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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 27, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/27\hfil Oscillation for hyperbolic equations]
{Oscillation of solutions to neutral nonlinear impulsive hyperbolic
 equations with \\ several delays}

\author[J. C. Yang, A. P. Liu, G. J. Liu \hfil EJDE-2013/27\hfilneg]
{Jichen Yang, Anping Liu, Guangjie Liu}  

\address{Jichen Yang \newline
School of Mathematics and Physics, China University of Geosciences,
Wuhan, Hubei, 430074,  China}
\email{ yjch\_0102@hotmail.com}

\address{Anping Liu (corresponding author) \newline
School of Mathematics and Physics, China University of Geosciences,
Wuhan, Hubei, 430074,  China}
 \email{wh\_apliu@sina.com}

\address{Guangjie Liu \newline
School of Mathematics and Physics, China University of Geosciences,
Wuhan, Hubei, 430074,  China}
\email{guangjieliu@foxmail.com}

\thanks{Submitted October 1, 2012. Published January 27, 2013.}
\subjclass[2000]{35B05, 35L70, 35R10, 35R12}
\keywords{Oscillation; hyperbolic equation; impulsive; neutral type; delay}

\begin{abstract}
 In this article, we study oscillatory properties of solutions to
 neutral nonlinear impulsive hyperbolic partial differential equations
 with several delays. We establish sufficient conditions for oscillation
 of all solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

The theory of partial differential equations can be applied to many fields,
such as to biology, population growth, engineering, generic repression,
control theory and climate model. In the last few years, the fundamental
theory of partial differential equations with deviating argument has
 undergone intensive development. The qualitative theory of this class
of equations, however, is still in an initial stage of development.
 Many srudies have been done under the assumption that the state variables
and system parameters change continuously. However, one may easily visualize
situations in nature where abrupt change such as shock and disasters may occur.
These phenomena are short-time perturbations whose duration is negligible
in comparison with the duration of the whole evolution process.
Consequently, it is natural to assume, in modeling these problems, that
these perturbations act instantaneously, that is, in the form of impulses.

In 1991, the first paper \cite{4} on this class of equations was published.
However, on oscillation theory of impulsive partial differential
equations only a few of papers have been published.
Recently, Bainov, Minchev, Liu and Luo \cite{1,2,6,7,8,9,10,11}
investigated the oscillation of solutions of impulsive partial differential
equations with or without deviating argument.
But there is a scarcity in the study of oscillation theory of nonlinear
impulsive hyperbolic equations of neutral type with several delays.

 In this article, we discuss oscillatory properties of solutions for the
 nonlinear impulsive hyperbolic equation of neutral type
 with several delays.
\begin{gather} \label{1.1}
\begin{aligned}
&\frac{\partial^2}{\partial t^2}\Big[u(t,x)+\sum^m_{i=1}g_iu(t-\tau_i,x)\Big]
\\
&= a(t)h(u)\Delta u-q(t,x)f(u(t,x))+\sum^l_{r=1}a_r(t)h_r(u(t-\sigma_r,x))
 \Delta u(t-\sigma_r,x)\\
&\quad -\sum^n_{j=1}q_j(t,x)f_j(u(t-\rho_j,x)),\quad t\neq t_k,\;
 (t,x)\in \mathbb{R}^+\times\Omega=G,
\end{aligned} \\
\label{1.2}
u(t^+_k,x)=h_k(t_k,x,u(t_k,x)),\quad k=1,2,\dots, \\
\label{1.3}
u_t(t^+_k,x)=p_k(t_k,x,u_t(t_k,x)),\quad k=1,2,\dots,
\end{gather}
with the boundary conditions
\begin{gather}\label{1.4}
u=0,\quad (t,x)\in\mathbb{R}^+\times\partial\Omega,\\
\label{1.5}
\frac{\partial u}{\partial n}+\varphi(t,x)u=0,\quad
 (t,x)\in\mathbb{R}^+\times\partial\Omega,
\end{gather}
and the initial condition
$$
u(t,x)=\Phi(t,x),\ \frac{\partial u(t,x)}{\partial t}=\Psi(t,x),\quad
 (t,x)\in[-\delta,0]\times\Omega.
$$
 Here $\Omega\subset\mathbb{R}^N$ is a bounded domain with boundary
$\partial\Omega$ smooth enough and $n$ is a unit exterior normal vector
of $\partial\Omega,\ \delta=\max\{\tau_i,\sigma_r,\rho_j\}$,
$\Phi(t,x)\in C^2([-\delta,0]\times\Omega,\mathbb{R})$,
$\Psi(t,x)\in C^1([-\delta,0]\times\Omega,\mathbb{R})$.

 This article is organized as follows. in Section 2, we  study the
oscillatory properties of solutions for problems \eqref{1.1}, \eqref{1.4}.
In Section 3, we discuss oscillatory properties of solutions for problems
\eqref{1.1}, \eqref{1.5}.

 We will use the following conditions:
\begin{itemize}
\item[(H1)] $a(t),a_i(t)\in PC(\mathbb{R}^+,\mathbb{R}^+)$,
 $\tau_i,\sigma_r,\rho_j$ are positive constants,
 $q(t,x),q_j(t,x)$ are functions in $C(\mathbb{R}^+\times\bar{\Omega},(0,\infty))$,
 $g_i$ is a non-negative constant, $\sum^m_{i=1}g_i<1$, $i=1,2,\dots,m$,
$j=1,2,\dots,n$, $r=1,2,\dots,l$; where $PC$ denote the class of functions
which are piecewise continuous in $t$ with discontinuities of first kind
only at $t=t_k$, $k=1,2,\dots$, and left continuous at $t=t_k$, $k=1,2,\dots$.

\item[(H2)] $h(u),h_r(u)\in C^1(\mathbb{R},\mathbb{R})$,
 $f(u),f_r(u)\in C(\mathbb{R},\mathbb{R})$;
 $f(u)/u\geq C$  a positive constant,
$f_j(u)/u\geq C_j$ a positive constant, $u\neq0$;
 $h(0)=0$, $h_r(0)=0$, $uh'(u)\geq0$, $uh'_r(u)\geq0$,
$\varphi(t,x)\in C(\mathbb{R}^+\times\partial\Omega,\mathbb{R})$,
$h(u)\varphi(t,x)\geq0$, $h_r(u)\varphi(t-\sigma_r,x)\geq0$,
$r=1,2,\dots,l$, $0<t_1<t_2<\dots<t_k<\dots$, $\lim_{k\to\infty}t_k=\infty$.

\item[(H3)] $u(t,x)$ and their derivatives $u_t(t,x)$ are piecewise continuous
in $t$ with discontinuities of first kind only at $t=t_k$, $k=1,2,\dots$,
and left continuous at $t=t_k$, $u(t_k,x)=u(t^-_k,x)$,
$u_t(t_k,x)=u_t(t^-_k,x)$, $k=1,2,\dots$.

\item[(H4)] $h_k(t_k,x,u(t_k,x)),p_k(t_k,x,u_t(t_k,x))\in
 PC(\mathbb{R}^+\times\bar{\Omega}\times\mathbb{R},\mathbb{R})$,
$k=1,2,\dots$, and there exist positive constants
$\overline{a}_k, \underline{a}_k, \overline{b}_k, \underline{b}_k$ and
$\overline{b}_k\leq\underline{a}_k$ such that for $k=1,2,\dots,$
\[
\underline{a}_k\leq\frac{h_k(t_k,x,\eta)}{\eta}\leq\overline{a}_k,\quad
\underline{b}_k\leq\frac{p_k(t_k,x,\phi)}{\phi}\leq\overline{b}_k.
\]
 Let us construct the sequence $\{\bar{t}_k\}=\{t_k\}\cup\{t_{ki}\}
\cup\{t_{kr}\}\cup\{t_{kj}\}$, where
$t_{ki}=t_k+\tau_i,\ t_{kr}=t_k+\sigma_r,\ t_{kj}=t_k+\rho_j$ and
$\bar{t}_k<\bar{t}_{k+1}$, $i=1,2,\dots,m$, $r=1,2,\dots,l$,
$j=1,2,\dots,n$, $k=1,2,\dots$.
\end{itemize}

\begin{definition}\label{def1.1}\rm
By a solution of problems \eqref{1.1}, \eqref{1.4}
(\eqref{1.5}) with initial condition, we mean that any function $u(t,x)$
for which the following conditions are valid:
\begin{itemize}
\item[(1)] If $-\delta\leq t\leq0$, then $u(t,x)=\Phi(t,x)$,
$\frac{\partial u(t,x)}{\partial t}=\Psi(t,x)$.

\item[(2)] If $0\leq t\leq \bar{t}_1=t_1$, then $u(t,x)$ coincides with
the solution of the problems \eqref{1.1}--\eqref{1.3} and
\eqref{1.4} (\eqref{1.5}) with initial condition.

\item[(3)] If $\bar{t}_k<t\leq\bar{t}_{k+1}$,
$\bar{t}_k\in\{t_k\}\setminus(\{t_{ki}\}\cup\{t_{kr}\}\cup\{t_{kj}\})$,
then $u(t,x)$ coincides with the solution of the problems
\eqref{1.1}--\eqref{1.3} and \eqref{1.4} (\eqref{1.5}).

\item[(4)]  If $\bar{t}_k<t\leq\bar{t}_{k+1}$,
$\bar{t}_k\in\{t_{ki}\}\cup\{t_{kr}\}\cup\{t_{kj}\}$,
then $u(t,x)$ satisfies \eqref{1.4} (\eqref{1.5}) and coincides with
the solution of the  problem
\begin{gather*}
\begin{aligned}
&\frac{\partial^2}{\partial t^2}
\Big[u(t^+,x)+\sum^m_{i=1}g_iu((t-\tau_i)^+,x)\Big]\\
&=a(t)h(u(t^+,x))\Delta u(t^+,x)
 -q(t,x)f(u(t^+,x))\\
&\quad +\sum^l_{r=1}a_r(t)h_r(u((t-\sigma_r)^+,x))\Delta u((t-\sigma_r)^+,x)\\
&-\sum^n_{j=1}q_j(t,x)f_j(u((t-\rho_j)^+,x)),\quad
 t\neq t_k,\; (t,x)\in \mathbb{R}^+\times\Omega=G,
\end{aligned}\\
u(\bar{t}^+_k,x)=u(\bar{t}_k,x),\quad
u_t(\bar{t}^+_k,x)=u_t(\bar{t}_k,x),\\
\mathrm{for } \bar{t}_k\in(\{t_{ki}\}\cup\{t_{kr}\}\cup\{t_{kj}\})
\setminus\{t_k\},
\end{gather*}
or
\begin{gather*}
u(\bar{t}^+_k,x)=h_{k_i}(\bar{t}_k,x,u(\bar{t}_k,x)),\quad
u_t(\bar{t}^+_k,x)=p_{k_i}(\bar{t}_k,x,u_t(\bar{t}_k,x)),\\
\mathrm{for}\ \bar{t}_k\in(\{t_{ki}\}\cup\{t_{kr}\}\cup\{t_{kj}\})\cap\{t_k\},
\end{gather*}
where the number $k_i$ is determined by the equality $\bar{t}_k=t_{k_i}$.
\end{itemize}
\end{definition}

 We introduce the notation:
\begin{gather*}
\Gamma_k=\{(t,x):t\in(t_k,t_{k+1}),x\in\Omega\},\quad
\Gamma=\cup^\infty_{k=0}\Gamma_k,\\
\bar{\Gamma}_k=\{(t,x):t\in(t_k,t_{k+1}),x\in\bar{\Omega}\},\quad
\bar{\Gamma}=\cup^\infty_{k=0}\bar{\Gamma}_k,\\
 v(t)=\int_\Omega u(t,x)\,dx,\ q(t)=\min_{x\in\bar{\Omega}}q(t,x),\quad
 q_j(t)=\min_{x\in\bar{\Omega}}q_j(t,x).
\end{gather*}

\begin{definition}\label{def1.2}\rm
The solution $u\in C^2(\Gamma)\cap C^1(\bar{\Gamma})$ of problems \eqref{1.1},
 \eqref{1.4} (\eqref{1.5}) is called non-oscillatory in the domain $G$
if it is either eventually positive or eventually negative.
Otherwise, it is called oscillatory.
\end{definition}

\section{Oscillation properties for  \eqref{1.1}, \eqref{1.4}}

For the main result of this article, we need following lemmas.

\begin{lemma}\label{lem2.1}
Let $u\in C^2(\Gamma)\cap C^1(\bar{\Gamma})$ be a positive solution
of \eqref{1.1}, \eqref{1.4} in $G$, then function $w(t)$ satisfies
the impulsive differential inequality
\begin{gather}\label{2.1}
w''(t)+C\Big(1-\sum^m_{i=1}g_i\Big)q(t)w(t)
+\sum^n_{j=1}C_j\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)w(t-\rho_j)\leq0,\quad
 t\neq t_k,\\
\label{2.2}
\underline{a}_k\leq\frac{w(t^+_k)}{w(t_k)}\leq\overline{a}_k,\quad k=1,2,\dots,\\
\label{2.3}
\underline{b}_k\leq\frac{w'(t^+_k)}{w'(t_k)}\leq\overline{b}_k,\quad k=1,2,\dots,
\end{gather}
where $w(t)=v(t)+\sum^m_{i=1}g_iv(t-\tau_i)$.
\end{lemma}

\begin{proof}
Let $u(t,x)$ be a positive solution of the problem \eqref{1.1}, \eqref{1.4}
in $G$. Without loss of generality, we may assume that there exists a
$T>0$, $t_0>T$ such that $u(t,x)>0$, $u(t-\tau_i,x)>0$, $i=1,2,\dots,m$,
$u(t-\sigma_r,x)>0$, $r=1,2,\dots,l$, $u(t-\rho_j,x)>0$, $j=1,2,\dots n$,
for any $(t,x)\in[t_0,\infty)\times\Omega$.

 For $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$, integrating \eqref{1.1} with
respect to $x$ over $\Omega$ yields
\begin{align*}
&\frac{d^2}{dt^2}\Big[\int_\Omega u(t,x)\,dx+\sum^m_{i=1}g_i
\int_\Omega u(t-\tau_i,x)\,dx\Big]\\
&= a(t)\int_\Omega h(u)\Delta u\,dx  -\int_\Omega q(t,x)f(u(t,x))\,dx\\
&\quad +\sum^l_{r=1}a_r(t)\int_\Omega h_r(u(t-\sigma_r,x))
 \Delta u(t-\sigma_r,x)\,dx\\
&\quad -\sum^n_{j=1}\int_\Omega q_j(t,x)f_j(u(t-\rho_j,x))\,dx.
\end{align*}
By Green's formula and the boundary condition, we have
\begin{align*}
\int_\Omega h(u)\Delta u\,dx
&= \int_{\partial\Omega} h(u)\frac{\partial u}{\partial n}ds-\int_\Omega h'(u)|\mathrm{grad}u|^2\,dx\\
&= -\int_\Omega h'(u)|\mathrm{grad}u|^2\,dx\leq0,
\end{align*}
\[
\int_\Omega h_r(u(t-\sigma_r,x))\Delta u(t-\sigma_r,x)\,dx\leq0.
\]
From condition (H2), we can easily obtain
\begin{gather*}
\int_\Omega q(t,x)f(u(t,x))\,dx \geq Cq(t)\int_\Omega u(t,x)\,dx,\\
\int_\Omega q_j(t,x)f_j(u(t-\rho_j,x))\,dx
 \geq C_jq_j(t)\int_\Omega u(t-\rho_j,x)\,dx.
\end{gather*}
 From the above it follows that
\[
\frac{d^2}{dt^2}\Big[v(t)+\sum^m_{i=1}g_iv(t-\tau_i)\Big]
+Cq(t)v(t)+\sum^n_{j=1}C_jq_j(t)v(t-\rho_j)\leq0.
\]
Set $w(t)=v(t)+\sum^m_{i=1}g_iv(t-\tau_i)$, we have $w(t)\geq v(t)$,
 then we can obtain
\[
w''(t)+Cq(t)v(t)+\sum^n_{j=1}C_jq_j(t)v(t-\rho_j)\leq0,\quad t\neq t_k.
\]
 From this inequality, we have $w''(t)<0$, and $w'(t)>0,\ w(t)>0$ for
$t\geq t_0$. In fact, if $w'(t)<0$, there exists $t_1\geq t_0$ such
that $w'(t_1)<0$. Hence we have
\begin{gather*}
  w(t)-w(t_1)=\int^t_{t_1}w'(s)ds\leq\int^t_{t_1}w'(t_1)ds=w'(t_1)(t-t_1),\\
\lim_{t\to+\infty}w(t)=-\infty.
\end{gather*}
This is a contradiction, so $w'(t)>0$. Because
\begin{align*}
    v(t) &= w(t)-\sum^m_{i=1}g_iv(t-\tau_i) \\
         &= w(t)-\sum^m_{i=1}g_i\Big[w(t-\tau_i)-\sum^m_{i=1}g_iv(t-2\tau_i)\Big]\\
         &= w(t)-\sum^m_{i=1}g_iw(t-\tau_i)+\sum^m_{i=1}g_i
\Big[\sum^m_{i=1}g_iv(t-2\tau_i)\Big]\\
         &\geq \Big(1-\sum^m_{i=1}g_i\Big)w(t),
\end{align*}
\[
  v(t-\rho_j)\geq \Big(1-\sum^m_{i=1}g_i\Big)w(t-\rho_j).
\]
Hence, we obtain
\[
  w''(t)+C\Big(1-\sum^m_{i=1}g_i\Big)q(t)w(t)
+\sum^n_{j=1}C_j\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)w(t-\rho_j)\leq0,\quad
 t\neq t_k.
\]
 For $t\geq t_0,\ t=t_k,\ k=1,2,\dots$, from \eqref{1.2}, \eqref{1.3} and
condition (H4), we obtain
\begin{gather}\label{2.4}
\underline{a}_k\leq\frac{u(t^+_k,x)}{u(t_k,x)}\leq\overline{a}_k,\\
\label{2.5}\underline{b}_k\leq\frac{u_t(t^+_k,x)}{u_t(t_k,x)}\leq\overline{b}_k.
\end{gather}
According to the $v(t)=\int_\Omega u(t,x)\,dx$, we obtain
\[
\underline{a}_k\leq\frac{v(t^+_k)}{v(t_k)}\leq\overline{a}_k,\quad
\underline{b}_k\leq\frac{v'(t^+_k)}{v'(t_k)}\leq\overline{b}_k.
\]
Because $w(t)=v(t)+\sum^m_{i=1}g_iv(t-\tau_i),$ we finally have
\[
\underline{a}_k\leq\frac{w(t^+_k)}{w(t_k)}\leq\overline{a}_k,\quad
\underline{b}_k\leq\frac{w'(t^+_k)}{w'(t_k)}\leq\overline{b}_k.
\]
Hence we obtain that $w(t)$ is a solution of impulsive differential
 inequality \eqref{2.1}--\eqref{2.3}. This completes the proof.
\end{proof}

\begin{lemma}[{\cite[Theorem 1.4.1]{5}}]\label{lem2.2}
Assume that
\begin{itemize}
\item[(A1)]  the sequence $\{t_k\}$ satisfies $0<t_0<t_1<t_2<\dots$,
 $\lim_{k\to\infty}t_k=\infty$;
\item[(A2)]  $m(t)\in PC^1[\mathbb{R}^+,\mathbb{R}]$ is left continuous at
  $t_k$ for $k=1,2,\dots$;
\item[(A3)]  for $k=1,2,\dots$ and $t\geq t_0$,
\begin{gather*}
m'(t)\leq p(t)m(t)+q(t),\ \ t\neq t_k,\\
m(t^+_k)\leq d_km(t_k)+e_k,
\end{gather*}
where $p(t),q(t)\in C(\mathbb{R}^+,\mathbb{R}),\ d_k\geq0$ and $e_k$ are
constants. $PC$ denote the class of piecewise continuous function
from $\mathbb{R}^+$ to $\mathbb{R}$, with discontinuities of the
first kind only at $t=t_k,\ k=1,2,\dots$.
\end{itemize}
 Then
\begin{align*}
m(t)&\leq m(t_0)\prod_{t_0<t_k<t}d_k\exp\Big(\int^t_{t_0}p(s)ds\Big)
 +\int^t_{t_0}\prod_{s<t_k<t}d_k\exp\Big(\int^t_sp(r)dr\Big)q(s)ds\\
&\quad +\sum_{t_0<t_k<t}\prod_{t_k<t_j<t}d_j\exp\Big(\int^t_{t_k}p(s)ds\Big)e_k.
\end{align*}
\end{lemma}

\begin{lemma}[{\cite{11}}]\label{lem2.3}
Let $w(t)$ be an eventually positive (negative) solution of the differential
inequality \eqref{2.1}--\eqref{2.3}. Assume that there exists $T\geq t_0$
such that $w(t)>0\ (w(t)<0)$ for $t\geq T$. If
\begin{equation}\label{2.6}
\lim_{t\to+\infty}\int^t_{t_0}\prod_{t_0<t_k<s}
\frac{\underline{b}_k}{\overline{a}_k}ds=+\infty
\end{equation}
hold, then $w'(t)\geq0\ (w'(t)\leq0)$ for
$t\in[T,t_l]\cup\big(\cup^{+\infty}_{k=l}(t_k,t_{k+1}]\big)$, where
$l=\min\{k:t_k\geq T\}$.
\end{lemma}

 The following theorem is the first main result of this article.

\begin{theorem}\label{thm2.4}
If condition \eqref{2.6} and the following condition holds,
\begin{equation}\label{2.7}
\lim_{t\to+\infty}\int^t_{t_0}\prod_{t_0<t_k<s}
\frac{\underline{a}_k}{\overline{b}_k}F(s)ds=+\infty,
\end{equation}
where
$$
F(s)=C\Big(1-\sum^m_{i=1}g_i\Big)q(s)
+\sum^n_{j=1}C_j\Big(1-\sum^m_{i=1}g_i\Big)q_j(s)\exp(-\delta z(t_0)).
$$
 Then each solution of \eqref{1.1}--\eqref{1.3}, \eqref{1.4} oscillates in $G$.
\end{theorem}

\begin{proof}
Let $u(t,x)$ be a non-oscillatory solution of \eqref{1.1}, \eqref{1.4}.
Without loss of generality, we can assume that there exists $T>0$, $t_0\geq T$,
such that $u(t,x)>0$, $u(t-\tau_i,x)>0$, $i=1,2,\dots,m$,
$u(t-\sigma_r,x)>0$, $r=1,2,\dots,l$, $u(t-\rho_j,x)>0$, $j=1,2,\dots,n$
for any $(t,x)\in[t_0,\infty)\times\Omega$. From Lemma \ref{lem2.1},
we know that $w(t)$ is a solution of \eqref{2.1}--\eqref{2.3}.

 For $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$, define
\begin{equation}\label{2.8}
z(t)=\frac{w'(t)}{w(t)},\quad t\geq t_0.
\end{equation}
 From Lemma \ref{lem2.3}, we have $z(t)\geq0$, $t\geq t_0, w'(t)-z(t)w(t)=0$.
We may assume that $w(t_0)=1$, thus in view of \eqref{2.1}--\eqref{2.3}
we have that for $t\geq t_0$,
\begin{gather}\label{2.9}
w(t)=\exp\Big(\int^t_{t_0}z(s)ds\Big),\\
\label{2.10}w'(t)=z(t)\exp\Big(\int^t_{t_0}z(s)ds\Big),\\
\label{2.11}w''(t)=z^2(t)\exp\Big(\int^t_{t_0}z(s)ds\Big)+z'(t)
\exp\Big(\int^t_{t_0}z(s)ds\Big).
\end{gather}
We substitute \eqref{2.9}--\eqref{2.11} into \eqref{2.1} and  obtain
\begin{align*}
&z^2(t)\exp\Big(\int^t_{t_0}z(s)ds\Big)
 +z'(t)\exp\Big(\int^t_{t_0}z(s)ds\Big)\\
&+C\Big(1-\sum^m_{i=1}g_i\Big)q(t)\exp\Big(\int^t_{t_0}z(s)ds\Big)\\
&+\sum^n_{j=1}C_j\Big(1-\sum^m_{i=1}g_i\Big)q_i(t)
 \exp\Big(\int^{t-\rho_j}_{t_0}z(s)ds\Big)\leq0.
\end{align*}
Hence we have
\begin{align*}
&z^2(t)+z'(t)+C\Big(1-\sum^m_{i=1}g_i\Big)q(t)\\
&+\sum^n_{j=1}C_j\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)
 \exp\Big(-\int^t_{t-\rho_j}z(s)ds\Big)\leq0,\quad t\neq t_k,
\end{align*}
then we have
\begin{align*}
&z'(t)+C\Big(1-\sum^m_{i=1}g_i\Big)q(t)\\
&+\sum^n_{j=1}C_j\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)\exp
\Big(-\int^t_{t-\rho_j}z(s)ds\Big)\leq0,\quad t\neq t_k.
\end{align*}
 From above inequality and condition $\overline{b}_k\leq \underline{a}_k$,
we know that $z(t)$ is non-increasing, then $z(t)\leq z(t_0)$, for $t\geq t_0$,
 we obtain
\[
z'(t)+C\Big(1-\sum^m_{i=1}g_i\Big)q(t)+\sum^n_{j=1}C_j
\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)\exp(-\delta z(t_0))\leq0,\quad t\neq t_k.
\]
From \eqref{2.2}, \eqref{2.3} and \eqref{2.8}, we obtain
\[
z(t^+_k)=\frac{w'(t^+_k)}{w(t^+_k)}
\leq\frac{\overline{b}_kw'(t_k)}{\underline{a}_kw(t_k)}
=\frac{\overline{b}_k}{\underline{a}_k}z(t_k),
\]
so we can easily obtain
\begin{gather*}
z'(t)\leq-C\Big(1-\sum^m_{i=1}g_i\Big)q(t)
 -\sum^n_{j=1}C_j\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)\exp(-\delta z(t_0))\quad
  t\neq t_k,\\
z(t^+_k)\leq\frac{\overline{b}_k}{\underline{a}_k}z(t_k).
\end{gather*}
Let
$$
-F(t)=-C\Big(1-\sum^m_{i=1}g_i\Big)q(t)-\sum^n_{j=1}C_j
\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)\exp(-\delta z(t_0)).
$$
Then according to Lemma \ref{lem2.2}, we have
\begin{align*}
z(t)&\leq z(t_0)\prod_{t_0<t_k<t}\frac{\overline{b}_k}{\underline{a}_k}+\int^t_{t_0}\prod_{s<t_k<t}\frac{\overline{b}_k}{\underline{a}_k}\left(-F(s)\right)ds\\
&= \prod_{t_0<t_k<t}\frac{\overline{b}_k}{\underline{a}_k}
\Big[z(t_0)-\int^t_{t_0}\prod_{t_0<t_k<s}
 \frac{\underline{a}_k}{\overline{b}_k}F(s)ds\Big]
< 0.
\end{align*}
Since $z(t)\geq0$, this is a contradiction. The proof is complete.
\end{proof}

\section{Oscillation properties of the problem \eqref{1.1}, \eqref{1.5}}

For the second main theorem, we need following lemma.

\begin{lemma}\label{lem3.1}
Let $u\in C^2(\Gamma)\cap C^1(\bar{\Gamma})$ be a positive solution of
\eqref{1.1}, \eqref{1.5} in $G$, then function $w(t)$ satisfies the
 impulsive differential inequality
\begin{gather}\label{3.1}
w''(t)+C\Big(1-\sum^m_{i=1}g_i\Big)q(t)w(t)+\sum^n_{j=1}C_j
\Big(1-\sum^m_{i=1}g_i\Big)q_j(t)w(t-\rho_j)\leq0,\quad t\neq t_k,\\
\label{3.2}\underline{a}_k\leq\frac{w(t^+_k)}{w(t_k)}\leq\overline{a}_k,\quad
  k=1,2,\dots,\\
\label{3.3}\underline{b}_k\leq\frac{w'(t^+_k)}{w'(t_k)}\leq\overline{b}_k,\quad
 k=1,2,\dots,
\end{gather}
where $w(t)=v(t)+\sum^m_{i=1}g_iv(t-\tau_i)$.
\end{lemma}

\begin{proof}
Let $u(t,x)$ be a positive solution of the problem \eqref{1.1}, \eqref{1.5}
in $G$. Without loss of generality, we may assume that there exists a $T>0$,
$t_0>T$ such that $u(t,x)>0$, $u(t-\tau_i,x)>0$,
$i=1,2,\dots,m$, $u(t-\sigma_r,x)>0$, $r=1,2,\dots,l$,
$u(t-\rho_j,x)>0$, $j=1,2,\dots n$, for any
$(t,x)\in[t_0,\infty)\times\Omega$.
 For $t\geq t_0$, $t\neq t_k$, $k=1,2,\dots$, integrating \eqref{1.1}
with respect to $x$ over $\Omega$ yields
\begin{align*}
&\frac{d^2}{dt^2}\Big[\int_\Omega u(t,x)\,dx
+\sum^m_{i=1}g_i\int_\Omega u(t-\tau_i,x)\,dx\Big]\\
&= a(t)\int_\Omega h(u)\Delta u\,dx
-\int_\Omega q(t,x)f(u(t,x))\,dx\\
&\quad +\sum^l_{r=1}a_r(t)\int_\Omega h_r(u(t-\sigma_r,x))
 \Delta u(t-\sigma_r,x)\,dx\\
&\quad -\sum^n_{j=1}\int_\Omega q_j(t,x)f_j(u(t-\rho_j,x))\,dx.
\end{align*}
By Green's formula and the boundary condition, we have
\begin{align*}
\int_\Omega h(u)\Delta u\,dx
&= \int_{\partial\Omega} h(u)\frac{\partial u}{\partial n}ds
 -\int_\Omega h'(u)|\mathrm{grad}u|^2\,dx\\
&= -\int_{\partial\Omega}h(u)\varphi(t,x)u\,ds
-\int_\Omega h'(u)|\mathrm{grad}u|^2\,dx\\
&\leq -\int_\Omega h'(u)|\mathrm{grad}u|^2\,dx\leq0,
\end{align*}
\[
\int_\Omega h_r(u(t-\sigma_r,x))\Delta u(t-\sigma_r,x)\,dx\leq0.
\]
The rest of the proof is similar to the one in Lemma \ref{lem2.1}.
 We omit it.
\end{proof}

 The following theorem is the second main result of this article.

\begin{theorem}\label{thm3.2}
If conditions \eqref{2.6} and \eqref{2.7} hold, then each solution
of \eqref{1.1}--\eqref{1.3}, \eqref{1.5} oscillates in $G$.
\end{theorem}

 The proof of the above theorem is similar to that of 
Theorem \ref{thm2.4}. We omit it.

\section{Examples}

\begin{example}\label{exa4.1}\rm
Consider the equation
\begin{gather*}
\begin{aligned}
\frac{\partial^2}{\partial t^2}\Big[u(t,x)+\frac{1}{2}u(t-\frac{\pi}{2},x)\Big]
&=u^2\Delta u-ue^{u^2}+e^tu^2(t-\frac{\pi}{2},x)\Delta u(t-\frac{\pi}{2},x)\\
&\quad -(x^2+1)e^tu(t-\frac{3\pi}{2},x)e^{u^2(t-\frac{3\pi}{2},x)},
\end{aligned}\\
  t>1,\quad t\neq 2^k,\quad (t,x)\in \mathbb{R}^+\times\Omega=G,\\
u((2^k)^+,x)=(4+\sin 2^k\cos x)u(2^k,x),\quad k=1,2,\dots, \\
u_t((2^k)^+,x)=(2+\sin 2^k\cos x)u_t(2^k,x),\quad k=1,2,\dots,
\end{gather*}
with the boundary condition
\[
u=0,\quad (t,x)\in\mathbb{R}^+\times\partial\Omega,
\]
where $a(t)=1$, $a_1(t)=e^t$, $\tau_1=\frac{\pi}{2}$,
$\sigma_1=\frac{\pi}{2}$, $\rho_1=\frac{3\pi}{2}$, $h(u)=u^2$,
$h_1(u)=u^2$, $f(u)=ue^{u^2}$, $f_1(u)=ue^{u^2}$, $q(t,x)=1$,
$q_1(t,x)=(x^2+1)e^t$, $g_1=\frac{1}{2}$, $t_k=2^k$.
It is easy to verify that the condition (H1)--(H4) and the conditions of
Theorem \ref{thm2.4} are satisfied. Hence the all solutions of
 above problem  oscillate.
\end{example}

\begin{example}\label{exa4.2} \rm
Consider the equation
\begin{gather*}
\begin{aligned}
&\frac{\partial^2}{\partial t^2}
\Big[u(t,x)+\frac{1}{2}u(t-\frac{\pi}{2},x)\Big]\\
&=u^2\Delta u-ue^{u^2}+e^tu^2(t-\frac{\pi}{2},x)\Delta u(t-\frac{\pi}{2},x)
 -(x^2+1)e^tu(t-\frac{3\pi}{2},x)e^{u^2(t-\frac{3\pi}{2},x)},
\end{aligned}\\
 t>1,\quad t\neq 3k,\quad (t,x)\in \mathbb{R}^+\times\Omega=G, \\
u((3k)^+,x)=(4+\sin 3k\cos x)u(3k,x),\quad k=1,2,\dots, \\
u_t((3k)^+,x)=(2+\sin 3k\cos x)u_t(3k,x),\quad k=1,2,\dots,
\end{gather*}
with the boundary condition
\[
\frac{\partial u}{\partial n}+t^2x^2u=0,\quad
(t,x)\in\mathbb{R}^+\times\partial\Omega,
\]
where $a(t)=1$, $a_1(t)=e^t$, $\tau_1=\frac{\pi}{2}$,
$\sigma_1=\frac{\pi}{2}$, $\rho_1=\frac{3\pi}{2}$,
$h(u)=u^2$, $h_1(u)=u^2$, $f(u)=ue^{u^2}$, $f_1(u)=ue^{u^2}$,
$q(t,x)=1$, $q_1(t,x)=(x^2+1)e^t$, $g_1=\frac{1}{2}$, $t_k=3k$,
$\varphi(t,x)=t^2x^2$. It is easy to verify that the condition
$\mathrm{H})$ and condition of Theorem \ref{thm3.2} are satisfied.
 Hence the all solutions of the above problem  oscillate.
\end{example}

\subsection*{Acknowledgments}
This research was partially supported by grants 
from the National Basic Research Program of China,
nos. 2010CB950904 and 2011CB710604.


\begin{thebibliography}{99}

\bibitem{1} D. D. Bainov, E. Minchev;
 Oscillation of solutions of impulsive nonlinear parabolic
differential-difference equations.
 \emph{International Journal of Theoretical Physics}, 1996, {\bf 35}(1): 207--215.

\bibitem{2} D. D. Bainov, E. Minchev;
 Forced oscillations of solutions of impulsive nonlinear parabolic
differential-difference equations.
\emph{Journal of the Korean Mathematical Society}, 1998, {\bf 35}(4): 881--890.

\bibitem{3} B. T. Cui, Y. Q. Liu, F. Q. Deng;
Some oscillation problems for impulsive hyperbolic differential systems
with several delays. \emph{Applied Mathematics and Computation}, 2003,
{\bf 146}(2--3):667--679.

\bibitem{4} L. Erbe, H. Freedman, X. Liu, J .H. Wu;
Comparison principles for impulsive parabolic equations
with application to models of single species growth.
\emph{Journal of the Australian Mathematical Society}, 1991, {\bf 32}, 382--400.

\bibitem{5} V. Lakshmikantham, D. D. Bainov, P. S. Simeonov;
Theory of impulsive differential equations.
\emph{Singapore, World Scientific}, 1989.

\bibitem{6} A. P. Liu, M. X. He;
Oscillatory properties of the solutions of nonlinear delay hyperbolic
differential equations of neutral type.
\emph{Applied Mathematics and Mechanics}, 2002, {\bf 23}(6): 678--685.

\bibitem{7} A. P. Liu, T. Liu, Q. X. Ma, M. Zou;
Oscillation of nonlinear hyperbolic equation with impulses.
\emph{International Journal of Mathematical Analysis}, 2007,
{\bf 1}(15): 719--725.

\bibitem{8} A. P. Liu, Q.X. Ma, M. X. He;
Oscillation of nonlinear impulsive parabolic equations of neutral type.
 \emph{Rocky Mountain Journal of Mathematics}, 2006, {\bf 36}(3): 1011--1026.

\bibitem{9} A. P. Liu, L. Xiao, M. X. He;
 Oscillation of nonlinear hyperbolic differential equations with impulses.
\emph{Nonlinear Oscillations}, 2004, {\bf 7}(4):425--431.

\bibitem{10} A. P. Liu, L. Xiao, T. Liu;
Oscillation of nonlinear impulsive hyperbolic equations with several delays.
\emph{Electronic Journal of Differential Equations}, 2004, {\bf 2004} (24): 1--6.

\bibitem{11} J. W. Luo; Oscillation of hyperbolic partial differential equations
with impulses. \emph{Applied Mathematics and Computation}, 2002, {\bf 133}:
309--318.

\bibitem{12} D. P. Mishev, D. D. Bainov;
A necessary and sufficient condition for oscillation of neutral type hyperbolic
equations[J]. \emph{Journal of Computational and Applied Mathematics}, 1992,
 {\bf 42}(2): 215--220.

\bibitem{13} Y. Shoukaku, N. Yoshida;
Oscillations of nonlinear hyperbolic equations with functional arguments
via Riccati method. \emph{Applied Mathematics and Computation}, 2010, {\bf 217}:
 143--151.

\bibitem{14} J. R. Yan, C. H. Kou; Oscillation of solutions of impulsive delay
differential equations[J]. \emph{Journal of Mathematical Analysis and Applications},
 2001, {\bf 254}: 358--370.

\bibitem{15} Y. H. Yu, B. Liu, Z. R. Liu;
Oscillation of solutions of nonlinear partial differential equations of
neutral type. \emph{Acta Mathematica Sinica, New Series}, 1997,
{\bf 13}(4): 563--570.

\end{thebibliography}

\end{document}