# L-shadowing lemma for the Cauchy equation

K. Lee  Department of Mathematics, Chungnam National University, Daejeon,
34134, Republic of Korea<EMAIL_ADDRESS>and  C.A. Morales  Beijing Advanced
Innovation Center for Future Blockchain and Privacy Computing, Beihang
University, Beijing, China<EMAIL_ADDRESS>
###### Abstract.

We prove that if the Cauchy problem $\dot{u}=Au$ in a Banach space is
hyperbolic, then the problem has the L-shadowing property. Conversely, if the
space is finite-dimensional and the L-shadowing property is satisfied, then
the problem is hyperbolic. This generalizes a previous result by Ombach [24,
25] for linear homeomorphisms. Some short applications are given.

###### Key words and phrases:

Shadowing property, Semigroup of linear operators, Banach space

###### 2020 Mathematics Subject Classification:

Primary 54G99; Secondary 37B05

##  1\. Introduction

The concept of the shadowing property was originally introduced by Bowen in
the discrete case [6], and its extension to flows was studied by Franke and
Selgrade [12], as well as later by Thomas [31]. This property holds a
fundamental role within the dynamical literature, as demonstrated, for
instance, in [18]. A variation known as the L-shadowing property, initially
termed the ”S-limit shadowing property” by Lee and Sakai [21], was
subsequently independently rediscovered and named by Carvalho and Cordeiro
[7]. Research into the shadowing property of partial differential equations
(PDEs) has been undertaken. For instance, a study by Pilyugin [27]
demonstrated this property in relation to the time-$t$ mapping of the Chafee-
Infante equation $u_{t}=\Delta u+bu+f(u)$ on the open interval $(0,\pi)$. This
was observed under Dirichlet conditions in $H^{1}_{0}(0,\pi)$ within a
vicinity of the global attractor. However, as of now, no established outcomes
exist regarding the L-shadowing property within the realm of PDEs.

In this paper, we initiate the study of the L-shadowing property in PDEs
through the abstract Cauchy problem

(1.1) $\dot{u}=Au$

in a (complex) Banach space $X$. Specifically, we address the conditions under
which it exhibits the L-shadowing property. Our main results establish that if
the problem (1.1) is hyperbolic, it inherently satisfies the L-shadowing
property. Furthermore, in cases where the L-shadowing property is met and the
dimension of $X$ is finite, the problem is hyperbolic. These findings draw
inspiration from analogous results obtained by Ombach in the realm of linear
homeomorphisms [24, 25]. To provide practical context, we also offer a
selection of applications stemming from these results. Hereafter, we proceed
to elaborate on these contributions in detail.

A _semiflow_ of a metric space $X$ is a one-parameter family of continuous
maps $T=\\{T(t):X\to X\\}_{t\geq 0}$ that satisfies the following conditions:
$T(0)=I$ (the identity map on $X$), and $T(s+t)=T(s)\circ T(t)$ for all
$s,t\geq 0$. The individual map $T(t)$ for a given $t\geq 0$ is called the
_time $t$-map_.

The shadowing property for semiflows, which we will adopt here, is a natural
forward-in-time modification of the corresponding definition for flows
introduced by Komuro [19]. This definition is similar to the ones used by Grag
and Das [14], as well as by [15]. For the original definition of shadowing for
flows, refer to Franke, Selgrade and Thomas [12, 31].

Let $T$ be a semiflow on a metric space $X$. Given $\delta,R>0$ a sequence
$(x_{i},t_{i})_{i\geq 0}\subset X\times(0,\infty)$ is called
$(\delta,R)$-pseudo orbit of $T$ if

$t_{i}\geq R\quad\mbox{ and }\quad
d(T(t_{i})x_{i},x_{i+1})<\delta,\quad\forall i\geq 0.$

Define

$\hat{t}_{0}=0\quad\mbox{ and
}\quad\hat{t}_{i}=\sum_{j=0}^{i-1}t_{j},\quad\quad\mbox{ for }i>0$

We introduce the useful notation

$x_{0}*t=\phi_{t-\hat{t}_{i}}(x_{i}),\quad\quad\forall i\geq
0,\,\hat{t}_{i}\leq t<\hat{t}_{i+1}.$

######  Definition 1.

A semiflow $T$ has the shadowing property if for every $\epsilon>0$ there are
$\delta,R>0$ such that every $(\delta,R)$-pseudo orbit of $T$ there are $x\in
X$ and an increasing homeomorphism $h:[0,\infty)\to[0,\infty)$ such that

$d(T(h(t))x,x_{0}*t)<\epsilon\quad\quad\forall t\geq 0.$

The following definition is the forward-in-time variation of the L-shadowing
property proposed in [10, 21].

######  Definition 2.

A semiflow $T$ has the L-shadowing property if for every $\epsilon>0$ there
are $\delta,R>0$ such that every $(\delta,R)$-pseudo orbit
$(x_{i},t_{i})_{i\geq 0}$ of $T$ satisfying

$\lim_{i\to\infty}d(T(t_{i})x_{i},x_{i+1})=0,$

there are $x\in X$ and an increasing homeomorphism $h:[0,\infty)\to[0,\infty)$
such that

$d(T(h(t))x,x_{0}*t)<\epsilon\quad\quad(\forall t\geq 0)\quad\mbox{ and
}\quad\lim_{t\to\infty}d(T(t)x,x_{0}*t)=0.$

Now, we focus on the case when $X$ is a Banach space with the norm
$\|\cdot\|$. By a semigroup of linear operators of $X$ we mean a semiflow $T$
of $X$ such that $T(t)$ is a bounded linear operator of $X$, for all $t\geq
0$. We can associate a map $A:D(A)\to X$ with domain $D(A)$ defined by

$D(A)=\\{x\in X:\lim_{t\to 0}\frac{T(t)x-x}{t}\mbox{ exists}\\}\mbox{ and
}A(x)=\lim_{t\to 0}\frac{T(t)x-x}{t},\quad\forall x\in D(A).$

It is clearly linear but not necessarily bounded. It is often called
infinitesimal generator of $T$. We say that $T$ is a $C_{0}$-semigroup if

$\lim_{t\to 0}\|T(t)x-x\|=0,\quad\quad\forall x\in X.$

We will assume that the linear operator $A$ in the Cauchy problem (1.1) is the
infinitesimal generator of a $C_{0}$-semigroup of linear operators of $X$. The
standard conditions for this are (Theorem 5.3 in [26]): $A$ is closed, $D(A)$
is dense in $X$, and the resolvent set of $A$ contains the ray
$(\omega,\infty)$ for some $\omega>0$. Additionally, there exists $M>0$ such
that

$\|(\lambda
I-A)^{-n}\|\leq\frac{M}{(\lambda-\omega)^{n}},\quad\quad\forall\lambda\in(\omega,\infty),\,n\in\mathbb{N}.$

It should be noted that the L-shadowing property may hold or not for a given
$C_{0}$-semigroup. To illustrate this, consider the following two examples.

######  Example 1.

The _trivial semigroup_ (i.e., $T(t)=I$ for every $t\geq 0$) does not have the
L-shadowing property. This can be concluded from the arguments presented in
the proof of Theorem 2.3.2 in [1]. On the other hand, the semigroup $e^{tA}$
generated by a bounded linear operator $A:X\to X$, where the spectrum of $A$
belongs to $\\{z\in\mathbb{C}:\text{Im}(z)<0\\}$, does have the L-shadowing
property. This can be deduced from Lemma 1 below and Theorem 4.3 p. 118 in
[26] (compare with the shadowing property case in Remark 3.4 of [14] or [30]).

Based on this example we introduce the following auxiliary concept.

######  Definition 3.

The Cauchy problem (1.1) is said to have the L-shadowing property if $A$ is
the infinitesimal generator of a $C_{0}$-semigroup with the L-shadowing
property.

Similarly we define the shadowing property for the Cauchy problem. This
definition raises the question of whether spectral properties on $A$ can be
used to characterize when the corresponding Cauchy problem has the L-shadowing
properties. We provide a partial answer using the following concepts.

Recall that an operator $B:X\to X$ on a Banach space $X$ is hyperbolic if its
spectrum $\sigma(B)$ does not intersect the unit complex circle $S^{1}$.
Drawing upon references such as [2], [3], [11], or Definition 1 in [29], we
adopt the following definition.

######  Definition 4.

A semigroup of linear operators $T$ on a Banach space is hyperbolic if its
time-one map $T(1)$ is a hyperbolic operator. We define the Cauchy problem
(1.1) to be hyperbolic if $A$ generates a hyperbolic $C_{0}$-semigroup.

(Note the difference between this notion of hyperbolicity and that for
evolution equations in Section 5.3 in [26].)

We are now in position to state our result. It gives the desired necessary and
sufficient conditions for the Cauchy problem (1.1) to have both the shadowing
and L-shadowing properties.

######  Theorem.

If the Cauchy problem (1.1) in a Banach space $X$ is hyperbolic, then the
problem has the L-shadowing property. Conversely, if the problem has the
L-shadowing property and $dim(X)<\infty$, then the problem is hyperbolic.

A suitable remark is as follows.

######  Remark 1.

It is worth to mention that the analogous result obtained by replacing
L-shadowing by shadowing in the above statement holds too (see [13] for a
possible reference about this fact).

The proof of the theorem also implies that all hyperbolic $C_{0}$-semigroups
have the strong shadowing property (as defined in Komuro [19]). An interesting
problem that arises from our investigation is whether there exists a
relationship between the shadowing and L-shadowing properties in the context
of the Cauchy problem. Specifically, we anticipate that every Cauchy problem
with the shadowing property also possesses the L-shadowing property. This
expectation is motivated by the observation that this implication holds true
for linear homeomorphisms (see [20]).

This paper is divided as follows. In Section 2 we prove three lemmas. In
Section 3 we prove the theorem and in Section 4 we give some applications.

##  2\. Preliminary lemmas

In this section we prove three lemmas. To motivate the first, we say that a
semiflow $T$ of a metric space $X$ is uniformly contracting [14] if there is
$\lambda>0$ such that $d(T(t)x,T(t)y)\leq e^{-t\lambda}d(x,y)$, for all
$x,y\in X,\,t\geq 0.$ It was mentioned in Remark 3.4 of [14] that every
uniformly contracting semiflow of a metric space has the shadowing property.
The following lemma deals with a slight more general class of semiflows:

######  Lemma 1.

Let $T$ be a semiflow of a complete metric space $X$. Suppose that there are
$K,\lambda>0$ such that

$d(T(t)x,T(t)y)\leq Ke^{-\lambda t}d(x,y),\quad\quad\forall x,y\in X,t\geq 0.$

Then, for every $\epsilon>0$ there are $\delta,R>0$ such that for every
$(\delta,R)$-pseudo orbit $(x_{i},t_{i})_{i\geq 0}$ satisfying

$\lim_{i\to\infty}d(T(t_{i})x_{i},x_{i+1})=0$

there is $x\in X$ such that

$d(T(t)x,x_{0}*t)\leq\epsilon\quad(\forall t\geq 0)\quad\mbox{ and
}\quad\lim_{t\to\infty}d(T(t)x,x_{0}*t)=0.$

###### Proof.

We will adapt the proof of Proposition 1 in Ombach [24] to the semiflow case.
Let $K,\lambda>0$ be given by the fact that $T$ is eventually contractive. Let
$\epsilon>0$. Take $R>0$ such that $Ke^{-\lambda R}<1$. Define

$\delta=(1-Ke^{-\lambda R})\frac{\epsilon}{K}.$

Let $(x_{i},t_{i})_{i\geq 0}$ be a $(\delta,R)$-pseudo orbit namely

$t_{i}\geq R\quad\mbox{ and }\quad
d(T(t_{i})x_{i},x_{i+1})<\delta,\quad\forall i\geq 0.$

Define

$E=\\{y=(y_{i})_{i\geq 0}:y_{i}\in X\mbox{ and
}d(x_{i},y_{i})\leq\frac{\epsilon}{K},\,\forall i\geq 0\\}.$

Since $X$ is complete, $E$ also is if endowed with the supremum metric

$D(y,z)=\sup_{i\geq 0}d(y_{i},z_{i}).$

For any $y\in E$ we define the sequence $\Gamma(y)=(\Gamma(y)_{i})_{i\geq 0}$
by

$\Gamma(y)_{i}=\left\\{\begin{array}[]{rcl}x_{0},&\mbox{if}&i=0\\\
T(t_{i-1})y_{i-1},&\mbox{if}&i>0.\end{array}\right.$

For any $y\in E$ one has
$d(x_{0},\Gamma(y)_{0})=d(x_{0},x_{0})=0\leq\frac{\epsilon}{K}$ and for all
$i\geq 0$,

(2.1) $d(x_{i},\Gamma(y)_{i})=d(x_{i},T(t_{i-1})y_{i-1})\leq
d(x_{i},T(t_{i-1})x_{i-1})+d(T(t_{i-1})x_{i-1},T(t_{i-1})y_{i-1}).$

This implies

$d(x_{i},\Gamma(y)_{i})\leq\delta+Ke^{-\lambda t_{i-1}}d(y_{i-1},x_{i-1})\\\
\leq(1-Ke^{-\lambda R})\frac{\epsilon}{K}+Ke^{-\lambda R}\frac{\epsilon}{K}\\\
=\frac{\epsilon}{K},\quad\quad\forall i>0,$

thus $\Gamma(y)\in E$ for all $y\in E$. So, we obtain a map $\Gamma:E\to E$.

Since $Ke^{-\lambda R}<1$ and

$\displaystyle D(\Gamma(y),\Gamma(z))$ $\displaystyle=$
$\displaystyle\sup_{i\geq 0}d(\Gamma(y)_{i},\Gamma(z)_{i})$ $\displaystyle=$
$\displaystyle\sup_{i\geq 1}d(T(t_{i-1})y_{i-1},T(t_{i-1})z_{i-1})$
$\displaystyle\leq$ $\displaystyle Ke^{-\lambda t_{i-1}}\sup_{i\geq
1}d(y_{i-1},z_{i-1})$ $\displaystyle\leq$ $\displaystyle Ke^{-\lambda
R}D(y,z)$

for all $y,z\in E$, we have that $\Gamma:E\to E$ is a contraction. Since $E$
is a complete metric space, the Banach contracting principle implies that
$\Gamma$ has a fixed point i.e. there is $y\in\Gamma$ such that $\Gamma(y)=y$.
It follows that

(2.2) $y_{0}=x_{0},\quad d(y_{i},x_{i})\leq\frac{\epsilon}{K}\quad\mbox{ and
}\quad T(t_{i-1})y_{i-1}=y_{i},\quad\forall i\geq 1.$

The first and third of the above expressions imply

$y_{i}=T(\hat{t}_{i})x_{0},\quad\quad\forall i\geq 0.$

By replacing in the second we obtain

$d(T(\hat{t}_{i})x_{0},x_{i})\leq\frac{\epsilon}{K},\quad\quad\forall i\geq
1.$

Then,

(2.3) $d(T(t)x_{0},T(t-\hat{t}_{i})x_{i})\leq
Ke^{-\lambda(t-t_{i})}d(T(\hat{t}_{i})x_{0},x_{i})\leq
K\frac{\epsilon}{K}=\epsilon$

for all $i\geq 0$ and $\hat{t}_{i}\leq t<\hat{t}_{i+1}$ so

$d(T(t)x_{0},x_{0}*t)\leq\epsilon,\quad\quad\forall t\geq 0.$

Define

$E_{0}=\\{y\in E:\lim_{i\to\infty}d(y_{i},x_{i})=0\\}.$

It follows that $E_{0}$ is a closed subspace of $E$. Applying (2.1) we obtain

$\lim_{i\to\infty}d(x_{i},\Gamma(y)_{i})\leq\lim_{i\to\infty}[d(x_{i},T(t_{i-1})x_{i-1})+d(x_{i-1},y_{i-1})]=0,\quad\forall
y\in E_{0},\,i\geq 0.$

It follows that $\Gamma(E_{0})\subset E_{0}$. Since $E_{0}$ is closed, the
fixed point $y$ in (2.2) belongs to $E_{0}$. Then, (2.3) implies

$\lim_{t\to\infty}d(T(t)x_{0},x_{0}*t)=0$

completing the proof. ∎

The second lemma is inspired by Proposition 3 in Ombach [24].

######  Lemma 2.

Let $T$ be a semigroup of a Banach space $X$ such that $T(t)$ is invertible
for all $t\geq 0$. Suppose that there are $\lambda,K>0$ such that

$\|(T(t))^{-1}\|\leq Ke^{-t\lambda},\quad\quad\forall t\geq 0.$

Then, for every $\epsilon>0$ there is $\delta>0$ such that for every
$(\delta,1)$-pseudo orbit $(x_{i},t_{i})_{i\geq 0}$ satisfying

$\lim_{i\to\infty}\|T(t_{i})x_{i}-x_{i+1}\|=0,$

there is $x\in X$ such that

$\|T(t)x-x_{0}*t\|\leq\epsilon\quad(\forall t\geq 0)\quad\mbox{ and
}\quad\lim_{t\to\infty}\|T(t)x-x_{0}*t\|=0.$

###### Proof.

Fix $\epsilon>0$. Take $\delta>0$ such that

(2.4)
$K\cdot\left(\sum_{i=0}^{\infty}(e^{-\lambda})^{k}\right)\cdot\delta<\epsilon.$

Let $(x_{i},t_{i})_{i\geq 0}$ be a $(\delta,1)$-pseudo orbit namely

$t_{i}\geq 1\quad\mbox{ and
}\quad\|T(t_{i})x_{i}-x_{i+1}\|\leq\delta,\quad\forall i\geq 0.$

It follows that

(2.5) $x_{i+1}=T(t_{i})x_{i}+h_{i},\quad\quad\forall i\geq 0,$

for some sequence $(h_{i})_{i\geq 0}$ satisfying $\|h_{i}\|\leq\delta$ for all
$i\geq 0$. By induction,

$x_{i}=\left(\prod_{j=0}^{i-1}T(t_{j})\right)x_{0}+\sum_{k=1}^{i}\left(\prod_{j=k}^{i-1}T(t_{j})\right)h_{k-1},\quad\quad\forall
i\geq 0$

(here we convey that $\sum_{a}^{b}=I$ when $a>b$). Then,

$x_{i}=T(\hat{t}_{i})[x_{0}+\sum_{k=1}^{i}(T(\hat{t}_{k})^{-1})h_{k-1}],\quad\quad\forall
i\geq 0.$

Since

$\|(T(\hat{t}_{k})^{-1})h_{k-1}\|\leq\|(T(\hat{t}_{k}))^{-1}\|\|h_{k-1}\|\leq
Ke^{-\lambda\hat{t}_{k}}\delta=Ke^{-\lambda\sum_{j=0}^{k-1}t_{j}}\delta\leq
K(e^{-\lambda})^{k}\delta$

and $e^{-\lambda}<1$, the series

$x=x_{0}+\sum_{k=1}^{\infty}(T(\hat{t}_{k})^{-1})h_{k-1}$

is convergent. Now, fix $i\geq 0$ and $\hat{t}_{i}\leq t<\hat{t}_{i+1}$. Then,

$\displaystyle T(t)x-T(t-\hat{t}_{i})x_{i}$ $\displaystyle=$ $\displaystyle
T(t)x_{0}+T(t)\sum_{k=1}^{\infty}(T(\hat{t}_{k})^{-1})h_{k-1}-T(t)\left[x_{0}+\sum_{k=1}^{i}(T(\hat{t}_{k})^{-1})h_{k-1}\right]$
$\displaystyle=$ $\displaystyle
T(t)\sum_{k=i+1}^{\infty}(T(\hat{t}_{k})^{-1})h_{k-1}$ $\displaystyle=$
$\displaystyle\sum_{k=i+1}^{\infty}(T(\hat{t}_{k}-t))^{-1}h_{k-1}$

so

(2.6)
$\|T(t)x-T(t-\hat{t}_{i})x_{i}\|\leq\left(\sum_{k=i+1}^{\infty}\|(T(\hat{t}_{k}-t))^{-1}\|\right)\cdot\sup_{k\geq
i}\|h_{k}\|$

Since

$\displaystyle\sum_{k=i+1}^{\infty}\|(T(\hat{t}_{k}-t)^{-1}\|$
$\displaystyle\leq$
$\displaystyle\sum_{k=i+1}^{\infty}Ke^{-\lambda(\hat{t}_{k}-t)}$
$\displaystyle<$ $\displaystyle
K\sum_{k=i+1}^{\infty}e^{-\lambda(\hat{t}_{k}-\hat{t}_{i+1})}$
$\displaystyle\leq$ $\displaystyle K\sum_{k=i+1}^{\infty}e^{-\lambda(k-i-1)}$
$\displaystyle=$ $\displaystyle K\sum_{k=0}^{\infty}(e^{-\lambda})^{k},$

(2.6) implies

$\|T(t)x-T(t-\hat{t}_{i})x_{i}\|\leq
K\cdot\left(\sum_{k=0}^{\infty}(e^{-\lambda})^{k}\right)\cdot\delta\overset{\eqref{coker}}{<}\epsilon.$

This proves

$\|T(t)x-x_{0}*t\|<\epsilon,\quad\quad\forall t\geq 0.$

Since

$\lim_{i\to\infty}\|T(t_{i})x_{i}-x_{i+1}\|=0,$

the sequence $(h_{i})_{i\geq 0}$ in (2.5) satisfies

$\lim_{i\to\infty}h_{i}=0.$

Fix $\Delta>0$ and $i_{\Delta}>0$ large so that

$K\cdot\left(\sum_{k=0}^{\infty}(e^{-\lambda)})^{k}\right)\cdot\sup_{k\geq
i}\|h_{k}\|<\Delta,\quad\quad\forall i\geq i_{\Delta}.$

Then, (2.6) implies $\|T(t)x-T(t-\hat{t}_{i})x_{i}\|<\Delta$ for all $i\geq
i_{\Delta}$ and $\hat{t}_{i}\leq t<\hat{t}_{i+1}$ hence

$\lim_{t\to\infty}\|T(t)x-x_{0}*t\|=0$

completing the proof. ∎

The third lemma is a direct consequence of the definition. Its proof is
included for completeness.

######  Lemma 3.

Let $T$ be a semigroup of linear operators of a Banach space $X$. If $T$ has
the L-shadowing property, and there is a direct sum $X=M\oplus N$ by closed
subspaces $M,N$ such that $T(t)M\subset M$ and $T(t)N\subset N$ for all $t\geq
0$, then the restricted semigroups $T|_{M}$ and $T|_{N}$ have the L-shadowing
property.

###### Proof.

Denote by $T^{M}=T|_{M}$ and $T^{N}|_{N}$ the restrictions of $T$ to $M$ and
$N$ respectively. We only need to prove that $T^{M}$ has the shadowing
property. By passing to projections if necessary we can assume

(2.7) $\|x\|=\max\\{\|x^{M}\|,\|x^{N}\|\\}\quad\mbox{ whenever }\quad
x=x^{M}\oplus x^{N}\in M\oplus N.$

Let $\epsilon>0$ and $\delta,R>0$ be given by the L-shadowing property of $T$.
Let $(x^{M}_{i},t_{i})_{i\geq 0}$ be a $(\delta,R)$-pseudo orbit of $T^{M}$
satisfying $\lim_{i\to\infty}\|T^{M}(t_{i})x_{i}-x_{i+1}\|=0$. It follows from
(2.7) that it is also a $(\delta,R)$-pseudo orbit of $T$ satisfying
$\lim_{i\to\infty}\|T(t_{i})x_{i}-x_{i+1}\|=0$. Then, there are $x\in X$ and
an increasing homeomorphism $h:[0,\infty)\to[0,\infty)$ such that

(2.8) $\|T(h(t))x-x^{M}_{0}*t\|\leq\epsilon\quad\quad(\forall t\geq
0)\quad\mbox{ and }\quad\lim_{t\to\infty}\|T(h(t))x-x^{M}_{0}*t\|=0.$

Writing $x=x^{M}+x^{N}$ one has

$\displaystyle\|T^{M}(h(t))x^{M}-T^{M}(t-\hat{t}_{i})x^{M}_{i}\|$
$\displaystyle\leq$
$\displaystyle\max\\{\|T^{M}(h(t))x^{M}-T^{M}(t-\hat{t}_{i})x^{M}_{i}\|,$
$\displaystyle\|T^{N}(h(t))x^{N}\|\\}$
$\displaystyle\overset{\eqref{anis}}{=}$
$\displaystyle\|T^{M}(h(t))x^{M}-T^{M}(t-\hat{t}_{i})x^{M}_{i}+T^{N}(h(t))x^{N}\|$

for all $i\geq 0$ and $\hat{t}_{i}\leq t<\hat{t}_{i+1}$ proving

$\|T^{M}(h(t))x^{M}-x^{M}_{0}*t\|\leq\|T(h(t))x-x^{M}_{0}*t\|,\quad\quad\forall
t\geq 0.$

Then, (2.8) implies

$\|T^{M}(h(t))x^{M}-x^{M}_{0}*t\|\leq\epsilon\quad\quad(\forall t\geq
0)\quad\mbox{ and }\quad\lim_{t\to\infty}\|T^{M}(h(t))x^{M}-x^{M}_{0}*t\|=0$

completing the proof. ∎

##  3\. Proof of the theorem

First we assume that the Cauchy problem (1.1) is hyperbolic. That is, $A$
generates a $C_{0}$-semigroup $T$ on $X$ with $\sigma(T(1))\cap
S^{1}=\emptyset$. Then, by Proposition 1.15 p. 305 in [11], there are a direct
sum $X=M\oplus N$ by closed subspaces $M,N$ and numbers
$\lambda_{M},\lambda_{N},K_{M},K_{N}>0$ such that $T(t)M=M$, $T(t)N=N$,
$T(t)|_{N}:N\to N$ is invertible (for all $t\geq 0$),

(3.1) $\|T(t)|_{M}\|\leq K_{M}e^{-t\lambda_{M}}\quad\mbox{ and
}\quad\|(T(t)|_{N})^{-1}\|\leq K_{N}e^{-t\lambda_{N}}\quad\quad\forall t\geq
0.$

Fix $\epsilon>0$. By (3.1) we can apply lemmas 2 and 3 to get
$\delta_{M},R_{M}>0$ and $\delta_{N}>0$ satisfying the conclusion of these
lemmas for the restrictions $T^{M}=T|_{M}$ and $T^{N}=T|_{N}$ respectively.
Take

$\delta=\min\\{\delta_{M},\delta_{N}\\}\quad\mbox{ and }\quad
R=\max\\{R_{M},1\\}.$

Let $(x_{i},t_{i})_{i\geq 0}$ be a $(\delta,R)$-pseudo orbit of $T$ satisfying

(3.2) $\lim_{i\to\infty}\|T(t_{i})x_{i}-x_{i+1}\|=0,$

For all $i\geq 0$ one has $x_{i}=x^{M}_{i}+x^{N}_{i}\in M\oplus N$. By (2.7)
one has

$\max\\{\|T^{M}(t_{i})x^{M}_{i}-x^{M}_{i+1}\|,\|T^{N}(t_{i})x^{N}_{i}-x^{N}_{i+1}\|\\}=\|T(t_{i})x_{i}-x_{i+1}\|\leq\delta,\quad\quad\forall
i\geq 0.$

So, $(x^{M}_{i},t_{i})_{i\geq 0}$ is a $(\delta_{M},R_{M})$-pseudo orbit of
$T^{M}$ and $(x^{N}_{i},t_{i})_{i\geq 0}$ is a $(\delta_{N},1)$-pseudo orbit
of $T^{N}$. Thus, (3.2) together with lemmas 2 and 3 provide $x^{M}\in M$ and
$x^{N}\in N$ such that

$\|T^{*}(t)x^{*}-T^{*}(t-\hat{t}_{i})x^{*}_{i}\|\leq\epsilon\quad\quad\forall*\in\\{M,N\\},\,i\geq
0,\,\hat{t}_{i}\leq t<\hat{t}_{i+1},$

and

$\lim_{i\to\infty}\|T^{*}(t_{i})x^{*}_{i}-x^{*}_{i+1}\|=0,\quad\quad\forall*\in\\{M,N\\}.$

Therefore, $x=x^{M}+x^{N}\in X$ and $h:[0,\infty)\to[0,\infty)$ defined by
$h(t)=t$ for $t\geq 0$ satisfy

$\displaystyle\|T(h(t))x-T(t-\hat{t}_{i})x_{i}\|$ $\displaystyle=$
$\displaystyle\max\\{\|T^{M}(t)x^{M}-T^{M}(t-\hat{t}_{i})x^{M}_{i}\|,$
$\displaystyle\|T^{N}(t)x^{N}-T^{N}(t-\hat{t}_{i})x^{N}\|\\}$
$\displaystyle\leq$ $\displaystyle\epsilon,\quad\forall i\geq
0,\,\hat{t}_{i}\leq t<\hat{t}_{i+1},$

proving $\|T(t)x-x_{0}*t\|\leq\epsilon$ for all $t\geq 0$ and

$\lim_{t\to\infty}\|T^{*}(t)x^{*}-x_{0}^{*}*t\|=0,\quad\quad\forall*\in\\{M,N\\}$

so

$\lim_{t\to\infty}\|T(h(t))x-x_{0}*t\|=\lim_{t\to\infty}\max\\{\|T^{M}(t)x^{M}-x^{M}_{0}*t\|,\|T^{N}(t)x^{N}-x_{0}^{N}*t\|\\}=0$

proving that (1.1) has the L-shadowing property.

Now, we suppose $dim(X)<\infty$ and that (1.1) has the L-shadowing property.
In such a case, the semigroup $T$ is uniformly continuous i.e.

$\lim_{t\to 0^{+}}\|T(t)-I\|=0$

and so $T(t)=e^{tA}$ for all $t\geq 0$ (see [11]).

Suppose by contradiction that (1.1) is not hyperbolic. Then, $\sigma(T(1))\cap
S^{1}\neq\emptyset$ namely there are $x_{0}\in X$ with $\|x_{0}\|=1$ and
$\lambda\in S^{1}$ such that $Ax_{0}=\lambda x_{0}$. Let $M$ be the subspace
of $X$ generated by $x_{0}$. It follows easily that $Ax=\lambda x$ for all
$x\in M$. Since $T(t)=e^{tA}$ one has that

(3.3) $T(t)x=e^{t\theta i}x,\quad\quad\forall t\geq 0,x\in M$

where $\theta=Im(\lambda)$ and $i=\sqrt{-1}$.

Note that this semiflow (in fact, a flow) does not have the L-shadowing
property. This follows from the fact that it is chain transitive (i.e., we can
go from one point to another through pseudo orbits) but not transitive (since
the orbits are confined to circles). However, since $dim(X)<\infty$, we can
apply the Jordan form to prove that $M$ has a complementary subspace $N$ (i.e.
$X=M\oplus N$) such that $A(N)\subset N$. This implies $T(t)M\subset M$ and
$T(t)N\subset N$ for all $t\geq 0$. Then, $T|_{M}$ has the L-shadowing
property by Lemma 3 which is absurd since (3.3) implies that $T|_{M}$ is the
semiflow $e^{t\theta i}$. This completes the proof of the theorem.

##  4\. Applications

Let us give two short applications of the theorem. The first is a spectral
sufficient condition for the Cauchy problem on Hilbert spaces to have the
shadowing properties under consideration.

######  Corollary.

If the Cauchy problem (1.1) in an Hilbert space satisfies

$\sigma(A)\cap(i\mathbb{R})=\emptyset\quad\mbox{ and }\quad\sup_{\lambda\in
i\mathbb{R}}\|(\lambda I-A)^{-1}\|<\infty,$

then the problem has the L-shadowing property.

###### Proof.

The proof follows from theorem and Theorem 1.18 p. 307 in [11]. ∎

We can also obtain some dynamical conclusion from the shadowing properties.
More precisely, we say that $x\in X$ is a nonwandering point of the Cauchy
problem (1.1) if for all neighborhood $U$ of $x$ and $R>0$ there are $y\in U$
and $t\geq R$ such that $T(t)y\in U$, where $T$ is the semigroup generated by
$A$.

######  Proposition 1.

If the Cauchy problem given by equation (1.1) is hyperbolic, then $0$ is the
only nonwandering point of the problem.

###### Proof.

It is clear that $0$ is nonwandering so we just need to prove that any
nonwandering point $x$ (say) vanishes. For this, let us take $\epsilon>0$. On
the other hand, it follows from the proof of the theorem that the problem has
the shadowing property (see Remark 1). So, let $\delta,R>0$ with
$0<\delta<\epsilon$ be given by this property for $\frac{\epsilon}{2}$ and the
semigroup $T$. Since $x$ is nonwandering, there are $x_{0}\in X$ with
$\|x_{0}-x\|<\frac{\delta}{2}$ and $t_{*}\geq R$ such that
$\|T(t_{*})x_{0}-x\|<\frac{\delta}{2}$. Since

$\|T(t_{*})x_{0}-x_{0}\|\leq\|T(t_{*})x_{0}-x\|+\|x-x_{0}\|<\frac{\delta}{2}+\frac{\delta}{2}=\delta,$

the sequence $(x_{i},t_{i})_{i\geq 0}$ defined by
$(x_{i},t_{i})=(x_{0},t_{*})$ for all $i\geq 0$ is a $(\delta,R)$-pseudo
orbit. Then, there are $y\in X$ and an increasing homeomorphism
$h:[0,\infty)\to[0,\infty)$ such that

(4.1) $\|T(h(t))y-x_{0}*t\|\leq\frac{\epsilon}{2},\quad\quad\forall t\geq 0.$

But $\hat{t}_{i}=it_{*}$ so $x_{0}*t=T(t-it_{*})x_{0}\in\\{T(s)x_{0}:0\leq
s\leq t_{*}\\}$ which is bounded thus there is $0<K<\infty$ such that
$\|T(t)y\|<K$ for all $t\geq 0$.

By Proposition 1.15 p. 305 in [11], there are a direct sum $X=M\oplus N$ by
closed subspaces $M,N$ and numbers $\lambda_{M},\lambda_{N},K_{M},K_{N}>0$
such that $T(t)M=M$, $T(t)N=N$, $T(t)|_{N}:N\to N$ is invertible,

$\|T(t)|_{M}\|\leq K_{M}e^{-t\lambda_{M}}\quad\mbox{ and
}\quad\|(T(t)|_{N})^{-1}\|\leq K_{N}e^{-t\lambda_{N}}\quad\quad\forall t\geq
0.$

Write $y=y^{M}+y^{N}\in M\oplus N$. It follows that $\\{T(t)y^{N}:t\geq 0\\}$
is bounded. Since $\|T(t)y^{N}\|\geq K^{-1}_{N}e^{t\lambda_{N}}\|y^{N}\|$ for
all $t\geq 0$ we conclude that $y^{N}=0$. It follows that $y=y^{M}$ so

$\lim_{t\to\infty}T(t)y=0.$

But since $x_{0}*\hat{t}_{i}=x_{0}$ for all $i\geq 0$, (4.1) implies
$\|x_{0}\|\leq\frac{\epsilon}{2}$. Then,
$\|x\|\leq\|x-x_{0}\|+\|x_{0}\|\leq\frac{\delta}{2}+\frac{\epsilon}{2}<\epsilon$.
Letting $\epsilon\to 0$ we get $x=0$ completing the proof. ∎

Now, consider again the Cauchy problem (1.1). Given $\delta,R>0$ we call a
sequence $(x_{i},t_{0})_{i=0}^{k}$ $(\delta,R)$-chain for the problem if
$t_{i}\geq R$ and $\|T(t_{i})x_{i}-x_{i+1}\|<\delta$ ($\forall i\geq 0$) where
$T$ is the semigroup generated by $A$. In such a case, we say that the chain
is from $x_{0}$ to $x_{k}$. Given $x,y\in X$, we say that $x$ is
$(\delta,R)$-related to $y$ if there are $(\delta,R)$-chains from $x$ to $y$
and from $y$ back to $x$. We write $x\sim y$ if $x$ is $(\delta,R)$-related to
$y$ for all $\delta,R>0$. The chain recurrent set is the set of $x\in X$ such
that $x\sim x$.

Based on the previous proposition we obtain the following result.

######  Proposition 2.

If the Cauchy problem given by equation (1.1) is hyperbolic, then $0$ is the
only chain recurrent point of the problem.

###### Proof.

The problem is hyperbolic so $0$ is the only nonwandering point by Proposition
1. Then, the result follows since the problem has the L-shadowing property (by
the theorem) and this property easily implies that every chain recurrent point
is nonwandering. ∎

In particular, the hyperbolic Cauchy problems (1.1) are excluded from the
hypercyclic theory of semigroups [23].

We can also apply our theorem to some PDEs. An example is the heat equation in
a bounded domain $\Omega\subset\mathbb{R}^{N}$ with boundary $\partial\Omega$,

(4.2) $\left\\{\begin{array}[]{rll}u_{t}&=&\Delta u,\hbox{ in
}\Omega\times(0,\infty)\\\ u&=&0,\,\,\,\,\hbox{ on
}\partial\Omega\times(0,\infty)\\\ u(\cdot,0)&=&u_{0},\,\,\hbox{ in
}\Omega.\end{array}\right.$

Transform this equation to the Cauchy problem (1.1) with $X=L^{2}(\Omega)$,
and $A:D(A)\to X$ defined by

$D(A)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)\quad\mbox{ and }\quad Au=\Delta
u,\quad\forall u\in D(A).$

See p. 327 in [5]. It follows that $A$ is self-adjoint (hence normal), and
$L^{2}(\Omega)$ is Hilbert so $\sigma(e^{tA})$ is the closure of
$\\{e^{t\lambda}:\lambda\in\sigma(A)\\}$ for all $t>0$ [28] (here $e^{tA}$
denotes the $C_{0}$-semigroup generated by $A$). However, $\sigma(A)$ consists
of a sequence
$0<\lambda_{1}(\Omega)\leq\lambda_{2}(\Omega)\leq\lambda_{k}(\Omega)\leq\cdots\to\infty$
(c.f. Theorem 1.2.2 in [16]), so $\sigma(e^{tA})\cap S^{1}=\emptyset$ for
large $t>0$, thus $e^{tA}$ is hyperbolic for some (and then for all) $t>0$.
Therefore, the Cauchy problem is hyperbolic, and it has both the shadowing and
L-shadowing properties according to the theorem. As a result, we conclude that
the heat equation (4.2) also possesses the L-shadowing property. It is
possible that this result follows from different arguments (e.g. [17]).

Another equation is

(4.3) $\left\\{\begin{array}[]{rll}u_{t}&=&u_{x}-\theta u,\hbox{ in
}\mathbb{R}\times(0,\infty)\\\ u(\cdot,0)&=&u_{0},\quad\quad\,\,\hbox{ in
}\mathbb{R}\end{array}\right.$

with $\theta\neq 0$. Transform this equation into the Cauchy problem (1.1)
with $X=C_{b}(\mathbb{R})$ (the set of bounded uniformly continuous maps
$u:\mathbb{R}\to\mathbb{R}$ endowed with the supremum norm) and $A:D(A)\to X$
defined by

$D(A)=C^{1}_{b}(\mathbb{R})\quad\mbox{ and }\quad A(u)=u^{\prime}-\theta
u,\quad\forall u\in D(A),$

where $C^{1}_{b}(\mathbb{R})$ is the subspace of $C_{b}(\mathbb{R})$
consisting of all $u$ such that the classical derivative

$u^{\prime}(x)=\lim_{y\to x}\frac{u(y)-u(x)}{y-x}$

exists for all $x\in\mathbb{R}$ and belongs to $C_{b}(\mathbb{R})$. Since $A$
generates the $C_{0}$-semigroup $T$ of $X$ defined by

$(T(t)u)x=e^{-t\theta}u(x+t)\quad\quad(\forall x,t\geq 0)$

which is clearly hyperbolic, one has that the equation has the L-shadowing
property by the theorem.

An intriguing question arises from our theorem: Specifically, we wonder
whether every Cauchy problem with the shadowing property is hyperbolic.
However, our findings suggest that the answer to this question is negative. In
fact, this is proven to be false in the case of invertible linear operators on
Banach spaces. These operators can possess the shadowing property without
being hyperbolic. Recently, a conjecture has emerged, stating that an
invertible linear operator exhibits the shadowing property if and only if it
is generalized hyperbolic [9]. Consequently, it becomes imperative to restate
this conjecture within the framework of semigroups of linear operators.

First of all, we say that a linear operator of a Banach space $B:X\to X$ is
generalized hyperbolic if there are a direct sum $X=M\oplus N$ by closed
subspaces $M,N$ and $K,\lambda>0$ such that $B(M)\subset M$, $N\subset B(N)$,

$\|B^{n}x\|\leq Ke^{-n\lambda}\quad(\forall x\in M,\,n\geq 0)\quad\mbox{ and
}\quad\|B^{n}x\|\geq K^{-1}e^{n\lambda}\quad(\forall x\in N,\,n\geq 0).$

This definition clearly reduces to the original one [8] when $B$ is
invertible. As in [4] it can be proved that every generalized hyperbolic
operator $B$ as above has the (positive) shadowing property.

Based on this definition and the description of hyperbolic semigroups in p.
305 of [11] we propose the following definition.

######  Definition 5.

We say that a semigroup of linear operators $T$ on a Banach space $X$ is
generalized hyperbolic if there are a direct sum $X=M\oplus N$ by closed
subspace $M,N$ and $K,\lambda>0$ such that $T(t)M\subset M$ and $N\subset
T(t)N$ ($\forall t\geq 0$),

$\|T(t)x\|\leq Ke^{-t\lambda}\|x\|\quad(\forall x\in M,t\geq 0)\mbox{ and
}\|T(t)x\|\geq K^{-1}e^{t\lambda}\|x\|\quad(\forall x\in N,t\geq 0).$

We say that the Cauchy problem (1.1) is generalized hyperbolic if $A$
generates a generalized hyperbolic $C_{0}$-semigroup.

If the Cauchy problem (1.1) is hyperbolic, then it is generalized hyperbolic.
The converse is false by the following proposition.

######  Proposition 3.

There is a Cauchy problem (1.1) which is generalized hyperbolic but not
hyperbolic.

###### Proof.

Define $w:\mathbb{R}\to\mathbb{R}$ by $w(x)=e^{|x|}$ for all $x\in\mathbb{R}$.
Consider the Cauchy problem (1.1) in $X=L^{2}(\mathbb{R})$ with $A:D(A)\to
L^{2}(\mathbb{R})$ defined by

$D(A)=H^{1}(\mathbb{R}):=\\{u\in L^{2}(\mathbb{R}):u^{\prime}\in
L^{2}(\mathbb{R})\\}\quad\mbox{ and }\quad
A(u)=u^{\prime}-\frac{w^{\prime}}{w}u,\quad\forall u\in D(A)$

(derivatives in the weak sense).

Note that $A$ generates the $C_{0}$-semigroup

$(T(t)u)(x)=\frac{w(x)}{w(x+t)}u(x+t),\quad\quad\forall u\in
L^{2}(\mathbb{R}),\,x\in\mathbb{R},\,t\geq 0.$

To see this fix $u\in H^{1}(\mathbb{R})$. Given $\epsilon>0$ one has $1\leq
e^{t}<1+\epsilon$ for $t\geq 0$ small so straightforward computations show

$\displaystyle\left\|\frac{T(t)u-u}{t}-u^{\prime}+\frac{w^{\prime}}{w}u\right\|^{2}_{L^{2}}$
$\displaystyle\leq$ $\displaystyle
e^{t}\int\left|\frac{u(x+t)-u(x)}{t}-u^{\prime}(x)\right|^{2}dx+\epsilon^{2}\|u^{\prime}\|^{2}_{L^{2}}+$
$\displaystyle\|u\|^{2}_{L^{2}}\cdot\int\left|\frac{w(x+t)-w(x)}{t}-w^{\prime}(x)\right|^{2}dx+$
$\displaystyle(e^{t}-1)\|u\|_{L^{2}}^{2}.$

Then, the assertion follows by letting $\epsilon,t\to 0$ since the $L^{2}$ and
weak derivatives coincide (e.g. Exercise 1.9 p. 21 in [22]).

Now, define

$M=\\{u\in L^{2}(\mathbb{R}):u=0\mbox{ a.e. in }[0,\infty)\\}\mbox{ and
}N=\\{u\in L^{2}(\mathbb{R}):u=0\mbox{ a.e. in }(-\infty,0]\\}.$

Then, $M$ and $N$ are closed subspaces with $L^{2}(\mathbb{R})=M\oplus N$.
Clearly $T(t)M\subset M$ for all $t\geq 0$. On the other hand, each $T(t)$ is
invertible with inverse $T(t)^{-1}:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$
given by

$(T(t)^{-1}u)(x)=\frac{w(x)}{w(x-t)}u(x-t),\quad\quad\forall x\in\mathbb{R}.$

We can easily check $T(t)^{-1}N\subset N$ hence $N\subset T(t)N$ for all
$t\geq 0$. Since for all $u\in M$ and $t\geq 0$

$\|T(t)u\|^{2}_{L^{2}}=\int_{-\infty}^{\infty}\left(\frac{w(x)}{w(x+t)}\right)^{2}|u|^{2}dx=\int_{0}^{\infty}e^{-2t}|u|^{2}dx=e^{-2t}\|u\|^{2}_{L^{2}},$

one has $\|T(t)u\|_{L^{2}}=e^{-t}\|u\|_{L^{2}}$ for all $u\in M$ and $t\geq
0$. Likewise, $\|T(t)^{-1}u\|_{L^{2}}=e^{-t}\|u\|_{L^{2}}$ for all $u\in N$
and $\forall t\geq 0$. We then conclude that $T$ (and therefore its
corresponding Cauchy problem) are generalized hyperbolic.

To finish, let us prove that the problem is not hyperbolic. Indeed, take any
continuous function $u:\mathbb{R}\to\mathbb{R}$ with $u\neq 0$ whose support
$supp(u)=\\{x\in\mathbb{R}:u(x)\neq 0\\}$ is contained in a compact interval
of $(0,\infty)$. Clearly $u\in N$ so $T(t)^{-1}u\to 0$ in $L^{2}(\mathbb{R})$
as $t\to\infty$. It follows also that $\\{x\in\mathbb{R}:u(x+t)\neq
0\\}=supp(u)-t$ for all $t\geq 0$ so $T(t)u\in M$ for $t>0$ large. Then,
$T(t)u\to 0$ in $L^{2}(\mathbb{R})$ as $t\to\infty$. This implies that $u$ is
recurrent, and so, the problem cannot be hyperbolic by Proposition 2. This
completes the proof. ∎

Just as every generalized hyperbolic invertible operator in a Banach space
possesses the shadowing and L-shadowing properties [4, 20], we believe that
the same holds true for every generalized hyperbolic Cauchy problem. If this
belief is valid, it implies that the proposition mentioned earlier offers an
example of a nonhyperbolic Cauchy problem that exhibits both shadowing
properties.

## Acknowledgements

We thank the Sejong Institute for Mathematical Sciences (SIMS), Sejong, South
Korea, for its warm hospitality during the preparation of this work.

## Funding

Partially supported by the Basic Science Research Program through the NRF
funded by the Ministry of Education of the Republic of Korea (Grant Number:
2022R1l1A3053628).

## Declaration of competing interest

There is no competing interest.

## Data availability

No data was used for the research described in the article.

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situation presented to us. In this view, what society has taught us does not
matter because we can do whatever it is we want. For example, some people
would go to the store and steal a bag of chips. Whether because they want to,
or can’t afford it. They know it is wrong. It is widely accepted by society
that stealing is wrong. And yet some people exercise their free will to act
off of their atitudes of the immediate situation: they are hungry, they steal
the chips, their hunger is immedialy satisfied. In their view, the benefit of
stealing the chips was greater than the cost, therefore prompting their
attitudes to be shaped around their immediate situation rather than the views
of society. But they had the develop this view from somewhere, probably a
lesson from society (although a more negative lesson). Therefor, regardless of
a positive or negative attitude, our attitudes are developed based off of
society as opposed to our immediate situation because human thought is
developed from somewhere, most likely society.

Table 3. An example draft prototype generated by plain text editor (i.e.,
baseline) from a user study session Topic1 [Scientific research funding]:
Governments should not fund any scientific research whose consequences are
unclear.
Write a response in which you discuss your views on the policy and explain
your reasoning for the position you take. In developing and supporting your
position, you should consider the possible consequences of implementing the
policy and explain how these consequences shape your position.
---
Governments should not fund any scientific research whose consequences are
unclear.From an ethical standpoint, it is imperative that governments exercise
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potential risks to human life and the environment cannot be
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Node” is an important consideration when it comes to funding scientific
research with uncertain consequences. The development of new technologies and
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but they can also pose potential risks to human life and the environment.
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life, violate human rights, or negatively impact the environment. Governments
must, therefore, ensure that the research is conducted with the utmost care
and responsibility to prevent any adverse effects. In conclusion, ethical
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argument that governments must exercise caution when funding scientific
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should prioritize funding research that has clear and well-understood
consequences, rather than taking a gamble on research that could have
unintended and harmful outcomes. When it comes to funding scientific research,
it is crucial to consider the potential consequences. If the outcomes of the
research are unclear, it is difficult to determine the potential benefits or
harms that may result. In these cases, the negative consequences of funding
research with unclear consequences can be significant. For example, funding
research that could lead to the development of dangerous weapons or
technologies could have devastating consequences. Additionally, funding
research that could harm the environment or public health without fully
understanding the risks could also have severe negative consequences.
Therefore, it is important for governments to carefully evaluate the potential
consequences of funding scientific research before committing resources to it.
By looking back at precedents of similar decisions in the past, we can see the
importance of making informed decisions based on the potential consequences of
the research.[continue in next page¿¿]

[¿¿continue for last page] While it is important to consider precedents of
similar decisions in the past, it is also important to consider the potential
for innovation. The counter argument suggests that funding scientific research
with unclear consequences can lead to innovation. Many scientific
breakthroughs and advancements have come from research that was initially
deemed uncertain or risky. If governments only funded research with clear and
immediate results, they would miss out on potentially groundbreaking
discoveries. Additionally, it is difficult to predict the long-term
consequences of any scientific research, so refusing to fund research with
unclear consequences could limit progress and innovation in the future.
Therefore, it is important to weigh the potential risks and benefits of
funding scientific research and not solely rely on precedent as a deciding
factor. From the perspective of potential risks of scientific research, it is
imperative that governments exercise caution when funding research whose
consequences are unclear. When it comes to scientific research, it is
essential to consider the potential risks associated with it, especially the
potential harm to human health or the environment. Governments must exercise
caution when funding research whose consequences are unclear, as the
consequences can be catastrophic. The potential risks associated with
scientific research can be numerous, including the release of hazardous
chemicals or the creation of new diseases. In such cases, the consequences can
be devastating, and the government must take measures to ensure that the risks
are minimized. Therefore, it is crucial to evaluate the benefits and potential
risks of the research before funding it, as it can have far-reaching
consequences for the health and well-being of the population and the
environment. Ultimately, it is the responsibility of the government to ensure
that the research is conducted safely and that the risks are minimized to the
greatest extent possible. It is undeniable that scientific research has the
potential to be misused by individuals or groups with malicious intent. This
misuse could take many forms, from the development of new weapons to the
creation of dangerous new viruses. Given this reality, it is essential that
governments exercise caution when funding research whose consequences are
unclear. In particular, they must carefully consider the potential risks
associated with any research they choose to support, and take steps to
mitigate those risks as much as possible. This may involve setting strict
ethical and safety standards for researchers, or imposing regulations on the
use of certain technologies. Ultimately, the goal should be to promote
scientific progress while minimizing the potential for harm, and this requires
a careful balancing of risks and benefits at all stages of the research
process. The difficulty of predicting the long-term consequences of scientific
research is a major concern when it comes to funding scientific research. The
potential risks of scientific research cannot be fully understood until the
research has been conducted and the consequences become apparent. This is
especially true for research that involves new technologies or untested
hypotheses. Governments must exercise caution when funding research whose
consequences are unclear, as the long-term effects of such research may be
unpredictable and potentially harmful. Therefore, it is crucial that
governments take a proactive approach in assessing the potential risks of
scientific research before funding it, and ensure that researchers are
conducting their studies in a responsible and ethical manner. By doing so,
governments can help to minimize potential risks and ensure that the benefits
of scientific research outweigh the potential harms.
---

Table 4. An example draft prototype generated by GPT Playground from a user
study session Topic3 [Career-oriented education]:
Educational institutions should actively encourage their students to choose
fields of study that will prepare them for lucrative careers.
Write a response in which you discuss the extent to which you agree or
disagree with the claim. In developing and supporting your position, be sure
to address the most compelling reasons and/or examples that could be used to
challenge your position.
---
Educational institutions should actively encourage their students to choose
fields of study that will prepare them for lucrative careers.Career
development is a crucial aspect of a student’s education, and educational
institutions should actively encourage their students to choose fields of
study that will prepare them for lucrative careers. By doing so, these
institutions can help students achieve their financial goals and secure a
stable future. Encouraging students to choose fields that offer high-paying
jobs can also help address the issue of student debt. Students who graduate
with high-paying jobs are better equipped to pay off their loans and avoid the
financial burden that comes with student debt. Additionally, by promoting
lucrative careers, educational institutions can contribute to the overall
economic growth of a country. When students are trained in high-demand fields,
they are more likely to find employment and contribute to the growth of
various industries. Therefore, it is imperative that educational institutions
prioritize career development and encourage students to consider fields that
offer financial stability and growth opportunities.
One of the primary advantages of encouraging students to pursue fields of
study that prepare them for lucrative careers is the economic benefits that
students can enjoy throughout their lives. Encouraging students to pursue
fields of study that prepare them for lucrative careers not only benefits the
students themselves but also their families. With a higher income, students
can provide better financial support for their families, which can lead to an
improved quality of life. For example, they can afford better housing,
healthcare, and education for their children. Additionally, students can also
contribute to their communities by investing in local businesses and
philanthropic endeavors. By pursuing a career that offers a higher salary,
students can not only secure their own financial stability but also positively
impact the lives of those around them. Therefore, it is essential to encourage
students to consider the economic benefits of their chosen field of study, not
only for their own future but also for their families and communities. When
students pursue fields of study that prepare them for high-paying careers,
they are more likely to be able to pay off their student loans and other debts
in a timely manner. This is because they have the financial resources to do
so, which can relieve a significant amount of stress and anxiety. The ability
to pay off student loans and debt also allows students to invest in their
futures by saving for retirement, purchasing a home, or starting a business.
Furthermore, having a high-paying career can provide financial security and
stability, which can lead to a better quality of life overall. Therefore,
encouraging students to pursue fields of study that prepare them for lucrative
careers can provide economic benefits that can last a lifetime. Higher earning
potential and career growth opportunities are two significant benefits of
pursuing fields of study that prepare students for lucrative careers. By
investing in education and training in high-demand fields such as computer
science, engineering, finance, or healthcare, students can increase their
earning potential significantly. These fields offer not only high starting
salaries but also numerous opportunities for career advancement and growth.
Students who pursue these careers can expect to earn more throughout their
lives, which can lead to a more comfortable lifestyle and financial stability.
Furthermore, the skills and knowledge gained in these careers can open doors
to leadership positions and entrepreneurship, which can lead to even greater
financial rewards. In short, encouraging students to pursue fields of study
that prepare them for lucrative careers can provide them with lifelong
economic benefits and opportunities for professional growth. [continue in next
page¿¿]

[¿¿continue for last page] When it comes to choosing a field of study,
practicality should be a key consideration for students looking to maximize
their educational investment. Career counseling services provided by
educational institutions play a crucial role in helping students make
practical and informed decisions when it comes to choosing a field of study.
These services provide students with valuable information about the job
market, industry trends, and the skills and qualifications required to succeed
in various fields. By working with career counselors, students can assess
their strengths and weaknesses, identify potential career paths that align
with their interests and goals, and develop a plan to achieve their desired
career outcomes. Such guidance helps students make informed decisions about
their educational investments, ensuring that they choose a field of study that
not only aligns with their interests but also offers them the best
opportunities for career growth and financial stability. Therefore, students
should take advantage of career counseling services provided by educational
institutions to make practical and informed decisions when choosing their
field of study. The availability of internships and job placement programs
should be a crucial factor when students are selecting their field of study.
In today’s competitive job market, acquiring relevant work experience is just
as important as earning a degree. By choosing a practical field of study that
offers ample internship opportunities and job placement programs, students can
gain hands-on experience and develop a network of professional contacts that
can help them secure employment after graduation. Additionally, internships
can provide students with a glimpse into the day-to-day operations of their
chosen industry, allowing them to make more informed decisions about their
career path. Therefore, choosing a practical field of study that offers real-
world experience can be a smart investment for students seeking to maximize
their educational investment. I completely agree with your argument that
practicality should be a key consideration for students when choosing a field
of study. However, it is not solely the responsibility of the students to
ensure that their education aligns with the demands of the job market.
Educational institutions also have a responsibility to equip their students
with the skills and knowledge necessary to succeed in their chosen careers.
This means that universities and colleges should offer programs that are
tailored to the needs of the job market and provide opportunities for students
to gain practical experience through internships, co-op programs, and other
work-integrated learning opportunities. By doing so, educational institutions
can help students maximize their educational investment and increase their
chances of finding employment after graduation. Ultimately, both students and
educational institutions must work together to ensure that graduates are well-
prepared for the workforce and can make a positive impact on their chosen
industries.
While it is true that the job market can be unpredictable, this is not a valid
reason to shift the responsibility of practicality solely onto educational
institutions. Students must also take responsibility for their own education
and career prospects by researching job trends, networking with professionals
in their desired field, and gaining relevant experience through internships or
part-time jobs. Additionally, while the job market may be unpredictable,
certain industries and fields consistently have high demand and job
opportunities. Educational institutions should focus on providing programs and
opportunities that align with these in-demand industries and equip students
with the necessary skills to succeed in them. By solely relying on educational
institutions to tailor their programs to the job market, students may miss out
on valuable opportunities and fail to take proactive steps towards their own
career success. Therefore, it is important for both students and educational
institutions to work together and take responsibility for ensuring
practicality in education.
So overall, our conclusion is educational institutions should actively
encourage their students to choose fields of study that will prepare them for
lucrative careers.
---
Table 5. An example draft prototype generated by VISAR from a user study
session Figure 10. The visual outline of the example draft prototype shown in
Table 3