^{*}}\prod_{\begin{subarray}{c}l\neq
k\\\ \mathbf{v_{j}^{l}}\neq 0\end{subarray}}p_{l}\in\mathbb{Z}$

for all $k$ with $\mathbf{v_{j,k}}\neq 0.$ For the other $k,$ we trivially
have $t\mathbf{v_{j,k}}=0\in\mathbb{Z}.$ ∎

######  Lemma 7.6.

Consider the setup from Theorem 7.5. For $j=1,\cdots,N$ and $n=1,2,3$, the
coefficients $\hat{\lambda}_{j}^{(n)}\in\mathbb{R}$ are given by

$\displaystyle\hat{\lambda}_{j}^{(1)}$ $\displaystyle=b_{jj},$
$\displaystyle\hat{\lambda}_{j}^{(2)}$
$\displaystyle=-\sum^{N}_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}(v_{i}-v_{j})^{-1}b_{ji}b_{ij},$
$\displaystyle\hat{\lambda}_{j}^{(3)}$
$\displaystyle=\sum_{\begin{subarray}{c}i,l=1\\\ i,l\neq
j\end{subarray}}^{N}(v_{l}-v_{j})^{-1}(v_{i}-v_{j})^{-1}b_{jl}b_{li}b_{ij}-\sum_{\begin{subarray}{c}i=1\\\
i\neq j\end{subarray}}^{N}(v_{i}-v_{j})^{-2}b_{ij}b_{ji}b_{jj}.$

###### Proof of Lemma 7.6.

Recall

$\displaystyle\hat{\lambda}_{j}^{(2)}$
$\displaystyle=-\operatorname{tr}\big{(}BS_{j}BP_{j}\big{)},$
$\displaystyle\hat{\lambda}_{j}^{(3)}$
$\displaystyle=\operatorname{tr}\big{(}BS_{j}BS_{j}BP_{j}\big{)}-\operatorname{tr}\big{(}BS_{j}^{2}BP_{j}BP_{j}\big{)},$

where $P_{j}$ and $S_{j}$ are the matrices

$P_{j}=e_{j}\otimes e_{j}\quad\text{ and }\quad
S_{j}=\sum^{N}_{\begin{subarray}{c}i=1\\\ i\neq
j\end{subarray}}(v_{i}-v_{j})^{-1}(e_{i}\otimes e_{i}).$

For the following matrices, the column differing from all other columns is
always the $j$-th one. We have

$\displaystyle BP_{j}=\begin{pmatrix}0&\cdots&0&b_{1j}&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\cdots&0&b_{Nj}&0&\cdots&0\end{pmatrix},$ $\displaystyle
BS_{j}=\begin{pmatrix}(v_{1}-v_{j})^{-1}b_{11}&\cdots&(v_{j-1}-v_{j})^{-1}b_{1,j-1}&0&(v_{j+1}-v_{j})^{-1}b_{1,j+1}&\cdots&(v_{N}-v_{j})^{-1}b_{1N}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\\\
(v_{1}-v_{j})^{-1}b_{N1}&\cdots&(v_{j-1}-v_{j})^{-1}b_{N,j-1}&0&(v_{j+1}-v_{j})^{-1}b_{N,j+1}&\cdots&(v_{N}-v_{j})^{-1}b_{NN}\end{pmatrix},$
$\displaystyle BS_{j}BP_{j}=\begin{pmatrix}0&\cdots&0&\sum_{i\neq
j}(v_{i}-v_{j})^{-1}b_{1i}b_{ij}&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ 0&\cdots&0&\sum_{i\neq
j}(v_{i}-v_{j})^{-1}b_{Ni}b_{ij}&0&\cdots&0\end{pmatrix}.$

This implies

$\hat{\lambda}_{j}^{(2)}=-\operatorname{tr}\big{(}BS_{j}BP_{j}\big{)}=-\big{(}BS_{j}BP_{j}\big{)}_{jj}=-\sum_{i\neq
j}(v_{i}-v_{j})^{-1}b_{ji}b_{ij}$

and

$\displaystyle\operatorname{tr}\big{(}BS_{j}(BS_{j}BP_{j})\big{)}$
$\displaystyle=\big{(}BS_{j}BS_{j}BP_{j}\big{)}_{jj}=\sum_{l\neq
j}(v_{l}-v_{j})^{-1}b_{jl}\sum_{i\neq j}(v_{i}-v_{j})^{-1}b_{li}b_{ij}$
$\displaystyle=\sum_{i,l\neq
j}(v_{l}-v_{j})^{-1}(v_{i}-v_{j})^{-1}b_{jl}b_{li}b_{ij}.$ (156)

Concerning the computation of $\hat{\lambda}_{j}^{(3)},$ we also need to
compute

$\displaystyle(BP_{j})^{2}=\begin{pmatrix}0&\cdots&0&b_{1j}b_{jj}&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
0&\cdots&0&b_{Nj}b_{jj}&0&\cdots&0\end{pmatrix},$ $\displaystyle
BS_{j}^{2}=\begin{pmatrix}(v_{1}-v_{j})^{-2}b_{11}&\cdots&(v_{j-1}-v_{j})^{-2}b_{1,j-1}&0&(v_{j+1}-v_{j})^{-2}b_{1,j+1}&\cdots&(v_{N}-v_{j})^{-2}b_{1N}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\\\
(v_{1}-v_{j})^{-2}b_{N1}&\cdots&(v_{j-1}-v_{j})^{-2}b_{N,j-1}&0&(v_{j+1}-v_{j})^{-2}b_{N,j+1}&\cdots&(v_{N}-v_{j})^{-2}b_{NN}\end{pmatrix}.$

We obtain

$\displaystyle\operatorname{tr}\big{(}BS_{j}^{2}BP_{j}BP_{j}\big{)}=\big{(}BS_{j}^{2}BP_{j}BP_{j}\big{)}_{jj}=\sum_{i\neq
j}(v_{i}-v_{j})^{-2}b_{ji}b_{ij}b_{jj}$ (157)

and finally, combining (A) and (157) gives

$\hat{\lambda}_{j}^{(3)}=\sum_{i,l\neq
j}(v_{l}-v_{j})^{-1}(v_{i}-v_{j})^{-1}b_{jl}b_{li}b_{ij}-\sum_{i\neq
j}(v_{i}-v_{j})^{-2}b_{ji}b_{ij}b_{jj}.$

∎

######  Lemma 7.14.

Let $N=2.$ Assume that the transport directions satisfy 1, i.e. $v_{1}\neq
v_{2}.$ Let $\lambda_{1}(k),\lambda_{2}(k)$ be the eigenvalues of

$M(k)=-2\pi ikV+B.$

Then $\operatorname{Re}\lambda_{j}(k)$ is eventually constant if and only if
$B$ fulfills (at least) one of the following three conditions:

  1. (i)

$b_{12}=0$,

  2. (ii)

$b_{21}=0$,

  3. (iii)

$b_{11}=b_{22}$.

###### Proof of Lemma 7.14.

Wlog, consider $j=1.$ Let the real parts of $\lambda_{1}(k)$ be eventually
constant, i.e. constant for large $|k|.$ In particular,
$\hat{\lambda}_{1}^{(3)}=0$ has to hold due to the proof of Theorem 7.11.
Given the formula for $\hat{\lambda}_{1}^{(3)}$ from Lemma 7.6 and $N=2$, we
obtain

$\displaystyle 0$
$\displaystyle=\hat{\lambda}_{1}^{(3)}=\sum_{\begin{subarray}{c}i,l=1\\\
i,l\neq
1\end{subarray}}^{2}(v_{l}-v_{1})^{-1}(v_{i}-v_{1})^{-1}b_{1l}b_{li}b_{i1}-\sum_{\begin{subarray}{c}i=1\\\
i\neq 1\end{subarray}}^{2}(v_{i}-v_{1})^{-2}b_{i1}b_{1i}b_{11}$
$\displaystyle=(v_{2}-v_{1})^{-2}b_{12}b_{22}b_{21}-(v_{2}-v_{1})^{-2}b_{21}b_{12}b_{11}$
$\displaystyle=(v_{2}-v_{1})^{-2}b_{12}b_{21}(b_{22}-b_{11}).$

Hence, one of the three conditions of $(i),\,(ii)$ and $(iii)$ has to be
fulfilled.

Conversely, if $(i)$ holds true, i.e. $b_{12}=0$, the matrices $M(k)$ read

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&0\\\
0&v_{2}\end{pmatrix}+\begin{pmatrix}b_{11}&0\\\ b_{21}&b_{22}\end{pmatrix}$

and the eigenvalues of $M(k)$ are given by

$\lambda_{j}(k)=b_{jj}-2\pi ikv_{j}$

for $j=1,2.$ In particular, their real parts are constantly equal to $b_{jj}.$
In the case $(ii),$ $M(k)$ is an upper triangular matrix and the computation
is the same as in $(i).$
Given $b_{11}=b_{22}\eqqcolon b,$ the matrices $M(k)$ read

$M(k)=-2\pi ik\begin{pmatrix}v_{1}&0\\\
0&v_{2}\end{pmatrix}+\begin{pmatrix}b&b_{12}\\\ b_{21}&b\end{pmatrix}$

with

$\displaystyle\det(M(k)-\lambda I_{\mathbb{C}^{2\times 2}})$
$\displaystyle=(b-2\pi ikv_{1}-\lambda)(b-2\pi ikv_{2}-\lambda)-b_{12}b_{21}$
$\displaystyle=\lambda^{2}-\big{(}2b-2\pi
ik(v_{1}+v_{2})\big{)}\lambda+(b-2\pi ikv_{1})(b-2\pi ikv_{2})-b_{12}b_{21}$

Consequently, the eigenvalues of $M(k)$ are given by

$\displaystyle\lambda_{k}(k)$ $\displaystyle=b-\pi
ik(v_{1}+v_{2})\pm\sqrt{\big{(}b-\pi ik(v_{1}+v_{2})\big{)}^{2}-(b-2\pi
ikv_{1})(b-2\pi ikv_{2})+b_{12}b_{21}}$ $\displaystyle=b-\pi
ik(v_{1}+v_{2})\pm\sqrt{\pi^{2}k^{2}\big{(}4v_{1}v_{2}-(v_{1}+v_{2})^{2}\big{)}+b_{12}b_{21}}$
$\displaystyle=b-\pi
ik(v_{1}+v_{2})\pm\sqrt{-\pi^{2}k^{2}(v_{1}-v_{2})^{2}+b_{12}b_{21}}.$

For

$k^{2}>\frac{b_{12}b_{21}}{\pi^{2}(v_{1}-v_{2})^{2}},$

the term under the root is negative and $\operatorname{Re}\lambda_{j}(k)=b.$ ∎

##  Appendix B $C_{0}$-Semigroups, Generators and Abstract Cauchy Problems

Throughout this section, $(E,\|\cdot\|_{E})$ is a complex Banach space and
$\Omega\subseteq\mathbb{R}^{d}$ is an open domain.

In this chapter, we give a heuristic introduction of semigroups and explain
central definitions, generator theorems and solution concepts.

Consider the partial differential equation

$\begin{cases}\begin{array}[]{rrll}\partial_{t}u(t,\mathbf{x})+Au(t,\mathbf{x})&=&F(t,\mathbf{x},u(t,\mathbf{x}))&\qquad(t,\mathbf{x})\in(0,\infty)\times\Omega,\\\
u(0,\mathbf{x})&=&u_{0}(\mathbf{x})&\qquad\mathbf{x}\in\Omega\end{array}\end{cases}$
(158)

for a spacial differential operator $-A$, e.g. the Laplacian $\Delta,$
reasonable boundary conditions on $\partial\Omega$ and a function
$F\colon(0,\infty)\times\Omega\times\mathbb{C}\rightarrow\mathbb{C}$.
The basic idea in semigroup theory is to consider the mapping
$t\mapstochar\rightarrow u(t,\cdot)\in E$ for fixed initial value $u_{0}$ and
some Banach space $(E,\|\cdot\|_{E}).$ Then, $u(t)\coloneqq u(t,\cdot)\in E$
is the solution at time $t$ and (158) is interpreted as an ordinary
differential equation on $E,$ i.e. as

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+Au(t)&=&F(t,u(t))&\qquad t>0,\\\
u(0)&=&u_{0}\end{array}\end{cases}$ (159)

for a densely defined operator $-A$ on $E$ with domain $D(-A)$ and a function
$F\colon(0,\infty)\times E\rightarrow E.$ Equation (159) is called abstract
Cauchy problem and the family of operators $(T(t))_{t\geq 0}$ on $E$, which
maps an initial function $u_{0}$ to the solution for all $t\geq 0$, is called
the semigroup generated by $(-A,\,D(-A)).$ This family should clearly satisfy
$T(0)=I_{E},$ because the solution at time $t=0$ is the initial function
itself. Secondly, taking $t,s>0$ and restarting (159) after time $s$ with
initial function $T(s)u_{0}$ should give the same solution as if we did not
restart the equation. Thirdly, the solution map $t\mapstochar\rightarrow
T(t)u_{0}$ should fulfill a continuity property for every fixed initial
function. These considerations motivate the definition of a strongly
continuous semigroup.

######  Definition B.1 ([30, Chapter 1, Definition 1.1, p. 1 and Definition
2.1, p. 4]).

A one parameter family $(T(t))_{t\geq 0}$ of bounded linear operators from $E$
into $E$ is a strongly continuous semigroup if

  1. (i)

$T(0)=I_{E},$

  2. (ii)

$T(t+s)=T(t)T(s)\quad\text{ for all }t,s>0,$

  3. (iii)

$\displaystyle\lim_{t\searrow 0}T(t)u=u\quad\text{ for every }u\in E.$

######  Remark B.2.

Strongly continuous semigroups are often called $C_{0}$-semigroups and $(ii)$
is the so called semigroup property.

######  Definition B.3 ([30, Chapter 1, Definition 1.1, p. 1]).

Let $(T(t))_{t\geq 0}$ be a strongly continuous semigroup. The linear operator
$(-A,\,D(-A))$ defined by

$D(-A)=\bigg{\\{}u\in E\,\colon\,\lim_{t\searrow 0}\frac{T(t)u-u}{t}\text{
exists}\bigg{\\}}$

and

$-Au=\lim_{t\searrow 0}\frac{T(t)u-u}{t}\quad\text{ for }u\in D(-A)$

is the infinitesimal generator of the semigroup, $D(-A)$ is the domain of
$-A.$

In applications, one almost always starts with an abstract Cauchy problem
(159) and well-posedness comes down to the question, whether $-A$ generates a
$C_{0}$-semigroup. The domain of the operator has to be chosen carefully and
should take the boundary values of the PDE into account. With this in mind,
criteria which characterize infinitesimal generators of $C_{0}$-semigroups are
extremely useful and are given through the theorems of Hille-Yosida and Lumer-
Phillips.

######  Theorem B.4 ([30, Chapter 1, Theorem 2.2, p. 4]).

Let $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup. There exists constants
$\omega\geq 0$ and $M\geq 1$ such that

$\|T(t)\|_{\mathcal{L}(E)}\leq Me^{\omega t}\quad\text{ for all }t\geq 0.$
(160)

######  Definition B.5 ([30, Chapter 1, p. 8]).

A strongly continuous semigroup $(T(t))_{t\geq 0}$ which fulfills the estimate
(160) with $\omega=0$ is called uniformly bounded and if moreover $M=1$, it is
called a semigroup of contractions.

######  Theorem B.6 (Hille-Yosida, [30, Chapter 1, Theorem 5.3, p. 20]).

A linear operator $(-A,\,D(-A))$ is the infinitesimal generator of a
$C_{0}$-semigroup $(T(t))_{t\geq 0}$, satisfying
$\|T(t)\|_{\mathcal{L}(E)}\leq Me^{\omega t},$ if and only if

  1. (i)

$-A$ is closed and $D(-A)$ is dense in $E$,

  2. (ii)

The resolvent set $\rho(-A)$ of $-A$ contains the ray $(\omega,\infty)$ and

$\|R(\lambda,\,-A)^{n}\|_{\mathcal{L}(E)}\leq\frac{M}{(\lambda-\omega)^{n}}\quad\text{
for all }\lambda>\omega\text{ and }n\in\mathbb{N},$ (161)

where $R(\lambda,\,-A)=(\lambda+A)^{-1}$ is the resolvent.

######  Remark B.7.

This general version of the Hille-Yosida theorem is mainly of theoretical
importance. In practice, the estimate (161) is hard to check for all
$n\in\mathbb{N}.$ For contraction semigroups, there is a version of the Hille-
Yosida theorem which avoids this problem, see [30, Chapter 1, Theorem 3.1, p.
8].

######  Definition B.8 ([30, Chapter 1, Definition 4.1, p. 13 and Theorem 4.2,
p. 14]).

A linear operator $(-A,\,D(-A))$ is dissipative if

$\|(\lambda I_{E}+A)u\|_{E}\geq\lambda\|u\|_{E}\quad\text{ for all }u\in
D(-A)\text{ and all }\lambda>0.$

######  Theorem B.9 (Lumer-Phillips[30, Chapter 1, Theorem 4.3, p. 14]).

Let $(-A,\,D(-A))$ be a linear operator with dense domain $D(-A)$ in $E.$

  1. (i)

If $-A$ is dissipative and there is a $\lambda_{0}>0$ such that the range
$\operatorname{rg}(\lambda_{0}I_{E}+A)$ of $\lambda_{0}I_{E}+A$ is $E$, then
$-A$ is the infinitesimal generator of a $C_{0}$-semigroup of contractions.

  2. (ii)

Conversely, if $-A$ is the infinitesimal generator of a $C_{0}$-semigroup of
contractions on $E,$ then $\operatorname{rg}(\lambda I_{E}+A)=E$ for all
$\lambda>0$ and $-A$ is dissipative.

After ensuring that $(-A,\,D(-A))$ generates a strongly continuous semigroup,
the abstract Cauchy Problem (159) can be “solved”. There are specific solution
concepts for semigroup theory, since (158) was reformulated into a Banach
space setting and we state the basic concepts and results.
Let $T>0,$ possibly $T=\infty,$ and let $I=[0,T)\subseteq\mathbb{R}_{\geq 0}$
be an interval. For simplicity, we start with the cases $F\equiv 0$ and
$F(t,u)=F(t)$ in order to obtain a feeling for abstract Cauchy problems. In
the homogeneous case, the abstract Cauchy problem (159) on $I=\mathbb{R}_{\geq
0}$ reads

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+Au(t)&=&0&\qquad t>0,\\\
u(0)&=&u_{0}\end{array}\end{cases}$ (162)

and the next theorem validates the concept of semigroups.

######  Theorem B.10 ([30, Chapter 4, Theorem 1.3, p. 102]).

Let $(-A,\,D(-A))$ be a densely defined linear operator with non-empty
resolvent set $\rho(-A).$ The initial value problem (162) has a unique
solution $u\in C^{1}([0,\infty),\,E)$ for every $u_{0}\in D(-A)$ if and only
if $-A$ is the infinitesimal generator of a $C_{0}$-semigroup $(T(t))_{t\geq
0}.$ In this case, the solution is given by $u(t)=T(t)u_{0}.$

Hereafter, we will assume that $(-A,\,D(-A))$ generates the $C_{0}$-semigroup
$(T(t))_{t\geq 0}$ so that the homogeneous equation (162) has a unique
solution for $u_{0}\in D(-A)$.
The inhomogeneous abstract Cauchy problem on $I=[0,T)$ reads

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+Au(t)&=&F(t)&\qquad 0<t<T,\\\
u(0)&=&u_{0}\end{array}\end{cases}.$ (163)

######  Definition B.11 ([30, Chapter 4, Definition 2.1, p. 105]).

A function $u\colon I\rightarrow E$ is a classical solution on $I$ to the
abstract Cauchy problem (159) if

  1. (i)

$u\in C(I,\,E)\cap C^{1}((0,T),\,E),$

  2. (ii)

$u(t)\in D(-A)$ for all $t\in(0,T)$ and (159) is satisfied on $I$.

######  Definition B.12 ([30, Chapter 4, Definition 2.3, p. 106]).

Let $u\in E$ and $F\in L^{1}((0,T),\,E).$ The function $u\in C([0,T],\,E)$
given by

$u(t)=T(t)u+\int_{0}^{t}T(t-s)F(s)\,ds,\qquad t\in[0,T],$ (164)

is the mild solution to the initial value problem (163) on $[0,T].$

######  Remark B.13 ([30, Chapter 4, Corollary 2.2, p. 106]).

Under the assumption $F\in L^{1}((0,T),\,E)$, a classical solution coincides
with the mild solution and (163) has at most one solution. If (163) has a
solution, it is given by (164).

######  Theorem B.14 ([30, Chapter 4, Corollary 2.5, p. 107 and Corollary 2.6,
p. 108]).

The initial value problem (163) has a unique classical solution on $I$ for
every $u_{0}\in D(-A),$ if $F$ satisfies one of the following two properties:

  1. (i)

$F\in C^{1}([0,T],\,E),$

  2. (ii)

$F\in C((0,T),\,E)\cap L^{1}((0,T),\,E)\quad\text{ and }\quad F(s)\in
D(-A)\text{ for all }s\in(0,T)\text{ with }AF(\cdot)\in L^{1}((0,T),\,E).$

We now summarize the main results for the semilinear initial value problem

$\begin{cases}\begin{array}[]{rrll}\dot{u}(t)+Au(t)&=&F(t,u(t))&\qquad
0<t<T,\\\ u(0)&=&u_{0}\end{array}\end{cases}.$ (165)

The definition of classical solutions for (165) is the same as before, see
Definition B.11. Similar to the linear inhomogeneous case, classical solutions
are mild solutions in the sense of the next definition.

######  Definition B.15 ([30, Chapter 6, Definition 1.1, p. 184]).

A function $u\in C(I,\,E)$ which solves the integral equation

$u(t)=T(t)u_{0}+\int_{0}^{t}T(t-s)F(s,u(s))\,ds,\qquad t\in I,$

is called a mild solution to the initial value problem (165) on $I$ .

A typical Lipschitz assumption on $F$ ensures local existence of mild
solutions.

######  Definition B.16 ([30, Chapter 6, p. 185]).

A function $F\colon[0,\infty)\times E\rightarrow E$ is called locally
Lipschitz continuous in $u$, uniformly in $t$ on bounded intervals, if for
every $t^{\prime}\geq 0$ and $n\in\mathbb{N}$, there is a constant
$C(t^{\prime},n)$ such that

$\|F(t,u)-F(t,\widetilde{u})\|_{E}\leq C(t^{\prime},n)\|u-\widetilde{u}\|_{E}$

holds for all $u,\widetilde{u}\in E$ with
$\|u\|_{E},\,\|\widetilde{u}\|_{E}\leq n$ and $t\in[0,t^{\prime}].$

######  Theorem B.17 ([30, Chapter 6, Theorem 1.4, p. 185]).

Let $F\colon[0,\infty)\times E\rightarrow E$ be continuous in $t$ and locally
Lipschitz continuous in $u$, uniformly in $t$ on bounded intervals. Then for
every $u_{0}\in E,$ there is a time
$T_{\text{max}}=T_{\text{max}}(\|u_{0}\|_{E})>0,$ possibly $T_{\text{max}}=T,$
such that the initial value problem (165) has a unique mild solution $u$ on
$[0,T_{\text{max}}).$ Moreover, if $T_{\text{max}}<T,$ then

$\lim_{t\nearrow T_{\text{max}}}\|u(t)\|_{E}=\infty.$

######  Remarks B.18.

  1. (i)

The mild solution on $[0,T_{\text{max}})$ is often called maximal mild
solution.

  2. (ii)

If for each $t^{\prime}>0,$ there exists a constant $C(t^{\prime})>0$ such
that

$\|F(t,u)\|_{E}\leq C(1+\|u\|_{E})$

holds for all $t\in[0,t^{\prime}]$ and all $u\in E$, then (165) has a mild
solution on $[0,\infty),$ i.e. one can choose $T=\infty$ and it holds
$T_{\text{max}}=T.$

There are multiple criteria which ensure that a mild solution is a classical
solution (see [30, Chapter 6, p. 187ff.]) and they typically involve stronger
regularity of the right-hand side. We endow the domain $D(-A)$ with the graph
norm $\|u\|_{-A}=\|u\|_{E}+\|Au\|_{E},$ which turns $(D(-A),\|\cdot\|_{-A})$
to a Banach space due to the closedness of $-A,$ see Theorem B.6. Then, for
example, Theorem B.14 and Theorem B.17 imply the next statement.

######  Theorem B.19.

Let $F\in C([0,\infty)\times E,\,D(-A))$ be uniformly (in t) locally Lipschitz
continuous in $u$. Then for every $u_{0}\in D(-A),$ the initial value problem
(165) possesses a unique maximal strong solution

$u\in C([0,T_{\text{max}}),\,D(-A))\cap C^{1}((0,T_{\text{max}}),\,E).$

We finish our overview with a result concerning the continuous dependence of
the solution on the initial data.

######  Corollary B.20 ([30, Chapter 6, Theorem 1.2, p. 184]).

Let $F\in C([0,\infty)\times E,\,E)$ be globally Lipschitz continuous in $u.$
Then the mild solution depends Lipschitz continuously on $u_{0}\in E,$ i.e.
for all $T>0,$ the mapping $u_{0}\mapstochar\rightarrow u$ is Lipschitz
continuous from $E$ into $C([0,T],\,E)$.

##  Appendix C Perturbation and Approximation of $C_{0}$-Semigroups

We assume throughout this chapter, similar to the previous one, that
$(E,\,\|\cdot\|_{E})$ is a complex Banach space.

Many times in applications, the verification of the generator theorems from
Appendix B is a difficult task and cannot be performed in a direct way.
Therefore, one tries to build up the given operator and its semigroup from
simpler ones. Perturbation and approximation are the standard methods for this
approach [8, Chapter 3, Introduction, p. 157].
We present the main results for the perturbation of an infinitesimal generator
$(-A,\,D(-A))$ of a semigroup $(T(t))_{t\geq 0}$ with a bounded linear
operator $B\in\mathcal{L}(E).$

The following theorem summarizes multiple results from [8, 30], namely [8,
III. Theorem 1.3, p. 158] and [8, III. Corollary 1.7, p. 161]. The uniqueness
claim concerning properties $(ii)$ and $(iii)$ is proven in [30, Chapter 3,
Proposition 1.2, p. 77] and a proof of the (in our case obvious) equivalence
of the graph norms can be found in [8, III. Lemma 2.4, p. 171] for a more
general framework.

######  Theorem C.1.

Let $(-A,\,D(-A))$ generate the $C_{0}$-semigroup $(T(t))_{t\geq 0}$
satisfying $\|T(t)\|_{\mathcal{L}(E)}\leq Me^{\omega t}$ for all $t\geq 0$ and
constants $M\geq 1$ and $\omega\in\mathbb{R}.$ Let $B\in\mathcal{L}(E).$ Then
the sum $-A+B$ with $D(-A+B)=D(-A)$ generates a $C_{0}$-semigroup
$(R(t))_{t\geq 0}$ which fulfills

  1. (i)

$\|R(t)\|_{\mathcal{L}(E)}\leq Me^{(\omega+M\|B\|_{\mathcal{L}(E)})t},$

  2. (ii)

$R(t)u=T(t)u+\int_{0}^{t}T(t-s)BR(s)u\,ds,$

  3. (iii)

$R(t)u=T(t)u+\int_{0}^{t}R(t-s)BT(s)u\,ds$

for all $t\geq 0$ and $u\in E.$ The operator family $(R(t))_{t\geq 0}$ is the
only strongly continuous family of operators solving $(ii)$ and/or $(iii).$
The graph norms of $-A$ and $-A+B$ on $D(-A)$ are equivalent.

Aside from the implicit representation in Theorem C.1, there are other, more
useful formulas for the semigroup $(R(t))_{t\geq 0}.$

######  Theorem C.2 (Dyson-Phillips series, [8, III. Theorem 1.10, p. 163]).

Let $(-A,\,D(-A))$ generate the $C_{0}$-semigroup $(T(t))_{t\geq 0}$ and let
$B\in\mathcal{L}(E).$ The semigroup $(R(t))_{t\geq 0}$ generated by $-A+B$
with $D(-A+B)=D(-A)$ satisfies

$R(t)=\sum_{k=0}^{\infty}R_{k}(t)$ (166)

with

$R_{0}(t)=T(t),\quad R_{k+1}(t)u=\int_{0}^{t}T(t-s)BR_{k}(t)u\,ds$

for all $t\geq 0$ and $u\in E.$ The Dyson-Phillips series (166) converges in
$\mathcal{L}(E)$ and uniformly on compact subsets of $\mathbb{R}_{\geq 0}.$

######  Theorem C.3 (Lie-Trotter product forumla, [8, III. Corollary 5.8, p.
227]).

Assume that $(-A,\,D(-A))$ and $(B,\,D(B))$ generate the $C_{0}$-semigroups
$(T(t))_{t\geq 0}$ and $(S(t))_{t\geq 0}$ respectively, subject to the
stability bound

$\|(T(\tfrac{t}{n})S(\tfrac{t}{n}))^{n}\|_{\mathcal{L}(E)}\leq Me^{\omega t}$

for all $n\in\mathbb{N},\,t\geq 0$ and some $M\geq 1$ and
$\omega\in\mathbb{R}.$ Let $D\coloneqq D(-A)\cap D(B)$ and $(\lambda
I_{E}-(-A+B))D$ be dense in $E$ for some $\lambda>\omega.$ Then the sum $-A+B$
with domain $D(-A+B)\coloneqq D$ has a closure $\overline{-A+B}$ which
generates the $C_{0}$-semigroup $(R(t))_{t\geq 0}$ given by

$R(t)u=\lim_{n\rightarrow\infty}\left(T(\tfrac{t}{n})S(\tfrac{t}{n})\right)^{n}u$

for all $t\geq 0$ and $u\in E.$ For every fixed $u\in E,$ the convergence is
uniform on compact subsets of $\mathbb{R}_{\geq 0}.$

##  Appendix D Spectral Theory and Long-Term Behavior of $C_{0}$-Semigroups

Again, $(E,\,\|\cdot\|_{E})$ is assumed to be a complex Banach space.

The purpose of this chapter is to give an overview of the main theory we used
when studying the spectral properties and the long-term behavior of the
transport-reaction semigroups from Chapter 3.
It is necessary to recall definitions from spectral theory before presenting
statements on the asymptotics of strongly continuous semigroups as many
results characterize asymptotical properties in terms of spectral properties
of the semigroup.
As a guideline, the goal is often to deduce stability or more complex long-
term behavior of the semigroup $(T(t))_{t\geq 0}$ from spectral properties of
its generator $(-A,\,D(-A))$. To this end, it is necessary to study the
connection of the spectrum of the generator and the spectrum of the semigroup.
This connection should intuitively be given by

$``\quad\sigma(T(t))=e^{t\sigma(-A)}\quad",$ (167)

but generally, this formula is incorrect. Especially for hyperbolic equations,
proving a connection in the fashion of (167) is non-trivial.

In infinite dimensional spaces, a linear operator is no longer injective if
and only if it is surjective. Therefore, one needs a more refined definition
for the spectrum than in finite dimensions, where the spectrum of a matrix is
simply given by the set of its eigenvalues. The next definition and its remark
unite [8, IV. Definition 1.1, p. 239], [8, IV. Definition 1.6, p. 241], [8,
IV. Definition 1.8 and Lemma 1.9, p. 242] and [8, IV. Definition 1.11, p.
243].

######  Definition D.1.

Let $(-A,\,D(-A))$ be a closed operator. We define

  1. (i)

the spectrum $\sigma(-A)=\left\\{\lambda\in\mathbb{C}\,\colon\,\lambda
I_{E}+A\text{ is not bijective}\right\\},$

  2. (ii)

the resolvent set $\rho(-A)=\sigma(-A)^{c},$

  3. (iii)

the point spectrum
$\sigma_{p}(-A)=\left\\{\lambda\in\mathbb{C}\,\colon\,\lambda I_{E}+A\text{ is
not injective}\right\\},$

  4. (iv)

the approximate point spectrum

$\displaystyle\sigma_{ap}(-A)$
$\displaystyle=\\{\lambda\in\mathbb{C}\,\colon\,\lambda I_{E}+A\text{ is not
injective}\,\text{ or }\,R(\lambda+A)\text{ is not closed in }E\\}$
$\displaystyle=\\{\lambda\in\mathbb{C}\,\colon\,\exists(u^{(n)})_{n\in\mathbb{N}}\subset
D(-A)\colon\|u^{(n)}\|_{E}=1,\,\lambda u^{(n)}+Au^{(n)}\rightarrow 0\text{ as
}n\rightarrow\infty\\},$

  5. (v)

the residual spectrum
$\sigma_{r}(-A)=\\{\lambda\in\mathbb{C}\,\colon\,(\lambda I_{E}+A)D(-A)\text{
is not dense in E}\\}.$

######  Remarks D.2.

  1. (i)

For $\lambda\in\rho(-A),$ the inverse

$R(\lambda,-A)\coloneqq(\lambda+A)^{-1}$

is, by the closed graph theorem, a bounded linear operator on $E$ and is
called the resolvent (of $-A$ at the point $\lambda$).

  2. (ii)

Each $\lambda\in\sigma_{p}(-A)$ is called an eigenvalue and each $0\neq u\in
D(-A)$ satisfying $(\lambda+A)u=0$ is an eigenvector of $-A$ (corresponding to
$\lambda$).

######  Definition D.3 ([8, IV. Corollary 1.4, p. 241]).

Let $T\in\mathcal{L}(E).$ The quantity

$r(T)=\max\left\\{|\lambda|\,\colon\,\lambda\in\sigma(T)\right\\}$

is called the spectral radius of $T.$

######  Definition D.4 ([8, I. Definition 5.6, p. 40 and IV. Definition 2.1,
p. 250]).

Let $\mathcal{T}\coloneqq(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup on $E$ with
generator $(-A,\,D(-A)).$ The quantity

$\omega_{0}=\omega_{0}(\mathcal{T})=\inf\left\\{\omega\in\mathbb{R}\,\colon\,\exists
M_{\omega}\geq 1\,\forall t\geq 0\colon\|T(t)\|_{\mathcal{L}(E)}\leq
M_{\omega}e^{\omega t}\right\\}$

is called its growth bound and

$s(-A)=\sup\left\\{\operatorname{Re}\lambda\,\colon\,\lambda\in\sigma(-A)\right\\}$

is called the spectral bound of its generator.

######  Remark D.5.

Another common notation for the growth bound of a semigroup $(T(t))_{t\geq 0}$
with generator $(-A,\,D(-A))$ is $\omega_{0}(-A).$

The spectral radius of the semigroup operators, the growth bound of the
semigroup and the spectral bound of the generator are closely connected via
the next proposition.

######  Proposition D.6 ([8, IV. Proposition 2.2, p. 251]).

Let $\mathcal{T}=(T(t))_{t\geq 0}$ be a semigroup with generator
$(-A,\,D(-A)).$ One has

$-\infty\leq
s(-A)\leq\omega_{0}(\mathcal{T})=\inf_{t>0}\frac{1}{t}\log\|T(t)\|_{\mathcal{L}(E)}=\lim_{t\rightarrow\infty}\frac{1}{t}\log\|T(t)\|_{\mathcal{L}(E)}=\frac{1}{t_{0}}\log
r(T(t_{0}))<\infty$

for all $t_{0}>0.$ In particular, the spectral radius of the operator $T(t)$
is given by

$r(T(t))=e^{\omega_{0}t}\qquad\text{ for all }t\geq 0.$

######  Definition D.7 (equivalent to [8, V. Definition 1.1 (a), p. 296]).

A $C_{0}$-semigroup $(T(t))_{t\geq 0}$ is called uniformly exponentially
stable if there exist constants $M,\,\varepsilon>0$ such that

$\|T(t)\|_{\mathcal{L}(E)}\leq Me^{-\varepsilon t}$

holds for all $t\geq 0.$

An application of Proposition D.6 yields the following characterizations of
exponential stability.

######  Proposition D.8 ([8, V. Proposition 1.7, p. 299]).

Let $\mathcal{T}=(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup with generator
$(-A,\,D(-A)).$ Then the following assertions are equivalent:

  1. (i)

$(T(t))_{t\geq 0}$ is uniformly exponentially stable.

  2. (ii)

$\exists t_{0}>0$ such that $\|T(t_{0})\|_{\mathcal{L}(E)}<1.$

  3. (iii)

$\exists t_{1}>0$ such that $r(T(t_{1}))<1.$

  4. (iv)

$\omega_{0}(\mathcal{T})<0.$

If this is the case, $(ii)$ is valid for all sufficiently large $t_{0}>0,$
assertion $(iii)$ is true for all $t_{1}>0$ and we have $s(-A)<0.$

The above given criteria for uniform exponential stability have a huge
disadvantage: they rely on properties of the semigroup itself. Typically in
applications, only the generator is known. It is therefore desirable to
characterize uniform exponential stability directly in terms of the generator
[8, V. Introduction, p. 301].
One would hope that the inequality $s(-A)\leq\omega_{0}$ from Proposition D.6
is actually an equality. This would imply that

$s(-A)<0$

is equivalent to uniform exponential stability. Unfortunately, this is
generally incorrect and the situation is non-trivial. There are (even
positive) semigroups $(T(t))_{t\geq 0}$ with $s(-A)<0$ and $\omega_{0}\geq 0;$
see for example [8, IV. Counterexample 2.7, p. 254]. The equality can also
fail on less artificial spaces and there are counterexamples on reflexive
Banach spaces or even Hilbert spaces, cf. [8, IV. Counterexamples 3.3 and 3.4,
p. 271-275].

The so called weak spectral mapping theorem

$\sigma(T(t))\cup\\{0\\}=\overline{e^{t\sigma(-A)}}\cup\\{0\\}\quad\text{ for
all }t\geq 0$ (WSMT)

is one possibility to get around the problem.

######  Lemma D.9 ([8, V. Lemma 1.9, p. 301]).

If for a strongly continuous semigroup $(T(t))_{t\geq 0}$ and its generator
$(-A,\,D(-A))$ the weak spectral mapping theorem (WSMT) holds, then the growth
bound $\omega_{0}$ and the spectral bound $s(-A)$ coincide, i.e.

$s(-A)=\omega_{0}.$

In this case, the semigroup is uniformly exponentially stable if and only if
$s(-A)<0.$

The question remaining is which assumptions on the semigroup $(T(t))_{t\geq
0}$ ensure the existence of a weak spectral mapping theorem. The next result
shows that neither the point, nor the residual spectrum, cause any problems.

######  Theorem D.10 ([8, IV. Theorem 3.7, p. 277]).

For a generator $(-A,\,D(-A))$ of a strongly continuous semigroup
$(T(t))_{t\geq 0}$, we have the identities

$\displaystyle\sigma_{p}(T(t))\backslash\\{0\\}$
$\displaystyle=e^{t\sigma_{p}(-A)},$
$\displaystyle\sigma_{r}(T(t))\backslash\\{0\\}$
$\displaystyle=e^{t\sigma_{r}(-A)}$

for all $t\geq 0.$

Moreover, there is the spectral inclusion theorem.

######  Theorem D.11 (Spectral Inclusion Theorem, [8, IV. Theorem 3.6, p.
276]).

For the generator $(-A,\,D(-A))$ of a strongly continuous semigroup
$(T(t))_{t\geq 0}$, we have the inclusions

$\displaystyle\sigma(T(t))$ $\displaystyle\supseteq e^{t\sigma(-A)}$
$\displaystyle\sigma_{ap}(T(t))$ $\displaystyle\supseteq e^{t\sigma_{ap}(-A)}$

for all $t\geq 0.$

Special classes of semigroups fulfill a stronger version of (WSMT), the
spectral mapping theorem. A prominent example is the class of analytic
semigroups, which appears in the study of parabolic equations; see [8, II.
Definition 4.5, p. 101] for their definition.

######  Corollary D.12 ([8, IV. Corollary 3.12, p. 281]).

The spectral mapping theorem

$\sigma(T(t))\backslash\\{0\\}=e^{t\sigma(-A)}\quad\text{ for all }t\geq 0$
(SMT)

holds for the following class of strongly continuous semigroups:

  1. (i)

eventually compact semigroups,

  2. (ii)

eventually differentiable semigroups,

  3. (iii)

analytic semigroups,

  4. (iv)

uniformly continuous semigroups.

######  Remark D.13.

For the definitions of eventually compact, eventually differentiable and
uniformly continuous semigroups, we refer to [8, II. Definition 4.23, p. 117],
[8, II. Definition 4.13, p. 109] and [8, I. Definition 3.6, p. 16]
respectively.

Unfortunately, the transport-reaction semigroups appearing when studying the
linearized version of the hyperbolic equation

$\partial_{t}\begin{pmatrix}u_{1}\\\ \vdots\\\
u_{N}\end{pmatrix}+\begin{pmatrix}\mathbf{v_{1}}\cdot\nabla u_{1}\\\ \vdots\\\
\mathbf{v_{N}}\cdot\nabla u_{N}\end{pmatrix}=F(u)$ (168)

on the $d$-dimensional torus $\mathbb{T}^{d}$ with $N\in\mathbb{N}$ population
subgroups, transport directions
$\mathbf{v_{1}},\,\cdots,\mathbf{v_{N}}\in\mathbb{R}^{d}$ and a sufficiently
smooth reaction function $F$ are less regular than all classes of semigroups
from Corollary D.12. Having said this, they can be expanded to a strongly
continuous group $(T(t))_{t\in\mathbb{R}}$ and for these groups, there is the
following result.

######  Theorem D.14 ([8, IV. Theorem 3.16 and Exercise 3.22, p. 283ff.]).

Let $(T(t))_{t\in\mathbb{R}}$ be a polynomially bounded stongly continuous
group with generator $(-A,\,D(-A))$. Then the weak spectral mapping theorem

$\sigma(T(t))=\overline{e^{t\sigma(-A)}}$

for $t\in\mathbb{R}$ holds.

######  Remark D.15.

Polynomially bounded means that there is a polynomial
$P\colon\mathbb{R}\rightarrow\mathbb{R}$ with

$\|T(t)\|_{\mathcal{L}(E)}\leq P(t)\quad\text{ for all }t\in\mathbb{R}.$

Without any reactions, i.e. the homogeneous case of (168), the generated
transport group is bounded and therefore, Theorem D.14 applies. This is, to
some extent, the motivation for a weak spectral mapping theorem for the
linearized version of (168). It should be noted that the transport-reaction
groups generated in this case are almost never polynomially bounded. Depending
on the reactions, some components of the solution grow exponentially, either
for $t\rightarrow\infty$ or for $t\rightarrow-\infty.$

## References

  * [1] W. Arendt, A. Grabosch, G. Greiner, U. Moustakas, R. Nagel, U. Schlotterbeck, U. Groh, H. Lotz, and F. Neubrander.  One-parameter semigroups of positive operators, volume 1184.  Springer, 1986.
  * [2] W. Arendt and G. Greiner.  The spectral mapping theorem for one-parameter groups of positive operators on c0(x).  In Semigroup Forum, volume 30, pages 297–330. Springer New York, 1984.
  * [3] H. Brezis.  Functional analysis, sobolev spaces and partial differential equations.  Springer Science & Business Media, 2010.
  * [4] A. Bátkai, M. Fijavz, and A. Rhandi.  Positive Operator Semigroups, volume 257.  01 2017.
  * [5] Conifold.  Can matrix exponentials ever be negative?  https://math.stackexchange.com/questions/926943/can-matrix-exponentials-ever-be-negative-if-so-under-what-conditions, 2014\.  Accessed: 2021-04-05.
  * [6] N. Dunford and J. Schwartz.  Linear operators, part 1: general theory, volume 10.  John Wiley & Sons, 1988.
  * [7] R. Eftimie.  Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review.  Journal of mathematical biology, 65(1):35–75, 2012.
  * [8] K.-J. Engel and R. Nagel.  One-parameter semigroups for linear evolution equations.  In Semigroup forum, volume 63, pages 278–280. Springer, 2001.
  * [9] L. Evans.  Partial differential equations.  American Mathematical Society, Providence, R.I., 2010.
  * [10] A. Fick.  V. on liquid diffusion.  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 10(63):30–39, 1855.
  * [11] G. Folland.  Real analysis: modern techniques and their applications, volume 40.  John Wiley & Sons, 1999.
  * [12] F. Gantmakher.  The Theory of Matrices, Volume 2, volume 133.  American Mathematical Soc., 2000.
  * [13] A. Gierer and H. Meinhardt.  A theory of biological pattern formation.  Kybernetik, 12(1):30–39, 1972.
  * [14] D. Gillespie.  A rigorous derivation of the chemical master equation.  Physica A: Statistical Mechanics and its Applications, 188(1-3):404–425, 1992.
  * [15] L. Grafakos.  Classical fourier analysis, volume 2.  Springer, 2008.
  * [16] G. Greiner and M. Schwarz.  Weak spectral mapping theorems for functional differential equations.  Journal of Differential Equations, 94(2):205–216, 1991.
  * [17] K. Hadeler.  Reaction transport systems in biological modelling.  In Mathematics inspired by biology, pages 95–150. Springer, 1999\.
  * [18] T. Hillen.  A turing model with correlated random walk.  Journal of Mathematical Biology, 35(1):49–72, 1996.
  * [19] T. Hillen.  Hyperbolic models for chemosensitive movement.  Mathematical Models and Methods in Applied Sciences, 12(07):1007–1034, 2002.
  * [20] T. Hillen.  Existence theory for correlated random walks on bounded domains.  Canad. Appl. Math. Quart, 18(1):1–40, 2010.
  * [21] A. Holderrieth.  Matrix multiplication operators generating one parameter semigroups.  In Semigroup Forum, volume 42, pages 155–166. Springer, 1991.
  * [22] M. Kac.  A stochastic model related to the telegrapher’s equation.  The Rocky Mountain Journal of Mathematics, 4(3):497–509, 1974.
  * [23] T. Kato.  Perturbation theory for linear operators, volume 132.  Springer Science & Business Media, 2013.
  * [24] C. Kuttler.  Reaction-diffusion equations with applications.  In Internet seminar, 2011.
  * [25] K. Latrach, B. Lods, and M. Mokhtar-Kharroubi.  Weak spectral mapping theorems for c0-groups associated to transport equations in slab geometry.  Journal of mathematical analysis and applications, 342(2):1038–1051, 2008.
  * [26] P. Lax and L. Zalcman.  Complex proofs of real theorems, volume 58.  American Mathematical Soc., 2011.
  * [27] B. Lods and M. Sbihi.  Stability of the essential spectrum for 2d-transport models with maxwell boundary conditions.  Mathematical methods in the applied sciences, 29(5):499–523, 2006\.
  * [28] F. Lutscher.  Modeling alignment and movement of animals and cells.  Journal of mathematical biology, 45(3):234–260, 2002.
  * [29] F. Lutscher, A. Stevens, et al.  Emerging patterns in a hyperbolic model for locally interacting cell systems.  Journal of Nonlinear Science, 12(6):619–640, 2002.
  * [30] A. Pazy.  Semigroups of linear operators and applications to partial differential equations, volume 44.  Springer Science & Business Media, 2012.
  * [31] I. Primi, A. Stevens, and J. Velázquez.  Pattern forming instabilities driven by non-diffusive interaction.  Networks and Heterogeneous Media [electronic only], 1, 03 2013.
  * [32] J. Roe.  Elliptic operators, topology, and asymptotic methods.  CRC Press, 1999.
  * [33] H. Schaefer.  Banach lattices and positive operators.  Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2012.
  * [34] R. Schnaubelt.  Lecture notes on evolution equations.  Karlsruhe, April, 2011.
  * [35] R. Schnaubelt.  Lecture notes spectral theory.  Karlsruher Institut für Technologie, 33, 2012.
  * [36] J. Silvester.  Determinants of block matrices.  The Mathematical Gazette, 84(501):460–467, 2000.
  * [37] A. Stevens and J. Velazquez.  Partial differential equations and non-diffusive structures.  Nonlinearity, 21(12):T283, 2008.
  * [38] A. M. Turing.  The chemical basis of morphogenesis.  Bulletin of mathematical biology, 52(1-2):153–197, 1990.
  * [39] T. Woolley, R. Baker, and P. Maini.  Turing’s theory of morphogenesis: where we started, where we are and where we want to go.  In The Incomputable, pages 219–235. Springer, 2017.