\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 194, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/194\hfil Estimates for mild solutions]
{Estimates for mild solutions to semilinear Cauchy problems}

\author[K. Burazin, M. Erceg \hfil EJDE-2014/194\hfilneg]
{Kre\v{s}imir Burazin, Marko Erceg}  % in alphabetical order

\address{Kre\v{s}imir Burazin \newline
Department of Mathematics,
University of Osijek, Trg Ljudevita Gaja 6, Osijek, Croatia}
\email{kburazin@mathos.hr}

\address{Marko Erceg \newline
Department of Mathematics, Faculty of Science,
University of Zagreb, Bijeni\v{c}ka cesta 30, Zagreb, Croatia}
\email{maerceg@math.hr}

\thanks{Submitted July 29, 2014. Published September 18, 2014.}
\subjclass[2000]{47A50, 47D06, 47J35, 35K58, 35Q55}
\keywords{Semigroup; abstract Cauchy problem; blow-up; quenching time}

\begin{abstract}
 The existence (and uniqueness) results on mild solutions of the abstract
 semilinear Cauchy problems in Banach spaces are well known.
 Following the results of Tartar (2008) and Burazin (2008) in the case of
 decoupled hyperbolic systems, we give an alternative proof, which enables
 us to derive an estimate on the mild solution and its time of existence.
 The nonlinear term in the equation is allowed to be time-dependent.
 We discuss the optimality of the derived estimate by testing it on three
 examples: the linear heat equation, the semilinear heat equation that
 models dynamic deflection of an elastic membrane, and the semilinear
 Schr\"odinger equation with time-dependent nonlinearity, that appear
 in the modelling of numerous physical phenomena.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Let $X$ be a Banach space with a norm $\|\cdot\|_X$, and let $T>0$.
We consider the semilinear abstract Cauchy problem
\begin{equation} \label{ACP}
\begin{gathered}
\mathrm{u}'(t) - A\mathrm{u}(t) = \mathrm{f}(t,\mathrm{u}(t)) 
\quad \text{in } (0,T) \,, \\
\mathrm{u}(0) = \mathrm{g} \,,
\end{gathered}
\end{equation}
where $A:D(A)\subseteq X\to X$ is an infinitesimal generator of
a $C_0$-semigroup on $X$, $\mathrm{g}\in X$,
$\mathrm{f}:[0,T]\times X\to X$ and $\mathrm{u}$ is the unknown function.
Throughout the paper we follow the terminology
of \cite{Pa}. Let us denote by $(T(t))_{t\geq 0}$ the $C_0$-semigroup
 generated by $A$, and by $M\geq 1$ and
$\omega\in\mathbb{R}$ constants for which it holds
$$
(\forall t\geq 0) \quad \|T(t)\|_{\mathcal{L}(X)} \leq Me^{\omega t} \,.
$$
The open (closed) ball in $X$ centered at point $\mathrm{v}\in X$ with radius $r>0$
we will denote by $ \mathrm{B}_X(\mathrm{v},r)$ ($ \mathrm{B}_X[\mathrm{v},r]$).
For the open (closed) ball in $\mathbb{R}$, we will omit writing $\mathbb{R}$ 
and just use
$ \mathrm{B}(\mathrm{v},r)$ ($ \mathrm{B}[\mathrm{v},r]$).

For the right-hand side we assume that $\mathrm{f}$ is:
\begin{itemize}
\item[(i)]  Borel measurable (in both variables),
\item[(ii)] locally Lipschitz in $\mathrm{u}$:
$(\exists \Psi\in L_{\rm loc}^\infty(\mathbb{R})) (\forall r>0)
(\forall \mathrm{z}, \mathrm{w}\in \mathrm{B}_X[0,r])$,
$$
\|\mathrm{f}(t,\mathrm{z})-\mathrm{f}(t,\mathrm{w})\|_X \leq \Psi(r)\|\mathrm{z}-\mathrm{w}\|_X \quad
\text{(a.e. $t\in[0,T]$)} \,.		
$$
\end{itemize}
It is easy to see that (ii) implies that $\mathrm{f}$ is  locally bounded in $\mathrm{u}$.
More precisely,  as in \cite[Theorem 3]{B},  one can easily prove the
following theorem.

\begin{theorem} \label{thm1}
The function
$$
\Phi(t,u):=\sup_{\|\mathrm{u}\|_X\leq u}
\|\mathrm{f}(t,\mathrm{u})\|_X ,\quad  t\in[0,T]\,,\; u\in \mathbb{R}_0^+
$$
is (the smallest) local bound for $\mathrm{f}$:
$$
(\forall r>0)(\forall \mathrm{w}\in \mathrm{B}_X[0,r])\quad
\|\mathrm{f}(t,\mathrm{w})\|_X \leq \Phi(t,r) \quad \text{\rm (a.e. $t\in[0,T]$)}\,,
$$
and has the following properties:
\begin{itemize}
\item  $\Phi \in L_{\rm loc}^\infty([0,T]\times\mathbb{R}_0^+)$;

\item $\Phi\geq 0$  and $\Phi(t,\cdot)$ is non-decreasing for $t\in [0,T]$;

\item $\Phi$ is locally Lipschitz in $u$,
	 with the same $\Psi$ as in (ii):
\[
(\forall u,v \in \mathbb{R}_0^+) \quad u\geq v \implies
|\Phi(t,u)-\Phi(t,v)| \leq \Psi(u)|u-v|\quad \text{\rm (a.e. $t\in[0,T]$)}.
\]
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk1} \rm
The properties of the function $\Phi$ guarantee that the Cauchy problem
\begin{equation} \label{ODE-Phi}
\begin{gathered}
v'(t) = e^{-\omega t}\Phi(t,Me^{\omega t}v(t)) \\
v(0) = \|\mathrm{g}\|_X
\end{gathered}
\end{equation}
has the unique maximal solution $v\in W_{\rm loc}^{1,\infty}([0,S))$,
for some $S>0$ ($v$ is Lipschitz continuous
on every $[a,b]\subseteq [0,S)$).
This is a consequence of the function
$h(t,u):= e^{-\omega t}\Phi(t,Me^{\omega t}u)$  being locally bounded
on $[0,T]\times \mathbb{R}_0^+$ and
locally Lipschitz in the second variable (see appendix in \cite{B}
for more details regarding solutions of such problems).
\end{remark}

We use the following terminology from \cite{Pa}:
a function $\mathrm{u}\in C([0,S); X)$ is called
a \emph{mild solution} of \eqref{ACP} on $[0,S)$
(some authors use the term  \emph{weak solution}) if
\begin{equation} \label{CS}
\mathrm{u}(t) = T(t)\mathrm{g} + \int_0^t T(t-s)\mathrm{f}(s, \mathrm{u}(s)) \,ds \,,\quad t\in [0,S) \,.
\end{equation}

This article is organized as follows:
in the second section we derive an estimate on the mild solution of \eqref{ACP}
and its time of existence, while in the last section we apply the obtained
estimate on three examples:
the linear heat equation, the semilinear heat equation that models the dynamic
deflection of an elastic membrane in MEMS device, and the
semilinear Schr\" odinger equation with a
time-dependent nonlinearity, that models numerous physical phenomena in optics,
quantum physics, etc.

\section{Existence and uniqueness theorem}

The existence (and uniqueness) results of a mild solution of \eqref{ACP}
are well known \cite{Pa}, and their proofs usually
rely on some fixed point theorem. In the following theorem, which is the main
result of this paper, we provide an
alternative proof, which enables us to derive an estimate on the solution and
its time of existence. It tracks the idea of \cite[Ch.\ 14]{T},
in the case of decoupled hyperbolic systems,
which was further refined in \cite{B}. The proof follows steps
of \cite[Theorem 1]{B}, and uses a similar iterative procedure as it was used
in the classical proof.

\begin{theorem} \label{thm2}
Let the assumptions from the introductory section hold, and assume that
$v\in W_{\rm loc}^{1,\infty}([0,S))$ is
the maximal solution of \eqref{ODE-Phi} for some $S\in (0,T]$.
Then there exists the unique mild solution on $[0,S)$,
$\mathrm{u}\in C([0,S); X)$, of the problem \eqref{ACP}.
Additionally, $\mathrm{u}$ satisfies the estimate
\begin{equation} \label{E}
\|\mathrm{u}(t)\|_X \leq Me^{\omega t}v(t) \,, \quad t\in[0,S) \,.
\end{equation}
\end{theorem}

\begin{proof}
Let us prove the uniqueness first. Assume that $\mathrm{u}_1,\mathrm{u}_2\in C([0,S); X)$
are two mild solutions of \eqref{ACP}. From \eqref{CS} we have
\begin{align*}
\|\mathrm{u}_1(t)-\mathrm{u}_2(t)\|_X
&\leq \int_0^t \|T(t-s)\|_{\mathcal{L}(X)} \|\mathrm{f}(s,\mathrm{u}_1(s)) - \mathrm{f}(s,\mathrm{u}_2(s))\|_X \,ds \\
&\leq Me^{\omega t} \int_0^t \| \mathrm{f}(s,\mathrm{u}_1(s)) - \mathrm{f}(s,\mathrm{u}_2(s)) \|_X \,ds \,.
\end{align*}
Next, we use the fact that $\mathrm{f}$ is locally Lipschitz (ii)
and by Gronwall's inequality we get the result.
One needs to be aware that these steps have to be done on $[0,S-\epsilon)$
for an arbitrary $\epsilon>0$, because
our solutions can have a blow-up in $S$, hence possibly be unbounded.

For the existence we shall first inductively define $\mathrm{u}_n$, for $n\in\mathbb{N}$,
by using Picard's iterations.
Let us choose $\mathrm{u}_0\in C([0,S]; X)$ such that
$$
\|\mathrm{u}_0(t)\|_X \leq Me^{\omega t} v(t) \,, \quad t\in[0,S) \,,
$$
and then define
\begin{equation} \label{CSn}
\mathrm{u}_n(t) := T(t)\mathrm{g} + \int_0^t T(t-s)\mathrm{f}(s,\mathrm{u}_{n-1}(s)) \,ds \,, \quad t\in [0,S]\,.
\end{equation}
The existence of such a function $\mathrm{u}_0$ is trivial since any constant function
is admissible. To verify that
$\mathrm{u}_n$ is well-defined, it is enough to see that the function under the
integral sign is measurable.
Since $\mathrm{u}_0$ is continuous and $\mathrm{f}$ measurable we have
$s\mapsto \mathrm{f}(s,\mathrm{u}_0(s))$ is measurable, and so it
is $s\mapsto T(t-s)\mathrm{f}(s,\mathrm{u}_0(s))$. In fact,
$\mathrm{u}_1\in C([0,S]; X)$,
hence by the inductive argument
we have that $\mathrm{u}_n$ is well-defined and $\mathrm{u}_n\in C([0,S]; X)$
for $n\in\mathbb{N}$.
Moreover, for
$t\in[0,S)$,
\begin{align*}
\|\mathrm{u}_1(t)\|_X
&\leq \|T(t)\mathrm{g}\|_X + \int_0^t \|T(t-s)\mathrm{f}(s,\mathrm{u}_0(s))\|_X \,ds \\
&\leq Me^{\omega t}\|\mathrm{g}\|_X + Me^{\omega t} \int_0^t e^{-\omega s}
 \|\mathrm{f}(s,\mathrm{u}_0(s))\|_X \,ds \\
&\leq Me^{\omega t} \Big( \|\mathrm{g}\|_X
 + \int_0^t e^{-\omega s}\Phi(s,Me^{\omega s}v(s)) \,ds \Big) \\
&\leq Me^{\omega t}v(t) \,,
\end{align*}
where in the third inequality we used the local boundedness of
 $\mathrm{f}$ and $\mathrm{u}_0(s)\in \mathrm{B}_X[\mathrm{0},Me^{\omega s}v(s)]$
for $s\in[0,S)$. Inductively, the same estimate can be proved for each $\mathrm{u}_n$.

We now distinguish the following two cases depending on whether $v$ has a
 blow-up in $S$.
\smallskip

\noindent \textbf{I.} If $S=T$ (see appendix in \cite{B}),
then $v$ is bounded, so there exists a constant $P$ such that
$|\Psi(Me^{\omega\cdot}v(\cdot))|\leq P$ a.e.\ on $[0,S]$.
If we subtract the formulae for $\mathrm{u}_{n+1}$ and
$\mathrm{u}_n$ and use the locally Lipschitz property of $\mathrm{f}$, we get
(for $t\in [0,S]$)
\begin{align*}
\|\mathrm{u}_{n+1}(t) - \mathrm{u}_n(t)\|_X
&\leq Me^{\omega t}\int_0^t \|\mathrm{f}(s,\mathrm{u}_n(s))-\mathrm{f}(s,\mathrm{u}_{n-1}(s))\|_X \,ds \\
&\leq Me^{\omega t}\int_0^t \Psi(Me^{\omega s}v(s))\|\mathrm{u}_n(s)-\mathrm{u}_{n-1}(s)\|_X \,ds \\
&\leq Me^{\omega S} P \int_0^t \|\mathrm{u}_n(s)-\mathrm{u}_{n-1}(s)\|_X \,ds \,.
\end{align*}
Now one can easily prove by induction that
$$
\|\mathrm{u}_{n+1}(t) - \mathrm{u}_n(t)\|_X
\leq {RM^nP^n t^n \over n!}e^{n\omega S}
\leq {RM^nP^n S^n \over n!}e^{n\omega S} \,,
$$
where $R:=\|\mathrm{u}_1 - \mathrm{u}_0\|_{C([0,S];X) }$. As
$$
\sum_{j=0}^n \|\mathrm{u}_{j+1} - \mathrm{u}_j\|_{C([0,S];X) }
\leq \sum_{j=0}^n {RM^jP^j S^j \over j!} e^{j\omega S}
\leq R\sum_{j=0}^n{(MPSe^{\omega S})^j \over j!}
\leq R e^{MPSe^{\omega S}}\,,
$$
we conclude that the series with partial sums
$\sum_{j=0}^n (\mathrm{u}_{j+1} - \mathrm{u}_j) = \mathrm{u}_{n+1}-\mathrm{u}_0$
converge absolutely in the Banach space $C([0,S];X) $, which implies
that $\mathrm{u}_n$ converges and we denote its limit by $\mathrm{u}$.
It is obvious that $\mathrm{u}$ satisfies \eqref{E}, so it remains to verify
that $\mathrm{u}$ is a mild solution of \eqref{ACP}; i.e.,
that $\mathrm{u}$ satisfies \eqref{CS}.
By the locally Lipschitz property of $\mathrm{f}$ it follows that
\begin{align*}
&\big\| \int_0^t T(t-s)\mathrm{f}(s,\mathrm{u}_n(s))\,ds
- \int_0^t T(t-s)\mathrm{f}(s,\mathrm{u}(s))\,ds \big\|_X\\
&\leq MP e^{\omega S} \int_0^t \|\mathrm{u}_n(s)-\mathrm{u}(s)\|_X\,ds \\
&\leq MPS e^{\omega S} \|\mathrm{u}_n-\mathrm{u}\|_{C([0,S];X) } \,,
\end{align*}
and after passing to the limit in \eqref{CSn} we get the existence in this case.
\smallskip

\noindent \textbf{II.}
If $v$ has a blow-up in $S$, we can repeat the previous argument,
thus getting a mild solution $\mathrm{u}^{S_1}$ on $[0,S_1]$ for every $S_1\in (0,S)$.
The uniqueness of the mild solution provides that the function
$\mathrm{u}:[0,S)\to X$ given by
$$
\mathrm{u}(t) := \mathrm{u}^{S_1}(t) \,, \quad t\in [0,S_1)
$$
is well-defined.
It is then easy to see that $\mathrm{u}$ is the mild solution of \eqref{ACP} on
$[0,S)$, and that estimate \eqref{E} holds.
\end{proof}


\begin{remark} \label{rmk2} \rm
For the right hand side of \eqref{ACP} we can, instead of a function defined on
the whole $[0,T]\times X$, consider a function
$\mathrm{f}: [0,T]\times \mathrm{B}_X(\mathrm{0},b)\to X$, for some $b>0$.
In this case the function $\Phi$ from Theorem \ref{thm1} is defined on
$[0,T]\times [0,b)$ and the solution $v$  of \eqref{ODE-Phi} cannot blow-up,
but it can quench when $v$ approaches $b$.
The statement of Theorem \ref{thm2} is valid in this case as well,
 with some technical differences in its proof.
\end{remark}

\begin{remark} \label{rmk3} \rm
Note that Theorem \ref{thm2} also provides an estimate on the time of existence 
of the mild solution of \eqref{ACP}. Namely, the
mild solution of \eqref{ACP} \emph{exists at least as long as the solution
to} \eqref{ODE-Phi}.
\end{remark}

\begin{remark} \label{rmk4} \rm
Instead of $\Phi$ one can also consider some larger local bound for $\mathrm{f}$ in
\eqref{ODE-Phi}. However, an analogue of \cite[Theorem 2]{B} can be proven,
which states that the best possible estimate of type \eqref{E} will be given by the
smallest possible local bound for $\mathrm{f}$, i.e.\ the function $\Phi$.
\end{remark}

\begin{remark} \label{rmk5} \rm
The estimate in Theorem \ref{thm2} is not  optimal in general, as it can
be seen from examples in \cite{B}. Of course,
due to the generality of Theorem \ref{thm2}, this is expected. However, in some examples,
such as in our first example of linear
heat equation in the third section, our estimate appears to be  sharp. Actually,
this particular example illustrates
how  optimality can be lost due to the  imprecision in the bound
$$
(\forall r>0) (\forall \mathrm{w}\in \mathrm{B}_X[\mathrm{0},r])\quad\|\mathrm{f}(t,\mathrm{w})\|_X
\leq \Phi(t,r) \quad \text{(a.e. $t\in[0,T]$)} \,.
$$

This  non-optimality can also arise due to the imprecision of the estimate \\
$\|T(t)\|_{\mathcal{L}(X)}\leq Me^{\omega t}$.
In some examples this can happen even at $t=0$ (not if $(T(t))_{t\geq 0}$
is a semigroup of contractions, of course).
In such situations this estimate may be improved first by introducing
$\tilde A = A-\omega I$ (which is an infinitesimal
generator of a semigroup $(\tilde T(t))_{t\geq 0}$,
$\tilde T(t) = e^{-\omega t}T(t)$), and then replacing $\|\cdot\|_X$
by an equivalent norm $\|\cdot\|_{X,\tilde A} := \sup_{t\geq 0}\|\tilde T(t)\cdot\|_X$
on $X$ in which $(\tilde T(t))_{t\geq 0}$ is a semigroup of contractions.
\end{remark}

\begin{remark} \label{rmk6} \rm
Theorem \ref{thm2} can be stated also for non-autonomous (evolution) 
abstract systems
\begin{equation} \label{eACP}
\begin{gathered}
\mathrm{u}'(t) - A(t)\mathrm{u}(t) = \mathrm{f}(t,\mathrm{u}(t)) \quad \text{in } (0,T) \,, \\
\mathrm{u}(0) = \mathrm{g} \,,
\end{gathered}
\end{equation}
where $(A(t))_{t\in [0,T]}$ is a family of infinitesimal generators of
$C_0$-semigroups
$(S_t(s))_{s\geq 0}$ on X satisfying 
$\|S_t(s)\|_{\mathcal{L}(X)} \leq e^{\omega s}$
($\omega\in\mathbb{R}$ is independent of $t$) \cite[p.\ 131]{Pa},
 $D(A(t)) = D$ is independent of $t$,
and $A(t)\mathrm{v}$ is continuously differentiable in $X$
for every $\mathrm{v}\in V$.
Indeed, under the above assumptions there exists an  evolution system
$(U(t,s))_{t\geq s\geq 0}$ \cite[p.\ 145]{Pa},  and the mild solution on
 $[0,S)$ is given by
$$
\mathrm{u}(t) = U(t,0)\mathrm{g} + \int_0^t U(t,s)\mathrm{f}(s,\mathrm{u}(s)) ds \,,\quad t\in [0,S) \,.
$$
Since the evolution system $(U(t,s))_{t\geq s\geq 0}$ satisfies
$$
\| U(t,s) \|_{\mathcal{L}(X)} \leq e^{\omega(t-s)} \,,
$$
(see \cite[p.\ 135]{Pa}),  all the steps of the proof of Theorem \ref{thm2}
 remain the same.

One can consider even weaker assumptions on $(A(t))_{t\in [0,T]}$
\cite[Ch.\ 5]{Pa}, but checking
those assumptions can be very technical in general \cite[p.\ 225]{Pa}.
\end{remark}

\section{Applications}

In this section we will illustrate the result of Theorem \ref{thm2}.
There are many problems that can be placed in the
setting of Theorem \ref{thm2}, e.g. \cite[Ch.\ 8]{Pa}, \cite{CH},
and every (semilinear) problem that can be
written as a non-stationary Friedrichs system \cite{BE}.
We selected one classical example of linear equation and
two nonlinear examples which arise from particular problems in physics.


\subsection*{Example (Linear heat equation)}
This example appears to be quite illuminating regarding optimality of
our estimate. More precisely, it illustrates
how optimality of estimate \eqref{E} can depend on  precision of local bound
$\Phi$ for $\mathrm{f}$: let us consider a linear one-dimensional boundary value problem
\begin{equation} \label{lHE}
\begin{gathered}
\partial_t u(t,x) -\partial_{xx}u(t,x)
= \lambda u(t,x)  \quad \text{in } (0,T) \times (0,\pi), \\
u(\cdot,0) = u(\cdot,\pi) = 0, \\
u(0,\cdot) = u_0 \,,
\end{gathered}
\end{equation}
where $\lambda\in\mathbb{R}$ and $u_0\in L^2((0,\pi))$.

Using separation of variables we can obtain formula for the unique
global solution
$$
u(t,x) = \sum_{k=1}^\infty A_k e^{(\lambda-k^2\pi^2)t}\sin(k\pi x) \,,
$$
where
$$
A_k = 2\int_0^1 u_0(x) \sin(k\pi x) dx \,.
$$
By the Parseval identity from the above formulae one can derive the estimate
$$
\|u(t,\cdot)\|_{L^2((0,\pi))}
\leq \|u_0\|_{L^2((0,\pi))} e^{(\lambda-\pi^2)t} \,, \quad t\in[0,T] \,,
$$
which is sharp because in the case when $u_0$ equals the first eigenfunction
$\sin(\pi x)$ we have an equality.

Let us now consider the problem above in the abstract sense in the space
$X=L^2((0,\pi))$ and check how our estimate
\eqref{E} reads for this example.
We introduce notation for the unknown $\mathrm{u}(t):=u(t,\cdot)$ and the initial data
$\mathrm{u}_0:=u_0(\cdot)$, while $A:=\partial_{xx}$ defined on
$$
D(A) = \{ v\in H_0^1((0,\pi)) : \partial_{xx} v\in X \} \le X\,.
$$
According to \cite[Prop. 2.6.1]{CH}, $A$ is an infinitesimal generator of
a $C_0$-semigroup of contractions $(T(t))_{t\geq 0}$.
Moreover, by \cite[Prop.\ 3.5.5]{CH},
 we  have even sharper estimate
$$
\|T(t)\|_{\mathcal{L}(X)} \leq e^{-\lambda_1 t} \,,
$$
where $\lambda_1=\pi^2$ is the first eigenvalue of the operator
$-\partial_{xx}$ in $H_0^1((0,\pi))$, and thus we can
take $M=1$, $\omega =-\pi^2$.

Since the right hand side is linear in $\mathrm{u}$ and independent of $t$,
it is easy to see that the local bound from Theorem \ref{thm1} is
$\Phi(u) = |\lambda|u$, so the solution of \eqref{ODE-Phi} is
given by
$$
v(t) = \|\mathrm{u}_0\|_X e^{|\lambda|t} \,.
$$
The estimate \eqref{E} from Theorem \ref{thm2} then reads
$$
\|\mathrm{u}(t)\|_X \leq \|\mathrm{u}_0\|_X e^{(|\lambda|-\pi^2)t} \,, \quad t\in[0,T] \,.
$$
Compering the obtained estimate with the exact solution, we can see that
for $\lambda\geq 0$ we have a sharp result, but for $\lambda < 0$ there
is a big deviation, because the local bound
$\Phi$ does not distinguish the sign of the right hand side.

However, with a slight change in our abstract setting we can overcome an
appearing imperfection and get the sharp
estimate. Indeed, let us define $B:=A+\lambda I$, where $I$ is the identity
operator on $X$, which is also an
infinitesimal generator of a $C_0$-semigroup $(S(t))_{t\geq 0}$
satisfying $S(t)=e^{\lambda t}T(t)$
(see \cite[p.\ 12]{Pa}). In terms of the operator $B$ our abstract
equation becomes
$$
\mathrm{u}'(t) - B\mathrm{u}(t) = 0 \,.
$$
Since the right hand side is zero, hence $\Phi$ is zero, the solution
of \eqref{ODE-Phi} is
constant function $v(t) = \|\mathrm{u}_0\|_X$. Therefore in this case the
corresponding estimate \eqref{E} is given by
$$
\|\mathrm{u}(t)\|_X \leq \|\mathrm{u}_0\|_X e^{(\lambda-\pi^2)t} \,, \quad t\in[0,T] \,,
$$
where we have used
$$
\|S(t)\|_{\mathcal{L}(X)} \leq e^{\lambda t}\|T(t)\|_{\mathcal{L}(X)} \leq e^{(\lambda-\pi^2)} \,.
$$
The obtained estimate is indeed optimal, as it coincides with the one
derived from the explicit formula for the solution.

\begin{remark} \label{rmk7} \rm
The method from the previous example can be generalized for a wider class
of abstract Cauchy problems \eqref{ACP}: let a $C_0$-semigroup
$(T(t))_{t\geq 0}$ generated by $A$ for $\omega\in\mathbb{R}$ satisfies
$$
(\forall t\geq 0) \quad \|T(t)\|_{\mathcal{L}(X)} \leq e^{\omega t} \,,
$$
i.e.\ $M=1$, and the right hand side be of the form
$$
\mathrm{f}(t,\mathrm{u}(t)) = \lambda(t)\mathrm{u}(t)
+ \tilde{\mathrm{f}}(t,\mathrm{u}(t)) \,,
$$
for $\lambda\in C^1((0,T);\mathbb{R})$ bounded from above, i.e.\
$\lambda_\infty := \sup_{t\in [0,T]} \lambda(t) <\infty$.

Instead of directly applying Theorem \ref{thm2}, let us rewrite our system by introducing
a family of operators
$B(t) := A + \lambda(t)I$. Since $I$ is a bounded operator, for every
$t\in[0,T]$ we have $D(B(t)) = D(A)$
and a $C_0$-semigroup $(S_t(s))_{s\geq 0}$ associated to $B(t)$ satisfies
$$
\|S_t(s)\|_{\mathcal{L}(X)} = e^{\lambda(t)s} \|T(s)\|_{\mathcal{L}(X)}
\leq e^{\lambda_\infty s} e^{\omega s} = e^{\tilde\omega s} \,,
$$
where $\tilde\omega := \lambda_\infty+\omega$ does not depend on $t$.
Since $\lambda$ is continuously
differentiable, we have that the equivalent system to the starting \eqref{ACP}
\begin{gather*}
\mathrm{u}'(t) - B(t)\mathrm{u}(t) = \tilde{\mathrm{f}} (t,\mathrm{u}(t)) \quad \text{in } (0,T) \,, \\
\mathrm{u}(0) = \mathrm{g} \,,
\end{gather*}
satisfies the assumptions of Remark \ref{rmk6}, hence we have the unique solution
that satisfies the corresponding estimate \eqref{E}.

With this approach we indeed have a better estimate then with the direct
application of Theorem \ref{thm2}, since the obtained estimate distinguishes
the sign of $\lambda$ (as it can be seen in the previous example).
\end{remark}


\subsection*{Example (Nonlinear heat equation)}
Let $\Omega\subseteq\mathbb{R}^d$ be open and bounded with a Lipschitz boundary,
and let $T,b,p>0$ be positive constants.
We consider a semilinear initial-boundary value problem:
\begin{equation} \label{nlHE}
\begin{gathered}
\partial_t u(t,\mathbf{x}) -\Delta u(t,\mathbf{x})
 = {\gamma(\mathbf{x}) \over (b-u(t,\mathbf{x}))^p} \quad 
\text{in } (0,T)\times\Omega, \\
u\big|_{\partial\Omega} = 0, \\
u(0,\cdot) = u_0 \,,
\end{gathered}
\end{equation}
where $\gamma, u_0\in C_0(\Omega)$ (continuous functions on 
$\operatorname{cl}\Omega$
that are zero on the boundary of domain), while
$u:[0,T)\times\Omega\to \mathbb{R}$ is the unknown function.
Such an initial and boundary value problem models the dynamic
deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical
System (MEMS)  device (see \cite{GG} and references therein).

We will consider the problem above in the abstract sense in the space
$X=C_0(\Omega)$ with a sup norm; the unknown being
$\mathrm{u}(t):=u(t,\cdot)$, the initial data $\mathrm{u}_0:=u_0(\cdot)$,
and $\boldsymbol{\gamma}:=\gamma(\cdot)$. We take $A:=\Delta$
defined on
$$
D(A) = \{ v\in H_0^1(\Omega)\cap X : \Delta v\in X \} \le X\,,
$$
where $\Delta v$ is taken in the sense of distributions. One can notice that
this Laplace operator and one
used in the previous example (for $d=1$) are not the same, as they are acting on
different spaces. Here we have chose $C_0(\Omega)$
so that the right hand side would be Lipschitz, as it will be seen in the sequel.
According to \cite[Prop.\ 2.6.7]{CH}, $A$ is an infinitesimal generator of a
$C_0$-semigroup of contractions $(T(t))_{t\geq 0}$ (thus $M=1$, $\omega=0$).
Since the mapping $w\mapsto{1\over (b-w)^p}$ is locally Lipschitz on the open
 ball $\mathrm{B}(0,b)$, the right hand side in our equation is also locally Lipschitz
on $\mathrm{B}_X(\mathrm{0},b)$. The local bound from Theorem \ref{thm1} is
given by $\Phi(r):={\|\boldsymbol{\gamma}\|_X \over (b-r)^p}$ for $r\in [0,b)$.
Although the assumptions of Theorem \ref{thm2} for the right hand side are satisfied
only on $\mathrm{B}_X(\mathrm{0},b)$,
based on Remark \ref{rmk2}, the statement of Theorem \ref{thm2} is valid if
$\mathrm{u}_0\in \mathrm{B}_X(\mathrm{0},b)$.

The solution of \eqref{ODE-Phi},
$$
v(t)=b-\Bigl((b-\|\mathrm{u}_0\|_X)^{p+1}-(p+1)\|\boldsymbol{\gamma}\|_X t \Bigr)^{1/(p+1)},
$$
exists until time $T_1={(b-\|\mathrm{u}_0\|_X)^{p+1}\over (p+1)\|\boldsymbol{\gamma}\|_X}$ when
it quenches.
According to Theorem \ref{thm2}, the mild solution $\mathrm{u}$ of \eqref{nlHE} exists
on $[0,T_1)$ and we have
$$
\|\mathrm{u}(t)\|_{C_0(\Omega)} < v(t) \,,
$$
for $t\in[0,T_1)$.

As we have emphasized, the result of Theorem \ref{thm2} is not optimal in general,
 but in this particular example we
can say more regarding its optimality: in \cite[Theorem 2.1]{BD},
it was shown that under some additional
(rather technical) assumptions the classical solution of \eqref{nlHE} quenches
in some finite time $T_q$ which
satisfies the estimate
$$
0\le T_q-T_1
\le {1\over \|\boldsymbol{\gamma}\|_X}\Bigl(1+{C\over p+1}\Bigr)(b-\|\mathrm{u}_0\|_X)^{{4p+1\over 3}} +
	o\Bigl((b-\|\mathrm{u}_0\|_X)^{{4p+1\over 3}}\Bigr)\,,
$$
with some constant $C>0$ not depending on $\|\mathrm{u}_0\|_X$.
Thus the quenching time $T_q$ approaches $T_1$ as
$\|\mathrm{u}_0\|_X \to b$, so in that sense we can say that Theorem \ref{thm2}
gives \emph{good approximation} for the maximal
time of existence for the solution of \eqref{nlHE}.

\subsection*{Example (Nonlinear Schr\"odinger equation)}
Let us now consider an initial-value problem on $\mathbb{R}^d$, for
$d\leq 3$:
\begin{equation} \label{nlS}
\begin{gathered}
\partial_t u(t,\mathbf{x}) -i\Delta u(t,\mathbf{x})
= - \gamma(t) u(t,\mathbf{x}) -g(t)|u(t,\mathbf{x})|^2 u(t,\mathbf{x}) \quad \text{in }
(0,T)\times\mathbb{R}^d, \\
u(0,\cdot) = u_0 \,,
\end{gathered} 
\end{equation}
where $\gamma, g\in C([0,T];\mathbb{C})$, $u_0\in L^2(\mathbb{R}^d;\mathbb{C})$  and
$u:[0,T)\to \mathbb{C}$ is the unknown function.
Nonlinear Schr\"odinger equations appear in the modeling of numerous
physical phenomena, such as propagation
of laser beams in nonlinear media, plasma dynamics, mean field dynamics
of Bose-Einstein condensates, condensed
matter, etc. (see \cite{PTK} and references therein). As before, we shall
study \eqref{nlS} as an abstract problem, with
the unknown $\mathrm{u}(t):=u(t,\cdot)$, and the initial data $\mathrm{u}_0:=u_0(\cdot)$.

The operator $A:=i\Delta$ is an infinitesimal generator of a $C_0$-semigroup
 of unitary operators $(T(t))_{t\geq 0}$
on $L^2(\mathbb{R}^d;\mathbb{C})$ (thus $M=1$, $\omega=0$) with the domain
$D(A)=H^2(\mathbb{R}^d;\mathbb{C})$ \cite[p.\ 224]{Pa}. In order to
apply Theorem \ref{thm2}, it is necessary for the right hand side to be
locally Lipschitz in $\mathrm{u}$, which cannot be obtained
in $L^2(\mathbb{R}^d;\mathbb{C})$. Therefore, we shall restrict our problem to
$X:=D(A)=H^2(\mathbb{R}^d;\mathbb{C})$, equipped with the graph
norm of operator $A$. The equivalence of the graph norm and the
(standard) norm on $H^2(\mathbb{R}^d;\mathbb{C})$ gives us
that $X$ coincides with $H^2(\mathbb{R}^d;\mathbb{C})$, as a Banach space.

As in \cite[p.\ 190]{Pa},  we can conclude that $(T(t)\big|_X)_{t\geq 0}$
is also an unitary $C_0$-semigroup on $X$
generated by the part of $A$ in $X$, $A\big|_X:D(A\big|_X)\subseteq X \to X$,
defined by $A\big|_X \mathrm{u} := A\mathrm{u}$ on the domain
$$
D(A\big|_X):= \{\mathrm{u}\in D(A)\cap X  :  A\mathrm{u}\in X\}\leq X \,.
$$
Using the Sobolev emmbeding theorem, in \cite[Lemma 1.2.]{Pa}
it is shown that $\mathrm{w}\mapsto |\mathrm{w}|^2\mathrm{w}$
is locally Lipschitz in $X$. The proof relies on the continuous embedding of
 $H^2(\mathbb{R}^d;\mathbb{C})$ into $L^\infty(\mathbb{R}^d;\mathbb{C})$, which
is valid only for $d\leq 3$. As $\gamma$ and $g$ are bounded, we finally
obtain that the right hand side of \eqref{nlS} is
locally Lipschitz in $\mathrm{u}$.

With a stronger assumption on the inital data, $\mathrm{u}_0\in X$, we can apply
 Theorem \ref{thm2}.
As the local bound of the right hand side is $\Phi(t,r)=|\gamma(t)|r+|g(t)|r^3$,
we get a Bernoulli first-order ODE for \eqref{ODE-Phi}, with
the solution (for the non-trivial case $\|\mathrm{u}_0\|_X\ne 0$)
$$
v(t) = \Bigl( e^{-2\int_0^t|\gamma(\tau)|\,d\tau}
\Bigl( -2\int_0^t|g(s)|e^{2\int_0^s|\gamma(\tau)|\,d\tau}\,ds
	+\|\mathrm{u}_0\|_X^{-2} \Bigr) \Bigr)^{-1/2} \,,
$$
that has a blow-up in finite time $T_1$ (when the term inside inner parentheses
takes the value 0). Finally, we conclude
that for $\mathrm{u}_0\in H^2(\mathbb{R}^d;\mathbb{C})$ we have the existence of the unique mild
solution $\mathrm{u}\in C([0,T_1);H^2(\mathbb{R}^d;\mathbb{C}))$,
which is in fact the \emph{classical solution} \cite[p.\ 108]{Pa},
 and which satisfies \eqref{E} on $[0,T_1)$.

For some specific $\gamma$ and $g$ we are far from an \emph{optimal} result on
the maximal time of existence in this example.
More precisely, if we take $d=2$, $\gamma =0$ and $g(t)=-ik$, for some $k>0$,
it can be shown that our solution is global, i.e.\ it
exists for all times (see \cite[p.\ 233]{Pa}).
For more detailed exposition regarding globallity of solution for nonlinear
Schr\"odinger equation we refer to \cite[pp.\ 112--123]{CH}.

\subsection*{Acknowledgments}
This research was partially supported  by J. J. Strossmayer University of Osijek
trough project \emph{Evolution Friedrichs systems}, by the University of
Zagreb trough grant 4.1.2.14/2013, by DAAD trough project
\emph{Centre of Excellence for Applications of Mathematics},
and by Croatian Science Foundation under the project 9780.

Part of this research was carried out while both authors were visitng
the Basque Centre for Applied Mathematics within  the frame of the
FP7-246775 NUMERIWAVES project of the European Research Council.

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\end{document}