\documentclass[12pt]{article} \topmargin=-12mm \oddsidemargin=0mm \textwidth=148mm \textheight=225mm \parindent1.5em %\usepackage{bm} \usepackage{amssymb} \usepackage{latexsym} \usepackage{amsmath} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{ax}{Axiom} \newtheorem{defn}{Definition}[section] \newtheorem{rem}{Remark}[section] \def\loc{\text{loc}} \title{ {\bf Oscillation of nonlinear hyperbolic differential equations with impulses} \thanks{This work is supported by the National Natural Science Foundation of China(No.10071026) corresponding author: wh\_apliu@263.net(Anping Liu)}} \author{{\small {\bf Anping Liu$^1$ \hskip 5mm \small Li Xiao$^1$ \hskip 5mm \small Mengxing He$^2$}}\\ \date{} {\small $^{1}$\hskip 2mm {\small \textit{Department of Mathematics and Physics,}}} \\{\small\hskip 2mm {\small \textit{China University of Geosciences}, \textit{Wuhan, Hubei, 430074, P.R.China}}}\\{\small $^{2}$\hskip 2mm {\small \textit{Science School of Wuhan University of Science and Technology},}} \\ {\small \hskip 2mm \textit{Wuhan, Hubei, 430070, P.R.China}}} \maketitle {{\small\bf Abstract}} \vskip 2mm {\small In this paper, oscillatory properties of solutions for nonlinear impulsive hyperbolic differential equations are investigated and new sufficient conditions and necessary and sufficient condition for oscillations are established.} {\small Keywords: Impulse; Hyperbolic differential equation; Oscillation; Nonlinear \vskip 2mm MSC(2000): 35R12; 35L70} \vskip 6mm \noindent {\large {\bf 1.\hskip 2mm Introduction}} \vskip 2mm The theory of differential equations can be applied to many fields, such as to biology, population growth, engineering, medicine, physics and chemistry. In the last few years, A few of papers have been published on oscillation theory of partial differential equations. Many 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 modelling these problems, that these perturbations act instantaneously, that is, in the form of impulses. In 1991, the first paper on this class of equations [3] was published. The qualitative theory of this class of equations, however, is still in an initial stage of development. for instance, on oscillation theory of impulsive partial differential equations only a few of papers have been published. Recently, Bainov, Minchev, Deng, Fu and Luo [2,4,8,14] 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 partial differential equations. In this paper, we shall discuss the oscillatory properties of solutions for a class of nonlinear hyperbolic differential equations with impulses (1), under the boundary condition (4). It should be noted that the equation we discuss here is nonlinear. Up to now, we do not find the work for oscillations of these kind of problem. \begin{eqnarray*} && \frac{\partial^{2} u}{\partial t^{2}}=a(t)h(u)\Delta u-q(t,x)f(u(t,x)) \end{eqnarray*} \begin{equation} t\neq t_{k},\;\;\; (t,x)\in R_+\times \Omega =G \end{equation} \begin{equation} u(t_{k}^{+},x)-u(t_{k}^{-},x)=q_{k}u(t_{k},x),\;t=t_{k},\; k=1,2,\cdots \end{equation} \begin{equation} u_{t}(t_k^{+},x)dx-u_{t}(t_k^{-},x)dx=b_{k}u_{t}(t_k,x)dx. \end{equation} with the boundary condition \begin{equation} \frac{\partial u}{\partial n}=0,\;\;\;(t,x)\in R_+\times \partial \Omega , \end{equation}. Where $\Omega\subset R^N$ is a bounded domain with boundary $\partial \Omega $ smooth enough and $n$ is a unit exterior normal vector of $\partial \Omega $. Assume that the following conditions are fulfilled: \\ $H_{1})$ $a(t)\in PC(R_+,R_+),$ $q(t,x)\in C(R_+\times \overline {\Omega }, (0,\infty))$; where PC denote the class of functions which are piecewise continuous in $t$ with discontinuities of first kind only at $t=t_{k},k=1,2,\cdots$ and left continuous at $t=t_{k}, k=1,2,\cdots$. \\ $H_{2})$ $h'(u),f(u)\in C(R,R); f(u)/u\geq C=const.>0,$ for $u\neq0;uh'(u)\geq0,$ and $q_{k}>-1,b_{k}>-1, 0<t_{1}<t_{2}< \cdots<t_{k}<\cdots,\lim\limits_{t\rightarrow \infty }t_{k}= \infty$. \\ $H_{3})$ $ u(t,x)$ and their derivatives $u_{t}(t,x)$ is piecewise continuous in $t$ with discontinuities of first kind only at $t=t_{k},k=1,2,\cdots$ 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,\cdots $. {\bf Definition 1.} By a solution of problem (1), (4), we mean that any function $u(t,x)$ which satisfies the condition $H_{3})$ and coincides with the solution of the problem (1),(2),(3) and (4). We introduce the notations:$\Gamma_{k}=\{(t,x): t\in(t_{k},t_{k+1}),x\in\Omega \}, \Gamma=\bigcup^{\infty}_{k=0}\Gamma_{k}$, $\bar{\Gamma}_{k}=\{(t,x): t\in(t_{k},t_{k+1}),x\in\bar{\Omega} \}, \bar{\Gamma}=\bigcup^{\infty}_{k=0}\bar{\Gamma}_{k}$, $v(t)=\int_\Omega u(t,x)dx$ and $p(t)=min q(t,x)$, $x\in \bar{\Omega}.$ {\bf Definition 2.} The solution $u \in C^{2}(\Gamma) \bigcap C^{1}(\bar {\Gamma})$ of problem (1), (4) is called to be nonoscillatory in the domain $G$ if it is either eventually positive or eventually negative. Otherwise, it is called oscillatory. \vskip 6mm \noindent {\large {\bf 2.\hskip 2mm Oscillation properties of the problem(1), (4) }} \vskip 2mm The following is main theorem of this paper. The proof of theorem needs the following lemmas. {\bf Lemma 1.}\hskip 2mm \textit{Let $u \in C^{2}(\Gamma) \bigcap C^{1}(\bar{\Gamma})$ is a positive solution of the problem (1),(4) in $G$, then function $v(t)$ satisfies the impulsive differential inequality} \begin{equation} v''(t)+Cp(t)v(t)\leq0,\;,t\neq t_{k}, \end{equation} \begin{equation} v(t^{+}_{k})=(1+q_{k})v(t_{k})\;\;k=1,2,\cdots. \end{equation} \begin{equation} v'(t^{+}_{k})=(1+b_{k})v'(t_{k})\;\;k=1,2,\cdots. \end{equation} {\bf \textit{Proof}.}\hskip 2mm Let $u(t,x)$ is a positive solution of the problem (1),(4) in $G$. Without loss of generality, we may assume that $u(t,x)>0$ for any $(t,x)\in [t_0,\infty )\times \Omega.$ For $t\geq t_{0},\;t\neq t_{k},\;k=1,2,\cdots$, integrating (1) with respect to $x$ over $\Omega $ yield \begin{eqnarray*} \frac {d^{2}}{dt^{2}}[\int_\Omega udx]= a(t)\int_\Omega h(u)\Delta udx -\int_\Omega q(t,x)f(u(t,x))dx.\;(t\geq t_{0},\;t\neq t_{k}) \end{eqnarray*} By Green's formula and boundary condition we have \begin{eqnarray*} &&\int_\Omega h(u)\Delta udx=\int_{\partial \Omega }h(u)\frac {\partial u}{\partial n}ds-\int_\Omega h'(u)|gradu|^{2}dx\\&& \;\;\;\;\;\;\;\; \; \;\;\;\;\;\;\;\;\;\;\;\; \leq- \int_\Omega h'(u)|gradu|^{2}dx\leq 0. \end{eqnarray*} From condition $H_{2})$, we can easily obtain \begin{eqnarray*} \int_\Omega q(t,x)f(u(t,x))dx\geq Cp(t) \int_\Omega u(t,x)dx. \end{eqnarray*} It follows that from above \begin{equation} v'^{\prime}+Cp(t)v(t) \leq 0,\;\;(t\geq t_{0},\;t\neq t_{k}). \end{equation} Where $v(t)>0$. For $t>t_{0},\;t=t_{k},\;k=1,2,\cdots$, we have \begin{eqnarray*} &&\int_\Omega u(t_k^{+},x)dx- \int_\Omega u(t_k^{-},x)dx= q_{k}\int_\Omega u(t_k,x)dx. \\ &&\int_\Omega u_{t}(t_k^{+},x)dx- \int_\Omega u_{t}(t_k^{-},x)dx =b_{k}\int_\Omega u_{t}(t_k,x)dx. \end{eqnarray*} This implies \begin{equation} v(t_k^{+})=(1+q_{k})v(t_{k}). \end{equation} \begin{equation} v'(t^{+}_{k})=(1+b_{k})v'(t_{k})\;\;k=1,2,\cdots. \end{equation} Hence we obtain that $v(t)>0$ is a positive solution of differential inequality (5)-(7). This ends the proof of the lemma. {\bf Definition 3.} The solution $v(t)$ of differential inequality (5)-(7) is called eventually positive (negative) if there exists a number $t^{*}$ such that $v(t)>0 (v(t)<0)$ for $ t \geq t^{*}$. {\bf Lemma 2.} [2,Theorem 1.4.1 ]. \textit{Assume that} (i) $m(t)\in PC^{1}[R^{+},R]$ \textit{is left continuous at $t_{k}$ for $k=1,2,\cdots,$} (ii) \textit{for $k=1,2,\cdots,t \geq t_{0},$} \begin{eqnarray*} m^{\prime}(t)\leq p(t)m(t)+q(t) \;\;(t\neq t_{k}). \\ m(t^{+}_{k})\leq d_{k}m(t_{k})+e_{k} \;\;(t\neq t_{k}). \end{eqnarray*} \textit{where $ p(t),q(t)\in C(R^{+},R),d_{k} \geq 0 $ and $e_{k}$ are real constants,} $ PC^{1}[R^{+},R]=\{x:R^{+}\rightarrow R; x(t)$ \textit{is continuous and continuously differentiable everywhere except some} $t_{k}$ \textit{at} $ which\ x(t^{+}_{k}),x(t^{-}_{k}), x'(t^{+}_{k}) $ \textit{and} $x(t^{-}_{k})$ \textit{exist and} $x(t_{k})=x(t^{-}_{k}),x'(t_{k})= x'(t^{-}_{k}) \}$. \textit{Then} \begin{eqnarray*} && {} m(t)\leq m(t_{0})\prod\limits_{t_{0}<t_{k}<t}d_{k}exp( \int_{t_{0}}^t p(s)ds) +\int_{t_{0}}^t\prod\limits_{s<t_{k}<t}d_{k} exp(\int_{s}^t p(r)dr)q(s)ds \\&& \;\;\;\;\;\;\;\; +\sum\limits_{t_{0}<t_{k}<t} \prod\limits_ {t_{k}<t_{j}<t} d_{j}exp(\int_{t_{k}}^t p(s)ds)e_{k}. \end{eqnarray*} From lemma 2. we can obtain the following lemma 3. See also [14]. {\bf Lemma 3.} \textit{Let $v(t)$ be eventually positive(negative) solution of differential inequality (5)-(7). Assume that there exists $T \geq t_{0}$ such that $v(t)>0(v(t)<0)$ for $t \geq T$. If the following condition (11) hold,} \begin{equation} \lim \limits_{t \rightarrow + \infty } \int_{t_{0}}^t \prod\limits_{t_{0}<t_{k}<s} \frac{1+b_{k}}{1+q_{k}}ds=+\infty, \end{equation} \textit{then} $v'(t)\geq 0(v'(t) \leq 0)$ \textit{for} $t \in [T,t_{l}] \bigcup(\bigcup^{+\infty} _{k=l}(t_{k},t_{k+1}])$, \textit{where} $l=min\{k:t_{k}\geq T\}$. {\bf Theorem 1.} \textit{Let condition (11) and the following condition (12) hold,} \begin{equation} \lim \limits_{t \rightarrow + \infty } \int_{t_{0}}^t \prod\limits_{t_{0}<t_{k}<s} \frac{1+q_{k}}{1+b_{k}} p(s)ds=+\infty, \end{equation} \textit{then each solution of the problem (1)-(4) oscillates in $G$.} {\bf \textit{Proof}.} Let $u(t,x)$ be a nonoscillatory solution of (1),(4). Without loss of generality, we can assume that $u(t,x)>0$ for any $(t,x)\in [t_0,\infty )\times \Omega.$ From Lemma 2.1, we know that $v(t)$ is a positive solution of (5)-(7). Thus from Lemma 2.4, we can find that $v'(t)\geq 0$ for $t \geq t_{0}$. For $t\geq t_{0},\;t\neq t_{k},\;k=1,2,\cdots$, define \begin{eqnarray*} w(t)=\frac{v'(t)}{v(t)},\;\;t \geq t_{0}. \end{eqnarray*} Then we have $w(t)>0,\;t \geq t_{0}, v'(t)-w(t)v(t)=0$. We may assume that $v(t_{0})=1$, thus in view of (5)-(7) we have that for $t \geq t_{0}$ \begin{equation} v(t)=exp(\int_{t_{0}}^t w(s)ds) \end{equation} \begin{equation} v'(t)=w(t)exp(\int_{t_{0}}^t w(s)ds) \end{equation} \begin{equation} v''(t)=w^{2}(t)exp(\int_{t_{0}}^t w(s)ds)+w'(t)exp (\int_{t_{0}}^t w(s)ds) \end{equation} We substitute (13)-(15) into (5) and can obtain the following inequality \begin{eqnarray*} w^{2}(t)+w'(t)+Cp(t) \leq 0. \end{eqnarray*} From (6),(7) we get \begin{eqnarray*} w(t^{+}_{k})=\frac{v'(t^{+}_{k})}{v(t^{+}_{k})}=\frac {1+b_{k}}{1+q_{k}} w(t_{k}),\;k=1,2,\cdots. \end{eqnarray*} It follows that \begin{equation} w^{\prime}(t)\leq -Cp(t)\;\;(t\neq t_{k}). \end{equation} \begin{equation} w(t^{+}_{k})=\frac{1+b_{k}}{1+q_{k}}w(t_{k}) \;\;(t= t_{k}). \end{equation} By using Lemma 2.3, we obtain \begin{eqnarray*} && {} w(t)\leq w(t_{0})\prod\limits_{t_{0}<t_{k}<t} \frac{1+b_{k}}{1+q_{k}} +\int_{t_{0}}^t\prod\limits_{s<t_{k}<t}\frac{1+b_{k}} {1+q_{k}}(-Cp(s))ds \\&& \;\;\;\;\;\;\;\; =\prod\limits_{t_{0}<t_{k}<t} \frac{1+b_{k}}{1+q_{k}} \{w(t_{0})-\int_{t_{0}}^t\prod\limits_{t_{0}<t_{k}<s} \frac{1+q_{k}}{1+b_{k}}Cp(s)ds \}. \end{eqnarray*} Since $w(t)>0$, the last inequality contradicts (12). The proof of Theorem 1. is complete. It should be noted that obviously all solutions of problem (1),(4) are oscillatory if there exists a subsequence $n_{k}$ of $n$ such that $q_{n_{k}}<-1,$ for $k=1,2,\cdots$. So we only discuss the case of $ q_{k}>-1,$. \vskip 6mm \noindent {\large {\bf 3.\hskip 2mm Necessary and sufficient conditions}} \vskip 2mm \hskip 2mm In this section, we will establish necessary and sufficient conditions for oscillation of impulsive hyperbolic partial differential equation. We consider the following linear problem. \begin{eqnarray*} u_{tt}=a(t)\Delta u+p(t)u(t,x) \end{eqnarray*} \begin{equation} t\neq t_{k},\;(t,x)\in R_+\times \Omega =G \end{equation} \begin{equation} u(t_{k}^{+})-u(t_{k}^{-})=q_{k}u(t_{k},x),\;t=t_{k},\;k=1,2,\cdots \end{equation} \begin{equation} u_{t}(t_{k}^{+})-u_{t}(t_{k}^{-})=b_{k}u_{t}(t_{k},x),\;t=t_{k},\;k=1,2,\cdots \end{equation} with boundary condition (4). \vskip 2mm{\bf Theorem 2.}\hskip 2mm Necessary and sufficient condition of oscillations in domain $G$ for all solutions of the problems(18)-(20),(4) is that all solutions of the following impulsive differential equation (21)-(23) are oscillatory. \begin{equation} \frac{d^{2}v}{dt^{2}}+p(t)v(t)=0 \end{equation} \begin{equation} v(t_k^{+})-v(t_k^{-})=q_{k}v(t_{k}),k=1,2,\cdots. \end{equation} \begin{equation} v'(t_k^{+})-v'(t_k^{-})=b_{k}v'(t_{k}),k=1,2,\cdots. \end{equation} {\bf \textit{Proof}.}\hskip 2mm Sufficiency. Using reduction to absurdity. Let $u(t,x)$ be a nonoscillatory solution of the problem (18)-(20),(4). Without loss of generality, we may assume that there exists a $t_{0}\geq T$ such that $u(t,x)>0$ for any $(t,x)\in{[t_{0},+\infty)}\times\Omega$. For $t\geq t_{0},t\neq t_{k},k=1,2,\cdots$, integrating (18) with respect to $x$ over $\Omega $ yield \begin{equation} \frac {d^{2}}{dt^{2}}\int_\Omega udx=a(t)\int_\Omega \Delta udx-\int_\Omega p(t)u(t,x)dx. \end{equation} By Green's formula, we have \begin{eqnarray*} \int_\Omega \Delta udx =\int_{\partial \Omega }\frac {\partial u}{\partial n} ds=0 \end{eqnarray*} Denote $v(t)=\int_\Omega u(t,x)dx$, then $v(t)>0$. It follows that \begin{equation} \frac{d^{2}v}{dt^{2}}+p(t)v(t)=0 \end{equation} For $t\geq t_{0},t=t_{k},k=1,2,\cdots$, analogous to (9),(10) we have \begin{equation} v(t_k^{+})-v(t_k^{-})=q_{k}v(t_{k}) \end{equation} \begin{equation} v'(t_k^{+})-v'(t_k^{-})=b_{k}v'(t_{k}),k=1,2,\cdots. \end{equation} Hence we obtain that $v(t)>0$ satisfies equation (21)-(23). This means that impulsive differential equation (21)-(23) has a nonoscillatory solution. A contradiction. This ends the proof of sufficient condition. Necessity. Still using reduction to absurdity. Let $v(t)$ be a nonoscillatory solution of the equation (21)-(23). Without loss of generality, we may assume that there exists a $t_{1}$ large enough such that $v(t)>0$ for any $t\in{[t_{1},+\infty)}$. For $t\geq t_{1},t\neq t_{k},k=1,2,\cdots$, set $u(t,x)=v(t)$, we have $u(t,x)>0$ and we can easily obtain \begin{eqnarray*} \Delta u(t,x)=\Delta v(t)=0. \end{eqnarray*} Making use of this result, from equation (21). We obtain \begin{equation} \frac{d^{2} v(t)}{dt^{2}}+a(t)\Delta v(t)+p(t)v(t)=0 \end{equation} This means that $u(t,x)=v(t)$ satisfies equation (18). For $t\geq t_{1},t=t_{k},k=1,2,\cdots$, from the conditions (25),(26), it is easy to see that function $u(t,x)=v(t)$ satisfies (19),(20). And because $\frac{\partial v}{\partial x}=0$, $\;x\in\partial\Omega$,$u(t,x)=v(t)$ also satisfies boundary condition(4). This indicates that problem (18)-(20),(4) has a nonoscillatory solution. This is a contradiction. This ends the proof of Theorem 2. \vskip 6mm \noindent {\large {\bf 4.\hskip 2mm Remark}} \vskip 2mm {\bf Note .}\hskip 2mm The results of this paper, from the practical standpoint, is very convenient because these criterions only depend on the coefficients of the equations and impulsive term. Necessary and sufficient condition of oscillations reveal the relation between impulsive partial differential equation and impulsive differential equation. 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