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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 312, 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 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/312\hfil Nonexistence of non-trivial global weak 
solutions]
{Nonexistence of non-trivial global weak solutions for higher-order
nonlinear Schr\"odinger equations}

\author[A. Nabti \hfil EJDE-2015/312\hfilneg]
{Abderrazak Nabti}

\address{Abderrazak Nabti \newline
Laboratoire de Math\'ematiques, Image et Applications, EA 3165,
Universit\'e de La Rochelle, P\^ole sciences et Technologies,
Avenue Michel Cr\'epeau, 17000, La Rochelle, France}
\email{nabtia1@gmail.com}


\thanks{Submitted October 20, 2015. Published December 21, 2015.}
\subjclass[2010]{35Q55}
\keywords{Nonlinear Schr\"odinger equation; global solution; blowup}

\begin{abstract}
 We study the initial-value problem for the higher-order nonlinear
 Schr\"odinger equation
 $$
 i\partial_{t}u-(-\Delta)^{m}u=\lambda| u|^{p},
 $$
 subject to the initial data
 $$
 u(x,0)=f(x),
 $$
 where $u=u(x,t)\in\mathbb{C}$ is a complex-valued function,
 $(x,t)\in\mathbb{R}^{N}\times[0,+\infty)$, $p>1$, $m\geq 1$,
 $\lambda\in\mathbb{C}\backslash\{0\},$ and $f(x)$
 is a given complex-valued function. We prove nonexistence of
 a nontrivial global weak solution. Furthermore, we prove that 
 the $L^2$-norm of the local in time $L^2$-solution blows
 up at a finite time.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

We consider the higher-order nonlinear Schr\"odinger equation
\begin{equation}
i\partial_{t}u-(-\Delta)^{m}u=\lambda| u|^{p},\quad x\in\mathbb{R}^N,
\;t>0,\label{Eqrr1}
\end{equation}
supplemented with the initial data
\begin{equation}
u(x,0)=f(x),\quad x\in\mathbb{R}^N,\label{Eqrr2}
\end{equation}
where $u=u(x,t)$ is a complex-valued unknown function of $(x,t)$,
$\lambda=\lambda_1+i\lambda_2\in\mathbb{C}\backslash\{0\}$,
$\lambda_i\in\mathbb{R}\,(i=1,2)$, and $f=f(x)=f_1(x)+if_{2}(x)\in\mathbb{C}$,
$f_i(x)\in L^1(\mathbb{R})\,(i=1,2)$ are real-valued given functions.

Let us first recall some previous results on nonlinear Schr\"odinger equations
(NLS). Since there is a large amount of papers for NLS, we mention the ones
related to our result. Many authors have studied NLS with a gauge invariant
 power type nonlinearity
\begin{equation}
i\partial_t u+\Delta u=\lambda_0| u|^{p-1}u, \quad
 x\in\mathbb{R}^N,\,t\in\mathbb{R},\label{NLS}
\end{equation}
where $\lambda_0\in \mathbb{R}$, $p>1$. In the case of $1<p<1+\frac{4}{N}$,
Tsutsumi \cite{Tsu-A1} proved global existence of $L^2$-solution for an integral
equation associated to \eqref{NLS} with the initial condition
$u(x,0)=u_{0}(x)\in L^2$ without any size restriction (For other type of
solutions see e.g. \cite{Giniber} etc). It is also well known that when
$N\geq2,\,p\geq 1+\frac{4}{N}$ and $\lambda_0<0$, there are solutions
of \eqref{NLS} that blow up in finite time for certain initial data
(see, e.g. \cite{Blow}).
Ikeda and Wakasugi \cite{Ikeda-A1} studied the nonlinear Schr\"odinger
equation with nongauge invariant power nonlinearity
\begin{equation}
i\partial_t u+\frac{1}{2}\Delta u=\lambda| u|^p,\;x\in\mathbb{R}^N,\;t>0,\label{NLS1}
\end{equation}
subject to the initial data $u(x,0)=\varepsilon f(x)$, where
$f\in L^{2},\,\varepsilon>0$, $1<p<1+\frac{2}{N}$ and
$\lambda\in \mathbb{C}\backslash\{0\}$.
They proved the nonexistence of a non-trivial global weak solution for the
equation \eqref{NLS1} with some initial data but without any size and
coefficient restriction, which implies that ''small data global existence''
does not hold for \eqref{NLS1}. Furthermore, they also proved that the
$L^2$-norm of a time local $L^2$-solution with a suitable initial data blows
up in a finite time.

We will prove, using Banach fixed point theorem and Strichartz estimates
(see \cite{S1,Tsu-A1,Giniber,S4,S5}), a local existenceresult for
problem \eqref{Eqrr1}--\eqref{Eqrr2} for any initial data
$f\in L^2(\mathbb{R}^N)$. Moreover, by the test function method
(see \cite{Ft1,Ft2,Ft3,Ft4,Ft5,Ft6} and the references therein),
we will show nonexistence of non-trivial global weak solutions for
problem \eqref{Eqrr1}--\eqref{Eqrr2}. Next we prove that the $L^2$-norm
of the time local $L^2$-solution blows up at a finite time.

\section{Local existence}

 In this section, we prove the local existence and uniqueness of the
$L^2$-solution to the problem \eqref{Eqrr1}--\eqref{Eqrr2}.
It is well known that $(-\Delta)^m$ is a self-adjoint operator in
$L^2(\mathbb{R}^N)$ for every $m\geq1$, and it generates a strongly
continuous semigroup $S(t)$ on $L^2(\mathbb{R}^N)$ for $t>0$.
Using the semigroup theory (see, e.g. \cite{Yosida}), we can write
 problem \eqref{Eqrr1}--\eqref{Eqrr2} in the following equivalent integral equation
\begin{equation}
u(t)=S(t)f-i\int_{0}^{t}S(t-s)| u(s)|^{p}\,ds,\hspace*{4mm}t\geq 0.\label{Sg}
\end{equation}

For $S(t)=\exp(it(-\Delta)^{m})$, we have the following results.

\begin{lemma} \label{lem1}
Let $\rho$ and $r$ be positive numbers such that $\frac{1}{\rho}+\frac{1}{r}=1$
and $2\leq \rho\leq\infty$. For any $t>0$, $S(t)$ is a bounded operator
from $ L^{r}$ to $ L^{\rho}$. Moreover, it satisfies the important estimate
\begin{equation}
\| S(t)v\|_{ L^{\rho}(\mathbb{R}^N)}
\leq Ct^{-\frac{N}{2m}(\frac{1}{r}-\frac{1}{\rho})}\| v\|_{ L^{r}(\mathbb{R}^N)},
\quad v\in  L^{r}(\mathbb{R}^N),\;t>0,\label{Lem1}
\end{equation}
and for any $t>0$, the map $t\mapsto S(t)$ is strongly continuous.
For $\rho=2$, $S(t)$ is unitary and strongly continuous for $t>0$.
\end{lemma}

\begin{definition} \label{def2.2} \rm
The triple $(r,\rho,q)$ is called $\sigma$-admissible triple if
$\frac{1}{r}=\sigma(\frac{1}{q}-\frac{1}{\rho})$,
where $1<r\leq q \leq \infty$ and $\sigma>0$.
\end{definition}

Now, we give the following Strichartz estimate.

\begin{lemma} \label{lem2}
Let $(r,\rho,2)$ be $\frac{N}{2m}$-admissible.  Then
\begin{equation}
\| S(\cdot)v\|_{ L^{r}((0,T); L^{\rho}(\mathbb{R}^N))}
\leq C\| v\|_{L^2(\mathbb{R}^N)},\label{Lem2}
\end{equation}
where $C=C(N,p)$.
\end{lemma}

For the proof of Lemma \ref{lem1} see, e.g \cite{Giniber}.
 For Lemma \ref{lem2},  see Strichartz \cite{S5} and Ginibre and Velo \cite{Gvelo}.

Let $1<\rho$, $r<\infty$ and $a,b>0$. We set
\begin{align*}
E&:=\Big\{v(t)\in L^{\infty}((0,T);L^{2}(\mathbb{R}^N))
 \cap  L^{r}((0,T); L^{\rho}(\mathbb{R}^N)); \\
& \quad \| v(t)\|_{L^2(\mathbb{R}^N)}\leq a,\;
 \| v\|_{ L^{r}((0,T); L^{\rho}(\mathbb{R}^N))}\leq b\Big\};
\end{align*}
$E$ is a closed subset in $ L^{r}((0,T), L^{\rho}(\mathbb{R}^N))$.


\begin{theorem} \label{thm1}
Let $1<p< 1+\frac{4m}{N},\,\lambda\in\mathbb{C}\backslash\{0\}$ and
$f\in L^{2}(\mathbb{R}^N)$. Then there exists a positive time
$T=T(\| f\|_{L^2})>0$ and a unique solution
$u\in C([0,T);L^2(\mathbb{R}^N))\cap  L^{r}([0,T);L^{\rho}(\mathbb{R}^N))$
of the integral equation \eqref{Sg}, where $\rho$ and $r$ are defined by
 $\rho=p+1$ and $\frac{2m}{r}=\frac{N}{2}-\frac{N}{\rho}$.
\end{theorem}

\begin{proof}
We define the Banach space
\begin{align*}
E_T&:=\Big\{v(t)\in L^{\infty}(I_{T};L^{2}(\mathbb{R}^N))
 \cap L^{r}(I_T;L^{\rho}(\mathbb{R}^N));\\
& \quad\| v(t)\|_{L^{\infty}(I_T;L^2(\mathbb{R}^N))}
 \leq \| f\|_{L^2(\mathbb{R}^N)},\quad
 \| v\|_{L^{r}(I_T;L^{\rho}(\mathbb{R}^N))}
 \leq 2\delta\| f\|_{L^2(\mathbb{R}^N)}\Big\},
\end{align*}
where $I_T:= (0,T)$ and $\delta$ is the constant appearing in
\eqref{Lem2}, with $\rho=p+1$, $r=\frac{4m(p+1)}{N(p-1)}$ and $T$ is a small
positive constant to be determined later. Now, for every $u\in E_{T}$, we define
$$
\Psi(u):=S(t)f(x)-\lambda i\int_{0}^{t}S(t-s)| u|^{p}\,ds.
$$
As usual, we prove the  existence of local solutions using
 the Banach fixed point theorem.
\smallskip

\noindent$\bullet$ $\Psi$ is defined from $E_T$ to $E_T$:  Let $u\in E_{T}$. 
Setting
$$
\tilde{u}(t):=\begin{cases}
u(t), & \text{if } t\in I_T, \\
0, & \text{otherwise}.
\end{cases}
$$
Now, we have
\begin{align*}
&\| \Psi(u)\|_{ L^{r}(I_T; L^{\rho}(\mathbb{R}^N))}\\
& \leq  \delta\| f\|_{L^2(\mathbb{R}^N)}
+C\big\|\int_{0}^{t}(t-s)^{-\frac{N}{2m}(\frac{p}{\rho}-\frac{1}{\rho})}
 \| u(s)\|_{ L^{\rho}(\mathbb{R}^N)}^{p}\,ds\big\|_{ L^{r}(I_T)}\\
& \leq  \delta\| f\|_{L^2(\mathbb{R}^N)}
+C\big\|\int_{-\infty}^{+\infty}| t-s|^{-\frac{N}{2m}
 (\frac{p}{\rho}-\frac{1}{\rho})}\| \tilde{u}(s)\|_{ L^{\rho}
 (\mathbb{R}^N)}^{p}\,ds\big\|_{ L^{r}(\mathbb{R})}.
\end{align*}
By the generalized Young inequality  \cite{Holder}, we have
\begin{equation}
\begin{aligned}
\| \Psi(u)\|_{ L^{r}(I_T; L^{\rho}(\mathbb{R}^N))}
&\leq  \delta\| f\|_{L^2(\mathbb{R}^N)}
 +C\| \tilde{u}\|_{L^{\rho_1}(\mathbb{R}; L^{\rho}(\mathbb{R}^N))}^p \\
&\leq  \delta\| f\|_{L^2(\mathbb{R}^N)}
 +C\| u\|_{L^{\rho_1}(I_T; L^{\rho}(\mathbb{R}^N))}^p,
\end{aligned} \label{EqYoung}
\end{equation}
where $\rho_{1}=\frac{4mp(p+1)}{N+4m-(N-4m)p}$, and note that
$1<\rho_1<r$ for $1<p<1+\frac{4m}{N}$.
By H\"older's inequality we have, with
$\frac{1}{\rho_1}=\frac{1}{\rho_2}+\frac{1}{r}$,
\begin{equation}
\begin{aligned}
\| u\|_{L^{\rho_1}(I_T; L^{\rho}(\mathbb{R}^N))}
&\leq \Big(\int_{0}^{t}\,ds\Big)^{1/\rho_2}
 \| u\|_{ L^{r}(I_T, L^{\rho}(\mathbb{R}^N))} \\
&\leq  CT^{1/\rho_2}\| u\|_{ L^{r}(I_T, L^{\rho}(\mathbb{R}^N))},
\end{aligned}\label{EqHolder}
\end{equation}
where $\rho_2=\frac{4mp}{N+4m-Np}$ and $C=C(N,p)$.
Next, \eqref{EqYoung} and \eqref{EqHolder} give us
\begin{align*}
\| \Psi(u)\|_{ L^{r}(I; L^{\rho}(\mathbb{R}^N))}
&\leq \delta\| f\|_{L^2(\mathbb{R}^N)}
 +C_1T^{p/\rho_2}\| u\|_{ L^{r}(I; L^{\rho}(\mathbb{R}^N))}^p\\
&\leq  \delta\| f\|_{L^2(\mathbb{R}^N)}
 +2C_1T^{p/\rho_2}(2\delta)^{p-1}\| f\|_{L^2(\mathbb{R}^N)}^{p-1}
 \delta\| f\|_{L^2(\mathbb{R}^N)},
\end{align*}
where $C_1=C(N,p,\lambda)$. Now, if we choose $T$ small enough such that
$$
2C_1T^{p/\rho_2}(2\delta)^{p-1}\| f\|_{L^2(\mathbb{R}^N)}^{p-1}\leq 1,
$$
we conclude that $\| \Psi(u)\|_{ L^{r}(I; L^{\rho}(\mathbb{R}^N))} 
\leq \delta\| f\|_{L^{2}(\mathbb{R}^N)}$, and then $\Psi(u)\in E_T$.
\smallskip

\noindent$\bullet$ $\Psi$ is a contracting map.
For $u,\,v\in E_{T}$, repeating the same calculations as above, we obtain
\begin{align*}
& \| \Psi(u)-\Psi(v)\|_{ L^{r}(I; L^{\rho}(\mathbb{R}^N))} \\
&\leq C\big\| \int_{0}^{t}(t-s)^{-\frac{N}{2m}(\frac{p}{\rho}
 -\frac{1}{\rho})}\| | u|^{p}-| v|^{p}\|_{L^{\rho/p}(\mathbb{R}^N)}\,ds
 \big\|_{ L^{r}(I_T)}\\
&\leq C\big\| \int_{0}^{t}(t-s)^{-\frac{N}{2m}(\frac{p}{\rho}
 -\frac{1}{\rho})}\left(\| u\|_{ L^{\rho}(\mathbb{R}^N)}^{p-1}
 +\| v\|_{ L^{\rho}(\mathbb{R}^N)}^{p-1}\right)
 \| u(s)-v(s)\|_{ L^{\rho}(\mathbb{R}^N)}\,ds\big\|_{ L^{r}(I_T)}
 \\
&\leq C\left(\| u\|_{L^{\rho_1}(I_T, L^{\rho}(\mathbb{R}^N))}^{p-1}
 +\| v\|_{L^{\rho_1}(I_T, L^{\rho}(\mathbb{R}^N))}^{p-1}\right)
 \| u-v\|_{L^{\rho_1}(I_T, L^{\rho}(\mathbb{R}^N))}\\
&\leq C_2 T^{p/\rho_2}\left(\| u\|_{ L^{r}(I_T, L^{\rho}(\mathbb{R}^N))}^{p-1}
 +\| v\|_{ L^{r}(I_T, L^{\rho}(\mathbb{R}^N))}^{p-1}\right)
 \| u-v\|_{ L^{r}(I_T, L^{\rho}(\mathbb{R}^N))}\\
&\leq C_2 T^{p/\rho_2} 2 (2\delta\| f\|_{L^2(\mathbb{R}^N)})^{p-1}
 \| u-v\|_{ L^{r}(I_T, L^{\rho}(\mathbb{R}^N))}.
\end{align*}
If we choose $T$ so small such that 
$$
C_2 T^{p/\rho_2} 2 (2\delta\| f\|_{L^2(\mathbb{R}^N)})^{p-1}\leq \frac{1}{2},
$$
then we have
$$
\| \Psi(u)-\Psi(v)\|_{ L^{r}(I; L^{\rho}(\mathbb{R}^N))}
\leq \frac{1}{2}\| u-v\|_{ L^{r}(I_T, L^{\rho}(\mathbb{R}^N))}.
$$
By the Banach fixed point theorem, there exists a solution 
$u\in L^{\infty}(I_{T};L^{2}(\mathbb{R}^N)) \cap  L^{r}(I_T; L^{\rho}(\mathbb{R}^N))$ 
to problem \eqref{Eqrr1}--\eqref{Eqrr2} on $[0,T]$.

As usual, the solution can be extended to a maximal time of 
existence $T_{\rm max}>0$.
\smallskip

\noindent$\bullet$ Uniqueness of solution: We  show that the solution 
 of \eqref{Eqrr1}--\eqref{Eqrr2} is unique. Let $u$ and $v$ be two solutions in 
 $E_T$ for some $T>0$, we set
\[
t_1=\sup\{t\in[0,T_{\rm max}: u(t)=v(t)\}\,.
\]
If $t_1 = T_{\rm max}$, then $u(t) = v(t)$ on $[0, T_{\rm max}]$,
 which is the desired result.
If $t_1 < T_{\rm max}$, repeating the same calculations as before, 
and by the assumption on  $t_1$, we have
 \begin{align*}
&\| u-v\|_{ L^{r}((0,t_2); L^{\rho}(\mathbb{R}^N))}\\
&=\| u-v\|_{ L^{r}((t_1,t_2); L^{\rho}(\mathbb{R}^N))}\\
&\leq C\Big\| \int_{t_1}^{t_2}(t_2-t_1)^{-\frac{N}{2m}(\frac{p}{\rho}
-\frac{1}{\rho})}\| | u|^{p}-| v|^{p}\|_{L^{\rho/p}(\mathbb{R}^N)}\,ds
 \Big\|_{ L^{r}(I_T)}\\
&\leq C_2 (t_2-t_1)^{p/\rho_2}
 \Big(\| u\|_{ L^{r}((t_1,t_2), L^{\rho}(\mathbb{R}^N))}^{p-1}
 +\| v\|_{ L^{r}((t_1,t_2), L^{\rho}(\mathbb{R}^N))}^{p-1}\Big)\\
&\quad\times  \| u-v\|_{ L^{r}((t_1,t_2), L^{\rho}(\mathbb{R}^N))}\\
&\leq C_2 (t_2-t_1)^{p/\rho_2} 2 
 \big(2\delta\| f\|_{L^2(\mathbb{R}^N)}\big)^{p-1}
 \| u-v\|_{ L^{r}((t_1,t_2), L^{\rho}(\mathbb{R}^N))}.
\end{align*}
We can choose $t_2$ such that $t_2>t_1$ and 
\[
C_2 (t_2-t_1)^{p/\rho_2} 2 
\big(2\delta\| f\|_{L^2(\mathbb{R}^N)}\big)^{p-1}\leq \frac{1}{2}\,.
\]
Then we have
\[
 \| u-v\|_{ L^{r}(I; L^{\rho}(\mathbb{R}^N))}\leq 0,
\] 
which implies $u(t)=v(t)$ on $[t_1,t_2]$. 
This contradicts the assumption of $t_1$. Therefore, 
$u(t)=v(t)$ for $t\in[0,T_{\rm max}]$. 
\end{proof}

\section{Blow up of $L^2$-solutions}

We impose the following assumptions on the data
\begin{itemize} %\label{Ass} 
\item[(H1)] 
$f_{1}\in L^{1}(\mathbb{R}^{N})$, $\lambda_{2}\int_{\mathbb{R}^{N}}f_{1}(x)\, dx>0$,
for $f_{2}\in L^{1}(\mathbb{R}^{N})$,
$\lambda_{1}\int_{\mathbb{R}^{N}}f_{2}(x)\, dx<0$. 
\end{itemize}
Now, we want to derive a blow-up result for \eqref{Eqrr1}--\eqref{Eqrr2}. 

\begin{definition} \label{DefV}  \rm
Let $T>0$. We say that $u$ is a weak solution of \eqref{Eqrr1}--\eqref{Eqrr2} 
on $[0,T)$ if $u\in C([0,T];L^{p}_{loc}(\mathbb{R}^N)$ and satisfies
\begin{equation} \label{Deef}
\begin{aligned}
&\int_{0}^{T}\int_{\mathbb{R}^N}u(-i\partial_t\phi(x,t)-(-\Delta)^m\phi(x,t))\,dx\,dt \\
& =i\int_{\mathbb{R}^N}f(x)\phi(x,0)\,dx+\lambda\int_{0}^{T}
 \int_{\mathbb{R}^N}| u|^{p} \phi(x,t)\,dx\,dt
\end{aligned}
\end{equation}
for any $\phi\in C^{1,\infty}_{0}((0,T)\times\mathbb{R}^N)$, $\phi\geq 0$ 
and satisfying $\phi(\cdot,T)=0$. Moreover, if $T=+\infty$, $u$ 
is called a global weak solution for \eqref{Eqrr1}--\eqref{Eqrr2}.
\end{definition}

We note that an $L^2$-solution as in Theorem \ref{thm1} is always a
weak solution in the sense of Definition \ref{DefV}.

\begin{theorem} \label{thm3}
Let $1<p \leq p^{*}=1+\frac{2m}{N}$, $\lambda\in\mathbb{C}\backslash\{0\}$ 
and let $f$ satisfy (H1). Then problem \eqref{Eqrr1}--\eqref{Eqrr2}
has no global nontrivial weak solution.
\end{theorem}

We first prove the following lemma (see \cite{Poh1}).

\begin{lemma} \label{Poz}
Let $\psi\in L^{1}(\mathbb{R}^N)$ and $\int_{\mathbb{R}^N}\psi(x)\,dx<0$. 
Then there exists a test function $0\leq\varphi\leq 1$ such that
\begin{equation}
\int_{\mathbb{R}^N}\psi(x)\varphi(x)\,dx<0.\label{LP}
\end{equation}
\end{lemma}

\begin{proof}
We have
$$
\int_{\mathbb{R}^N}\psi\varphi\,dx
=\int_{| x|\leq R}\psi\varphi\,dx+\int_{R\leq | x|}\psi\varphi\,dx.
$$
Take a function $\varphi=\varphi_R(x),\,0\leq\varphi_R\leq1$, such that 
$\varphi_R(x)\equiv1$ for $| x|\leq R$. Then
\begin{equation}
\int_{\mathbb{R}^N}\psi\varphi_R\,dx
=\int_{| x|\leq R}\psi\,dx+\int_{R\leq | x|}\psi\varphi_R\,dx.\label{M}
\end{equation}
By the convergence of the integral $\int_{\mathbb{R}^N}|\psi|\,dx$, we have
$$
\big|\int_{R\leq | x|}\psi\varphi_R\,dx\big|
\leq\int_{R\leq | x|}| \psi |\,dx\to0\quad\text{as}\quad R\to+\infty.
$$
After passing to the limit as $R\to+\infty$ in \eqref{M}, we obtain
$$
\lim_{R\to+\infty}\int_{\mathbb{R}^N}\psi\varphi_R\,dx
=\lim_{R\to+\infty}\int_{| x|\leq R}\psi\,dx=\int_{\mathbb{R}^N}\psi\,dx<0.
$$
This implies the assertion of Lemma \ref{Poz}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
Suppose by contradiction that $u$ is a weak global solution to
 \eqref{Eqrr1}--\eqref{Eqrr2}.
Let $\Phi$ be a radial, smooth and non-increasing function on
 $[0,+\infty)$ such that
$$
\Phi(r)= \begin{cases}
1, & \text{if }0\leq r\leq1, \\
\searrow & \text{if } 1\leq r\leq2,\\
0,\, &\text{if } r\geq2.
\end{cases}
$$
Set 
$$
\phi_1(x):=\Phi\big(\frac{| x|}{BR}\big),\quad
\phi_{2}(t):=\Phi\big(\frac{t}{R^{2m}}\big),
$$
where $R,B>0$. We use the test function 
$$
\phi(x,t):=\phi_{1}(x)^{\ell}\phi_2(t)^{\sigma},\quad \ell,\,\sigma\gg1.
$$
The constant $B>0$ in the definition of $\phi_1$ is fixed and will be chosen later; 
it plays some role in the case $p=1+\frac{2m}{N}$, while in the case 
$p<1+\frac{2m}{N}$, we take $B=1$.

Let $Q:=[0, R^{2m})\times\mathbb{R}^N$. We consider only the case 
$\lambda_1>0$ and $\lambda_1\int_{\mathbb{R}^N}f_2\,dx<0$, since the other 
cases can be treated almost in the same way (see Remark \ref{rmk1}). 
Set
$$
I_R:=\int_{Q}| u|^p \phi\,dx\,dt.
$$
Now, using the identity \eqref{Deef}, and by taking the real part, we obtain
\begin{equation}
\lambda_1I_R-\int_{\mathbb{R}^N}f_{2}(x)\phi(x,0)\,dx
=\int_{Q}(\text{Im}\,u)\partial_t\phi\,dx\,dt
-\int_{Q}(\text{Re}\,u)(-\Delta)^{m}\phi\,dx\,dt.
\end{equation}
Furthermore, using the assumption (H1) on the initial condition $f$,
and Lemma \ref{Poz}, we obtain
\begin{equation}
\lambda_1I_R
\leq \int_{Q}| u|\,\phi_1^\ell|\partial_t\phi_2^{\sigma}|\,dx\,dt+\int_{Q}| u|| \Delta^m\phi_1^\ell|\,\phi_2^{\sigma}\,dx\,dt\equiv K_1+K_2.\label{EqV13}
\end{equation}
By applying $\varepsilon$-Young's inequality,
$XY\leq \varepsilon X^p+C(\varepsilon)Y^{ q}$, 
for $X\geq 0$, $Y\geq 0$, $p+ q=p q$, with $0 < \varepsilon \ll 1$, 
$C(\varepsilon)=(1/ q)(p\varepsilon)^{- q/p})$,
in $K_1$ and $K_2$, we obtain
\begin{align*}
(\lambda_1-2\varepsilon) I_R
\leq C(\varepsilon)\int_{Q}\phi_1^\ell \phi_{2}^{-\frac{\sigma}{p-1}}
 |\partial_t\phi_2^\sigma|^{ q}\,dx\,dt
+C(\varepsilon)\int_{Q}\phi_{1}^{-\frac{\ell}{p-1}}
|\Delta^m\phi_1^\ell|^{ q}\phi_2^\sigma\,dx\,dt.
\end{align*}
At this stage, we pass to the new variables $s=t/R^{2m}$ and $y=x/R$, 
to obtain the estimate
\begin{equation}
(\lambda_1-2\varepsilon)I_R \leq CR^{N+2m(1- q)}
(\mathcal{A}+\mathcal{B}),\label{Est}
\end{equation}
where
\begin{gather*}
\mathcal{A}:=\int_{\Omega_1}\int_{\Omega_2}\Phi(y)^{^\ell}\Phi(s)^{\sigma- q}
 |\Phi(s)^{\prime}|^{ q}\,dy\,ds<+\infty,\\
\mathcal{B}:=\int_{\Omega_1}\int_{\Omega_2}\Phi(y)^{-\frac{\ell}{p-1}}
 |\Delta^m(\Phi(y)^\ell)|^{ q}\Phi(s)^{\sigma}\,dy\,ds<+\infty, \\
\Omega_1:=\{s\geq 0: s\leq 2\},\quad
\Omega_2:=\{y\in\mathbb{R}^N:| y|\leq 2\}.
\end{gather*}
Note that inequality $p\leq p^{*}$ is equivalent to 
$\beta=N-\frac{2m}{p-1}\leq 0$. So, we have to distinguish two cases:

\noindent\textbf{Case (i):}
 $p<p^{*}\Longrightarrow \beta<0$. Passing to the limit in \eqref{Est} 
as $R\to+\infty$, we have
$$
\int_{0}^{+\infty}\int_{\mathbb{R}^N}| u|^p\phi\,dx\,dt=0\quad\Longrightarrow \quad
 u\equiv 0;
$$
this is a contradiction. 

\noindent\textbf{Case (ii):} $p=p^{*}\Longrightarrow \beta=0$. 
We estimate the first term in the right hand side of inequality \eqref{EqV13} 
using the H\"older inequality and the second term by the Young inequality 
as follows
\begin{align*}
(\lambda_1-\varepsilon)I_R 
&\leq  C(\varepsilon)\int_{Q}\phi_{1}^{-\frac{\ell}{p-1}}|\Delta^m\phi_1^\ell|^{ q}\phi_2^\sigma\,dx\,dt   \\
&\quad + \Big(\int_{C_R}\int_{\mathbb{R}^N}| u|^p\phi\,dx\,dt\Big)^{1/p}
\Big(\int_{C_R}\int_{\mathbb{R}^N}\phi_1^\ell \phi_{2}^{-\frac{\sigma}{p-1}}
|\partial_t\phi_2^\sigma|^{ q}\,dx\,dt\Big)^{1/q},
 \end{align*}
 where $C_R:=\{t\in[0,+\infty)\;:\;R^{2m}\leq t\leq 2R^{2m}\}$ 
is the support of $\partial_t\phi_2$. Note that
 $$
\lim_{R\to+\infty}\int_{C_R}\int_{\mathbb{R}^N}| u|^p\phi\,dx\,dt=0.
$$
 Now, introducing the new variables $s=t/R^{2m}$ and $y=x/BR$, we obtain
 \begin{equation}
 (\lambda_1-\varepsilon)I_R
 \leq  B^{N/ q} \mathcal{E}\Big(\int_{C_R}\int_{\mathbb{R}^N}
| u|^p\phi\,dx\,dt\Big)^{1/p}+ B^{-2m} \mathcal{B},\label{EqLimit}
 \end{equation}
 where
 $$
\mathcal{E}:= \Big(\int_{C_R}\int_{\Omega_2}\Phi(y)^{\ell}
\Phi(s)^{\sigma- q}|\Phi(s)^{\prime}|^{ q}\,dyds\Big)^{1/q}<+\infty.
 $$
 Passing to the limit first in \eqref{EqLimit} as $R\to+\infty$, 
and then $B\to+\infty$, we get
 $$
\int_{0}^{+\infty}\int_{\mathbb{R}^N}| u|^p\phi\,dx\,dt=0\quad
\Longrightarrow\quad u\equiv0;$$
which is a contradiction.
\end{proof}


\begin{remark} \label{rmk1} \rm
For the other cases, setting
$$
I_R:=\begin{cases}
-\int_{Q} \lambda_{1} {| u|}^{p} \phi(x,t)\,dx\,dt 
&\text{if } \lambda_{1}<0,\; \lambda_{1}  \int_{\mathbb{R}^N} f_{2}(x)\,dx<0,\\[4pt] 
 \int_{Q}\lambda_{2}| u|^p\phi(x,t)\,dx\,dt 
&\text{if } \lambda_{2}>0,\; \lambda_{2}\int_{\mathbb{R}^2N}f_{1}(x)\,dx>0,\\[4pt]
 -\int_{Q}\lambda_{2}| u|^p\phi(x,t)\,dx\,dt&\text{if }
 \lambda_{2}<0,\; \lambda_{2}\int_{\mathbb{R}^N}f_{1}(x)\,dx>0,
\end{cases}
$$
we can prove the same conclusion in the same manner as above. 
\end{remark}

Next, we will mention that an $L^2$-solution $u\in C([0,T];L^2(\mathbb{R}^N))$ 
is a weak solution in the sense of Definition \ref{DefV}.

\begin{proposition} \label{prop1}
Let $T>0$. If $u$ is an $L^2$-solution for problem \eqref{Eqrr1}--\eqref{Eqrr2} 
on $[0,T)$, then $u$ is also a weak solution on $[0,T)$ in the sense of 
Definition \ref{DefV}.
\end{proposition}

\begin{proof}
Let $T>0$, $f\in L^{2}(\mathbb{R}^N)$ and let 
$u\in C([0,T);L^{2}(\mathbb{R}^N))\cap  L^{r}((0,T); L^{\rho}(\mathbb{R}^N))$
 be a solution of \eqref{Sg}. Given $\phi \in C^{1,\infty}((0,T)\times\mathbb{R}^N)$ 
such that $\operatorname{supp}\phi:=\Omega$ is compact with
 $\phi(\cdot,T)=0$. Then after multiplying \eqref{Sg} by $\phi\equiv\phi(x,t)$ 
and integrating over $\mathbb{R}^N$, we obtain
\begin{equation*}
\int_{\Omega}u\,\phi\,dx\,dt
=\int_{\Omega}S(t)f(x)\phi\,dx
-\lambda i \int_{\Omega}\int_{0}^{t}S(t-s)|  u(s)|^{p}\,ds\phi\,dx.
\end{equation*}
So after differentiating in time, we obtain
\begin{equation}
\frac{d}{dt}\int_{\Omega}u\,\phi\,dx\,dt
= \int_{\Omega}\frac{d}{dt}(S(t)f(x)\phi)\,dx  
 - \lambda i \int_{\Omega}\frac{d}{dt}\int_{0}^{t}S(t-s)
|  u(s)|^{p}\,ds\phi\,dx.
\label{after}
\end{equation}
Now, using the properties of the semigroup $S(t)$ (see \cite{Banach}), we have
\begin{equation}
\begin{aligned}
\int_{\Omega}\frac{d}{dt}(S(t)f(x)\phi)\,dx
&= i\int_{\Omega}A(S(t)f(x))\phi\,dx+ \int_{\Omega} S(t)f(x)\partial_{t}\phi\,dx  \\
&=  i\int_{\Omega}S(t)f(x)A\phi\,dx+ \int_{\Omega} S(t)f(x)\partial_{t}\phi\,dx,
\end{aligned}\label{Semi}
\end{equation}
and
\begin{equation} \label{Semi1}
\begin{aligned}
&\int_{\Omega}\frac{d}{dt}\int_{0}^{t}S(t-s)F(u)\,ds\phi\,dx\\
&=i\int_{\Omega}\int_{0}^{t}A(S(t-s)F(u))\,ds\,\phi\,dx
 +\int_{\Omega}F(u)\phi\,dx
 +\int_{\Omega}\int_{0}^{t} S(t-s)F(u)\,ds\,\partial_{t}\phi\,dx \\
&= i \int_{\Omega}\int_{0}^{t}S(t-s)F(u)\,ds\,A\phi\,dx
 +\int_{\Omega}F(u)\phi\,dx
 +\int_{\Omega}\int_{0}^{t} S(t-s)F(u)\,ds\,\partial_{t}\phi\,dx,
\end{aligned}
\end{equation}
where $F(u):=| u(t)|^{p}$.
Thus, using \eqref{Sg}, \eqref{Semi} and \eqref{Semi1}, we conclude that 
\eqref{after} implies
\begin{equation*}
\frac{d}{dt}\int_{\Omega}u\,\phi\,dx\,dt
=\int_{\Omega}u\,\partial_t\phi\,dx\,dt
-i\int_{\Omega}u\,A\phi\,dx\,dt-i\lambda\int_{\Omega}F(u)\phi\,dx\,dt.
\end{equation*}
Finally,  by integrating in time over $[0,T]$ and using  
that $\phi(\cdot,T)=0$, we complete the proof.
\end{proof}

Let
\begin{align*}
T_m\equiv \sup\Big\{&T\in[0,+\infty)\,;\,\text{there exists a unique solution $u$ 
to \eqref{Sg}}\\
& \text{such that } u\in C([0,T);L^2(\mathbb{R}^N))
 \cap  L^{r}([0,T);L^{\rho}(\mathbb{R}^N)) \Big\}
\end{align*}
be the maximal existence time of $L^2$-solution, where 
$1<p\leq 1+\frac{2m}{N}$, $\rho=p+1$ and $\frac{2m}{r}=\frac{N}{2}-\frac{N}{\rho}$. 

\begin{theorem}
Let $1<p\leq 1+\frac{2m}{N}$, $\lambda=\lambda_1+i\lambda_2\in \mathbb{C} 
\backslash\{0\}$ and $f\in L^{2}(\mathbb{R}^N)$.
 If the initial data $f=f_1+if_2$ satisfies (H1), then the life span 
$T_m<+\infty$ and the $L^2$-norm of the solution blows up at $t=T_m$.
\begin{equation}
\liminf_{t\to T_m}\| u(t)\|_{L^2}=+\infty.\label{EqBlow}
\end{equation}
\end{theorem}

\begin{proof}
We assume the life span $T_m=+\infty$. Then $u$ is also a global weak solution 
of \eqref{Eqrr1}--\eqref{Eqrr2} in the sense of Definition \ref{DefV}. 
Then we can apply Theorem \ref{thm3} and obtain $u\equiv0$. On the other hand, 
by the identity \eqref{Deef}, we obtain
$$
\int_{\mathbb{R}^N}f_2(x)\phi_1(x)\,dx=0,
$$
which is a contradiction. Therefore, we have $T_m<+\infty$.

Next, we show a blowup of the $L^2$-norm for a local solution $u$ by 
using a contradiction argument again. First we assume
$$
\liminf_{_{t\to T_m}}\| u(t)\|_{L^2}<+\infty;
$$
then there exists a sequence $\{t_{n}\}_{n\geq 1}\subset [0,T_m)$ and 
a positive constant $M>0$ such that
\begin{gather}
\lim_{n\to+\infty}t_n=T_m,\label{Estn}\\
\sup_{n\in\mathbb{N}}\| u(t_n)\|_{L^2}\leq M.\label{Essup}
\end{gather}
Thus for any $t_n\in\{t_{n}\}_{n\geq 1}$, by the estimate 
\eqref{Essup} and the local existence theorem, there exists a positive
 constant $T(M)$ independent on $t_n$ such that we can construct a solution
$$
u\in X:=C([t_n,t_n+T(M));L^2(\mathbb{R}^N))
\cap L^{r}([t_n,t_n+T(M));L^{\rho}(\mathbb{R}^N));
$$
to the integral equation \eqref{Sg}. Moreover, since  the 
limit of $\{t_{n}\}_{n\geq 1}$ exists, we can take $t_n\in [0,T_m)$ 
such that $T_m-\frac{T(M)}{3}<t_n<T_m$. For this $t_n\in [0,T_m)$, 
we can also construct a solution $u\in X$. But the estimate $t_n+T(M)>T_m$ 
is a contradiction to the definition of $T_m$. Therefore we obtain
$$
\liminf_{_{t\to T_m}}\| u(t)\|_{L^2}=+\infty,
$$
which completes the proof.
\end{proof}


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\end{document}