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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 264, pp. 1--3.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/264\hfil Nonexistence of global solutions]
{Nonexistence of positive global solutions to the differential
equation $u''(t)-t^{-p-1} u^{p}=0$}

\author[A. Alsaedi, B. Ahmad \hfil EJDE-2016/264\hfilneg]
{Ahmed Alsaedi, Bashir Ahmad}

\address{Ahmed Alsaedi \newline
Department of Mathematics,
Faculty of Science,
King Abdulaziz University,
P.O. Box. 80203,  Jeddah 21589, Saudi Arabia}
\email{aalsaedi@hotmail.com}

\address{Bashir Ahmad \newline
Department of Mathematics,
Faculty of Science, King Abdulaziz University,
P.O. Box. 80203, Jeddah 21589, Saudi Arabia}
\email{bashirahmad\_qau@yahoo.com}

\thanks{Submitted July 28, 2016. Published September 28, 2016.}
\subjclass[2010]{34A12, 34A34}
\keywords{Nonlinear differential equation; test function; global solution; blow-up}

\begin{abstract}
 A blow-up result for positive solutions to the differential
 equation $u''(t)-t^{-p-1} u^{p}=0$ is derived. Our result is
 different from the one obtained in the  \cite{Li}, and our
 conditions are less restrictive.
\end{abstract}

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

\section{Introduction}

 Blow up of solutions for differential equations in finite time
is a well known phenomena. For details on the blow-up and 
the existence of global solution, we refer the reader
to the standard books \cite{Samarski,Straughan}.

In a recent work Li et al.\ \cite{Li}, discussed the nonexistence
of positive global solutions to a second-order initial value
problem
\begin{equation}\label{e1}
\begin{gathered}
   u''  - t^{-p-1}u^{p} =0,\quad   t >1, \; p \in (1, \infty), \\
    u(1)=u_0, \quad  u'(1)=u_1, \quad  u_0, u_1 \in \mathbb{R}.
\end{gathered}
\end{equation}
We remark that the existence and uniqueness of classical
local solutions for \eqref{e1} follows by standard arguments when
the function  $t^{-p-1}u^{p}$ with $p>1$, $u\ge 0$
and $t \ge 1$ is locally Lipschitz.

As shown in \cite{Li}, via the substitutions $u(t)=tv(t)$,
$v(t)=w(t)$, and $s=\ln t$,  problem \eqref{e1} is transformed into
\begin{gather}\label{e2}
  w_{ss}+w_s=w^{p},\\
\label{e3}
    u(0)=w_0=u_0, \quad u_1-u_0=w_1=u_1-u_0.
\end{gather}

 The objective of this note is to study the blow-up of
solutions for  problem \eqref{e1} via a test function approach. The
proof of our result is simpler, and different from the one
presented in \cite{Li}. Furthermore, we impose a condition only on 
$u_1$, that it is less restrictive than the conditions imposes 
on both $u_0$ and $u_1$ in \cite{Li}.



\section{Blow-up solution}

\begin{theorem} \label{thm2.1}
Assume that $u_1 \ge 0$. Then any solution of problem
\eqref{e2}-\eqref{e3} blows-up in a finite time.
\end{theorem}

\begin{proof}
Assume that a solution of  problem \eqref{e2}-\eqref{e3} is global.
Multiplying \eqref{e2} by a function
$\phi(s)$ of class $C^2$ such that $\phi(0)=1$, $\phi'(0)=0$,
$\phi(T)=0$, $\phi'(T)=0$, $T>0$ and integrating by parts, we obtain
\begin{equation}\label{e4}
  \int_0^T w^p \phi \,ds+u_1= -\int_0^T w \phi'\, ds+ \int_0^T w \phi''\,ds. 
\end{equation}
Writing
\[
 w |\phi'| =w \phi^{1/p}|\phi'|\phi^{-1/p}, \quad
 w |\phi''| =w \phi^{1/p}|\phi''|\phi^{-1/p}
\]
in \eqref{e4} and using the H\"{o}lder's inequality with
$\varepsilon$, we obtain
\begin{equation}
\begin{aligned}
\int_0^T w^p \phi \,ds+u_1 
&\le \varepsilon  \int_0^T w^p \phi\, ds 
 +C_{\varepsilon} \int_0^T |\phi'|^{p'} \phi^{-p'/p}\, ds \\
&\quad + \varepsilon  \int_0^T w^p \phi ds 
 +C_{\varepsilon} \int_0^T |\phi''|^{p'} \phi^{-p'/p}\, ds.
\end{aligned}\label{e5}
\end{equation}
Taking $\varepsilon=1/4$ (for example), we obtain
\begin{equation}\label{e6}
  \int_0^T w^p \phi \,ds+u_1 
\le C \Big(\int_0^T |\phi'|^{p'} \phi^{-p'/p} \,ds
+\int_0^T |\phi''|^{p'} \phi^{-p'/p}\,ds\Big).
\end{equation}
At this stage, we choose
\begin{equation}\label{laplacianinequality}
\phi(s)=\begin{cases}
1,   & 0 \le s \le  T/2, \\
 \searrow,   & T/2 \le s \le T, \\
0,   & s \ge T,
\end{cases}
\end{equation}
and introduce the change of variable $s= \tau T$ in the integrals
on the right hand side of \eqref{e6} to obtain
\begin{equation}\label{e7}
  \int_0^T w^p \phi ds+u_1 \le C \left(T^{-p'+1}+T^{-2p'+1}\right).
\end{equation}
As $p'>1$, letting $T \to + \infty$ in \eqref{e7}, we obtain
\begin{equation}
  \int_0^T w^p \phi \,ds+u_1 \le 0,
\end{equation}
which is a contradiction as $w>0$ and $u_1 \ge 0$. This completes
the proof.
\end{proof}

\section{Estimation of the blow-up time}

The solution cannot exist for $T >T_{*}$, where
\begin{equation}\label{e8}
T_*= \min\Big(\Big(\frac{2C}{u_1}\Big)^{\frac{1}{p'-1}},
\Big( \frac{2C}{u_1}\Big)^{\frac{1}{2p'-1} } \Big).
\end{equation}
Indeed, from \eqref{e7}, the  solution cannot exist for
\begin{equation}\label{e9}
u_1 \leq \big( T^{-p'+1  }  + T^{-2p'+1  } \big).
\end{equation}

Then, the estimate \eqref{e8} is obtained by considering the two cases
$T \leq 1$ and $T \geq 1$.

\begin{thebibliography}{0}

\bibitem{Li}  M.-R. Li,
T.-J. Chiang-Lin, Y.-S. Lee, D. W.-C. Miao;
\emph{Nonexistence of positive global solutions to the differential equation
$u''(t)-t^{-p-1} u^{p}=0$}, Electron. J. Differential Equations,
(2016), No. 189, 12 pp.

\bibitem{Samarski} A. A. Samarski, V. A. Galaktionov, S. P. Kurdyumov,  A. P.
Mikhailov;
 \textit{Blow-up in quasilinear parabolic equations},
(Translated from the 1987 Russian original) de Gruyter Expositions
in Mathematics, 19. Walter de Gruyter \& Co., Berlin, 1995.

\bibitem{Straughan} B. Straughan;
\emph{Explosive instabilities in mechanics}, Springer-Verlag, Berlin, 1998.

\end{thebibliography}


\end{document}