\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 59, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/59\hfil Oscillation of solutions] {Oscillation of solutions for forced nonlinear neutral hyperbolic equations with functional arguments} \author[Y. Shoukaku\hfil EJDE-2011/59\hfilneg] {Yutaka Shoukaku} \address{Yutaka Shoukaku \newline Faculty of Engineering, Kanazawa University, Kanazawa 920-1192, Japan} \email{shoukaku@t.kanazawa-u.ac.jp} \thanks{Submitted April 13, 2011. Published May 9, 2011.} \subjclass[2000]{34K11, 35B05, 35R10} \keywords{Forced oscillation; neutral hyperbolic equations; Riccati method; \hfill\break\indent interval criteria} \begin{abstract} This article studies the forced oscillatory behavior of solutions to nonlinear hyperbolic equations with functional arguments. Our main tools are the integral averaging method and a generalized Riccati technique. \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} In this work we consider the oscillatory behavior of solution to the hyperbolic equation \begin{equation} \label{eE} \begin{split} & \frac{\partial}{\partial t}\Big(r(t) \frac{\partial}{\partial t} \Big(u(x,t) + \sum_{i=1}^{l}h_i(t)u(x,\rho_i(t))\Big)\Big) - a(t)\Delta u(x,t)\\ & - \sum_{i=1}^{k}b_i(t)\Delta u(x,\tau_i(t)) + \sum_{i=1}^{m}q_i(x,t)\varphi_i(u(x,\sigma_i(t)))\\ & = f(x,t), \quad (x,t) \in \Omega \equiv G \times (0,\infty), \end{split} \end{equation} where $\Delta$ is the Laplacian in $\mathbb{R}^n$ and $G$ is a bounded domain of $\mathbb{R}^n$ with piecewise smooth boundary $\partial G$. We consider the boundary conditions \begin{gather} \label{eB1} u = \psi \quad \text{on } \partial G \times [0,\infty),\\ \label{eB2} \frac{\partial u}{\partial \nu} + \mu u = \tilde{\psi} \quad \text{on } \partial G \times [0,\infty), \end{gather} where $\nu$ denotes the unit exterior normal vector to $\partial G$ and $\psi, \tilde{\psi} \in C(\partial G \times (0,\infty); \mathbb{R})$, $\mu \in C(\partial G \times (0,\infty) ; [0,\infty))$. We use the following assumptions in this article: \begin{itemize} \item[(H1)] $r(t) \in C^1([0,\infty) ; (0,\infty))$, \\ $h_i(t) \in C([0,\infty) ; [0,\infty)) \ (i=1,2,\dots,l)$, \\ $a(t), b_i(t) \in C([0,\infty) ; [0,\infty))$ $(i=1,2,\dots,k)$, \\ $q_i(x,t) \in C(\overline{\Omega} ; [0,\infty))$ $(i=1,2,\dots,m)$, $f(x,t) \in C(\overline{\Omega} ; \mathbb{R})$; \item[(H2)] $\rho_i(t) \in C([0,\infty) ; \mathbb{R})$, $ \lim_{t \to \infty}\rho_i(t)=\infty$ $(i=1,2, \dots ,l)$, \\ $\tau_i(t) \in C([0,\infty) ; \mathbb{R})$, $ \lim_{t \to \infty}\tau_i(t)=\infty$ $(i=1,2, \dots, k)$, \\ $\sigma_i(t) \in C([0,\infty) ; \mathbb{R})$, $ \lim_{t \to \infty}\sigma_i(t)=\infty$ $(i=1,2, \dots ,m)$; \item[(H3)] $\varphi_i(s) \in C^1(\mathbb{R} ; \mathbb{R})$ $(i=1,2,\dots,m)$ are convex on $[0,\infty)$ and $\varphi_i(-s)=-\varphi_i(s)$ for $s \ge 0$. \end{itemize} By a \emph{solution} of \eqref{eE} we mean a function $u \in C^2(\overline{G} \times $ $[t_{-1},\infty)) \cap C(\overline{G} \times [\tilde{t}_{-1},\infty))$ which satisfies \eqref{eE}, where \begin{gather*} t_{-1}=\min\{0,\min_{1 \le i \le l}\{\inf_{t \ge 0} \rho_{i}(t)\}, \min_{1 \le i \le k}\{\inf_{t \ge 0} \tau_{i}(t)\} \}, \\ \tilde{t}_{-1}=\min\{0,\min_{1 \le i \le m}\{\inf_{t \ge 0} \sigma_{i}(t)\} \}. \end{gather*} A solution $u$ of \eqref{eE} is said to be \emph{oscillatory} in $\Omega$ if $u$ has a zero in $G \times (t,\infty)$ for any $t > 0$. \begin{definition} \label{def3} \rm We say that the pair of functions $(H_1,H_2)$ belongs to the class $\mathbb{H}$, if $H_1,H_2 \in C(D;[0,\infty))$ and satisfy $$ H_i(t,t)=0, \quad H_i(t,s) > 0 \quad\text{for } t > s \text{and }i=1,2, $$ where $D=\{(t,s):0 < s \le t < \infty\}$. Moreover, the partial derivatives $\partial H_1/\partial t$ and $\partial H_2/\partial s$ exist on $D$ and satisfy $$ \frac{\partial H_1}{\partial t}(s,t)=h_1(s,t) H_1(s,t), \quad \frac{\partial H_2}{\partial s}(t,s)=-h_2(t,s) H_2(t,s), $$ where $h_1, h_2 \in C_{\rm loc}(D;\mathbb{R})$. \end{definition} There are many articles devoted to the study of interval oscillation criteria for nonlinear hyperbolic equations with functional arguments by dealing with Riccati techniques; see for example \cite{c1,l1,l2,l3,r1,s1,s2,w1,w2,y1,z1,z2}. There are also some papers which deal with neutral hyperbolic or second order neutral differential equations, \cite{l3,r1,y1,z2}. However, it seems that very little is known about interval forced oscillations of the neutral hyperbolic equation \eqref{eE}. On the other hand, oscillation criteria of second order neutral differential equations have been studied by many authors. We make reference to result by Tanaka \cite{t1}, and extend them. The aim of this paper is to establish sufficient conditions for every solution of \eqref{eE} to be oscillatory by using Riccati techniques. Equation \eqref{eE} is naturally classified into two classes according to whether \begin{itemize} \item[(C1)] $\int_{t_0}^{\infty}\frac{1}{r(t)}dt = \infty$; or \item[(C2)] $\int_{t_0}^{\infty}\frac{1}{r(t)}dt < \infty$. \end{itemize} \section{Reduction to one-dimensional problems} In this section we reduce the multi-dimensional oscillation problems for \eqref{eE} to one-dimensional oscillation problems. It is known that the first eigenvalue $\lambda_1$ of the eigenvalue problem \begin{gather*} -\Delta w = \lambda w \quad\text{in } G ,\\ w = 0 \quad\text{on } \partial G \end{gather*} is positive, and the corresponding eigenfunction $\Phi(x)$ can be chosen so that $\Phi(x) > 0$ in $G$. The following notation will be used in this article. \begin{gather*} U(t) = K_{\Phi}\int_{G}u(x,t)\Phi(x)dx, \quad \tilde{U}(t) = \frac{1}{|G|}\int_{G}u(x,t)dx, \\ F(t) = K_{\Phi}\int_{G}f(x,t)\Phi(x)dx, \quad \tilde{F}(t) = \frac{1}{|G|}\int_{G}f(x,t)dx, \\ \Psi(t) = K_{\Phi}\int_{\partial G}\psi \frac{\partial \Phi}{\partial \nu}(x) dS, \quad \tilde{\Psi}(t) = \frac{1}{|G|}\int_{\partial G}\tilde{\psi} dS, \\ q_i(t) = \min_{x \in \overline{G}}q_i(x,t), \end{gather*} where $K_{\Phi}=(\int_{G}\Phi(x)dx)^{-1}$ and $|G|=\int_{G}dx$. \begin{theorem} \label{thm1} If the functional differential inequality \begin{equation} \label{e1} \frac{d}{dt}\Big(r(t) \frac{d}{dt}\Big(y(t) + \sum_{i=1}^{l}h_i(t)y(\rho_i(t))\Big)\Big) \\ + \sum_{i=1}^{m}q_i(t)\varphi_i(y(\sigma_i(t))) \le \pm G(t) \end{equation} has no eventually positive solution, then every solution of \eqref{eE}, \eqref{eB1} is oscillatory in $\Omega$, where $$ G(t) = F(t) - a(t)\Psi(t) - \sum_{i=1}^{k}b_i(\tau_i(t))\Psi(\tau_i(t)). $$ \end{theorem} \begin{proof} Suppose to the contrary that there is a non-oscillatory solution $u$ of \eqref{eE}, \eqref{eB1}. Without loss of generality we may assume that $u(x,t) > 0$ in $G \times [t_0,\infty)$ for some $t_0 >0$ because the case $u(x,t) <0$ can be treated similarly. Since (H2) holds, we see that $u(x,\rho_i(t)) > 0$ $(i=1,2,\dots,l)$, $u(x,\tau_i(t)) > 0$ $(i=1,2,\dots,k)$ and $u(x,\sigma_i(t)) > 0$ $(i=1,2,\dots,m)$ in $G \times [t_1,\infty)$ for some $t_1 \ge t_0$. Multiplying \eqref{eE} by $K_{\Phi}\Phi(x)$ and integrating over $G$, we obtain \begin{equation} \label{a1} \begin{split} & \frac{d}{dt}\Big(r(t) \frac{d}{dt}\Big(U(t) + \sum_{i=1}^{l}h_i(t)U(\rho_i(t))\Big)\Big) - a(t)K_{\Phi}\int_{G}\Delta u(x,t)\Phi(x)dx \\ &- \sum_{i=1}^{k}b_i(t) K_{\Phi}\int_{G}\Delta u(x,\tau_i(t))\Phi(x)dx + \sum_{i=1}^{m}K_{\Phi} \int_{G}q_i(x,t) \varphi_i(u(x,\sigma_i(t))) \Phi(x)dx \\ &= F(t), \quad t \ge t_1. \end{split} \end{equation} Using Green's formula, it is obvious that \begin{gather} \label{e3} K_{\Phi}\int_{G}\Delta u(x,t)\Phi(x)dx \le - \Psi(t), \quad t \ge t_1, \\ \label{e4} K_{\Phi}\int_{G}\Delta u(x,\tau_i(t))\Phi(x)dx \le - \Psi(\tau_i(t)), \quad t \ge t_1. \end{gather} An application of Jensen's inequality shows that \begin{equation}\label{a2} \sum_{i=1}^{m}K_{\Phi}\int_{G}q_i(x,t)\varphi_i(u(x, \sigma_i(t)))\Phi(x)dx \ge \sum_{i=1}^{m}q_i(t) \varphi_i(U(\sigma_i(t))) \end{equation} for $t \ge t_1$. Combining \eqref{a1}--\eqref{a2} yields $$ \frac{d}{dt}\Big(r(t) \frac{d}{dt}\Big(U(t) + \sum_{i=1}^{l}h_i(t)U(\rho_i(t))\Big)\Big) + \sum_{i=1}^{m}q_i(t) \varphi_i(U(\sigma_i(t))) \le G(t) $$ for $t \ge t_1$. Therefore, $U(t)$ is an eventually positive solution of \eqref{e1}. This contradicts the hypothesis and completes the proof. \end{proof} \begin{theorem} \label{thm2} If the functional differential inequality \begin{equation} \label{e2} \frac{d}{dt}\Big(r(t) \frac{d}{dt}\Big(y(t) + \sum_{i=1}^{l}h_i(t)y(\rho_i(t))\Big)\Big) + \sum_{i=1}^{m}q_i(t)\varphi_i(y(\sigma_i(t)))\le \pm \tilde{G}(t) \end{equation} has no eventually positive solution, then every solution of \eqref{eE}, \eqref{eB2} is oscillatory in $\Omega$, where $$ \tilde{G}(t) = \tilde{F}(t) + a(t)\tilde{\Psi}(t) + \sum_{i=1}^{k}b_i(\tau_i(t))\tilde{\Psi}(\tau_i(t)). $$ \end{theorem} \begin{proof} Suppose to the contrary that there is a non-oscillatory solution $u$ of \eqref{eE}, \eqref{eB2}. Without loss of generality we may assume that $u(x,t) > 0$ in $G \times [t_0,\infty)$ for some $t_0 >0$. Since (H2) holds, we see that $u(x,\rho_i(t)) > 0$ $(i=1,2,\dots,l)$, $u(x,\tau_i(t)) > 0$ $(i=1,2,\dots,k)$ and $u(x,\sigma_i(t)) > 0$ $(i=1,2,\dots,m)$ in $G \times [t_1,\infty)$ for some $t_1 \ge t_0$. Dividing \eqref{eE} by $|G|$ and integrating over $G$, we obtain \begin{equation} \label{b1} \begin{split} & \frac{d}{dt}\Big(r(t) \frac{d}{dt}\Big(\tilde{U}(t) + \sum_{i=1}^{l}h_i(t)\tilde{U}(\rho_i(t))\Big)\Big) - \frac{a(t)}{|G|}\int_{G}\Delta u(x,t)dx \\ &- \sum_{i=1}^{k}\frac{b_i(t)}{|G|}\int_{G}\Delta u(x,\tau_i(t))dx + \frac{1}{|G|}\sum_{i=1}^{m}\int_{G}q_i(x,t) \varphi_i(u(x,\sigma_i(t)))dx \\ &= \tilde{F}(t), \quad t \ge t_1. \end{split} \end{equation} It follows from Green's formula that \begin{gather} \label{e8} \frac{1}{|G|}\int_{G}\Delta u(x,t)dx \le \tilde{\Psi}(t), \quad t \ge t_1, \\ \frac{1}{|G|}\int_{G}\Delta u(x,\tau_i(t))dx \le \tilde{\Psi}(\tau_i(t)), \quad t \ge t_1. \end{gather} Applying Jensen's inequality, we observe that \begin{equation}\label{b2} \frac{1}{|G|}\sum_{i=1}^{m}\int_{G}q_i(x,t) \varphi_i(u(x,\sigma_i(t)))dx \ge \sum_{i=1}^{m}q_i(t) \varphi_i(\tilde{U}(\sigma_i(t))), \quad t \ge t_1. \end{equation} This together with \eqref{b1}--\eqref{b2} yield $$ \frac{d}{dt}\Big(r(t) \frac{d}{dt}\Big(\tilde{U}(t) + \sum_{i=1}^{l}h_i(t)\tilde{U}(\rho_i(t))\Big)\Big) + \sum_{i=1}^{m}q_i(t) \varphi_i(\tilde{U}(\sigma_i(t))) \le \tilde{G}(t) $$ for $t \ge t_1$. Hence $\tilde{U}(t)$ is an eventually positive solution of \eqref{e2}. This contradicts the hypothesis and completes the proof. \end{proof} \section{Second-order functional differential inequalities} We look for sufficient conditions so that the functional differential inequality \begin{equation}\label{3} \frac{d}{dt}\Big(r(t) \frac{d}{dt}\Big(y(t) + \sum_{i=1}^{l}h_i(t)y(\rho_i(t))\Big)\Big) + \sum_{i=1}^{m}q_i(t)\varphi_i(y(\sigma_i(t))) \le f(t) \end{equation} has no eventually positive solution, where $f(t) \in C([0,\infty) ; \mathbb{R})$. \subsection{Case: (C1) is satisfied} We assume the following hypotheses: \begin{itemize} \item[(H4)] For some $j \in \{1,2,\dots,m\}$, there exists a positive constant $\sigma$ such that $\sigma'_j(t) \ge \sigma$, $t \ge \sigma_j(t)$, $\varphi'_j(s) > 0$ and $\varphi'_j(s)$ is nondecreasing for $s > 0$; \item[(H5)] $\rho _{i}(t) \le t$ $(i=1,2,\dots ,l)$; \item[(H6)] $ \sum_{i=1}^{l}h_{i}(t) \le h < 1$ for some $h >0$; \item[(H7)] there exists $T\geq 0$ such that $T\leq a < b$ and $f(t) \le 0$ for all $t \in [a,b]$. \end{itemize} \begin{theorem} \label{thm3} Assume that {\rm (C1), (H4)--(H7)} hold. If the Riccati inequality \begin{equation}\label{re1} z'(t) + \frac{1}{2}\frac{1}{P_{K}(t)}z^{2}(t) \le -q_j(t) \end{equation} has no solution on $[T,\infty)$ for all large $T$, then \eqref{3} has no eventually positive solution, where $$ P_{K}(t) = \frac{r(\sigma_j(t))}{2 K (1-h) \sigma}. $$ \end{theorem} \begin{proof} Suppose that $y(t)$ is a positive solution of \eqref{3} on $[t_{0},\infty )$ for some $t_{0}>0$. From \eqref{3} there exist $j \in \{1,2,\dots,m\}$ and $a, b \ge t_0$ such that $f(t) \le 0$ on the interval $I \in [a,b]$, and so, $$ \frac{d}{dt}\Big( r(t)\frac{d}{dt}\Big( y(t)+\sum_{i=1}^{l}h_{i}(t) y(\rho_{i}(t))\Big) \Big) +q_j(t)\varphi_{j}(y(\sigma_j(t))) \le 0, \quad t \in I $$ for $t \ge t_0$. If we set the function $$ z(t) = y(t)+\sum_{i=1}^{l}h_{i}(t)y(\rho_{i}(t)), $$ then we see that \begin{equation}\label{c1} (r(t)z'(t))' \le -q_{j}(t)\varphi _{j}(y(\sigma_{j}(t))) \le 0, \quad t \ge t_{0}. \end{equation} Then we conclude that $z'(t) \ge 0$ or $z'(t) < 0$, $t \ge t_1$ for some $t_1 \ge t_0$. From the well known argument (cf. Yoshida \cite{y2}), we see that $z'(t) \ge 0$, $z(t) \ge 0$ and $$ y(\sigma_j(t)) \ge (1 - h) z(\sigma_j(t)), \ t \ge t_2 $$ for some $t_2 \ge t_1$. Setting $$ w(t) = \frac{r(t)z'(t)}{\varphi _{j}((1-h)z(\sigma_{j}(t)))}, $$ we show that \begin{equation} \label{c2} \begin{split} w'(t) &= \frac{(r(t)z'(t))'}{\varphi_{j}((1-h)z(\sigma_{j}(t)))} - (1-h)r(t)z'(t)\frac{\varphi _{j}'((1-h)z(\sigma_{j}(t))) z'(\sigma_{j}(t))\sigma_{j}'(t)}{\varphi_{j}^{2}((1-h) z(\sigma_{j}(t)))} \\ &\leq - q_j(t)\frac{\varphi_{j}(y(\sigma_j(t)))}{\varphi_{j}((1-h) z(\sigma_{j}(t)))} - \frac{(1-h) \sigma \varphi_j'((1-h)z(\sigma_j(t)))}{r(\sigma_j(t))} w^2(t), \quad t \ge t_2. \end{split} \end{equation} It follows from (H4) that \begin{equation}\label{c3} \varphi_j'((1-h)z(\sigma_j(t))) \ge \varphi_j'((1-h) k) \equiv K, \quad t \ge t_2. \end{equation} Combining \eqref{c3} and \eqref{c2}, we have \begin{equation}\label{c5} w'(t) + \frac{1}{2}\frac{1}{P_{K}(t)} w^{2}(t) \le -q_j(t), \quad t \ge t_2. \end{equation} That is, $w(t)$ is a solution of \eqref{3} on $[t_2,\infty)$. This is a contradiction and the proof is complete. \end{proof} \begin{itemize} \item[(H8)] There exists an oscillatory function $\theta(t)$ such that $$ \left(r(t) \theta'(t)\right)' = f(t) \quad \text{and} \quad \lim_{t \to \infty}\tilde{\theta}(t) = 0, $$ where $$ \tilde{\theta}(t) = \theta(t) - \sum_{i=1}^{l}h_i(t)\theta(\rho_i(t)). $$ \end{itemize} \begin{theorem} \label{thm4} Assume that {\rm (C1), (H4)--(H6), (H8)} hold. If the Riccati inequality \eqref{re1} has no solution on $[T,\infty )$ for all large $T$, then \eqref{3} has no eventually positive solutions. \end{theorem} \begin{proof} Suppose that $y(t)$ is a positive solution of \eqref{3} on $[t_{0},\infty )$ for some $t_{0}>0$. From \eqref{3} there exists $j \in \{1,2,\dots,m\}$ such that $$ \frac{d}{dt}\Big(r(t)\frac{d}{dt}\Big(y(t) +\sum_{i=1}^{l}h_{i}(t)y(\rho_{i}(t))\Big)\Big) + q_{j}(t)\varphi_{j}(y(\sigma _{j}(t))) \le f(t), \quad t \ge t_0. $$ Define the function $\tilde{z}(t)$ by $$ \tilde{z}(t) = y(t) + \sum_{i=1}^{l}h_i(t)y(\rho_i(t)) - \theta(t), $$ then it obvious that \begin{equation}\label{d1} (r(t) \tilde{z}'(t))' \le -q_j(t)\varphi_j(y(\sigma_j(t))) \le 0, \quad t \ge t_0, \end{equation} so that $\tilde{z}'(t) \ge 0$ or $\tilde{z}'(t) < 0$, $t \ge t_1$ for some $t_1 \ge t_0$. By standard arguments (cf. Yoshida \cite{y2}), we see that $\tilde{z}'(t) \ge 0$, $\tilde{z}(t) \ge 0$ and $$ y(t) \ge (1-h) \tilde{z}(t) + \tilde{\theta}(t), \quad t \ge t_2 $$ for some $t_2 \ge t_1$. Since (H8) holds, there exists a number $t_3 \ge t_2$ such that $$ |\tilde{\theta}(t)| \le \frac{(1-h) k}{2}, \quad t \ge t_3. $$ In view of $\tilde{z}(t) \ge k$, we observe that \begin{equation} \label{d2} y(t) \ge (1-h) \tilde{z}(t) - \frac{(1-h)k}{2} \\ \ge \frac{(1-h)k}{2} \equiv \tilde{k} > 0, \quad t \ge t_3. \end{equation} Setting $$ \tilde{w}(t) = \frac{r(t) \tilde{z}'(t)}{\varphi_j \big((1-h) \tilde{z}(\sigma_j(t)) - \tilde{k}\big)}, $$ for $t \ge t_3$, we have \begin{equation}\label{d3} \begin{split} \tilde{w}'(t) &= \frac{(r(t) \tilde{z}'(t))'}{\varphi_j \big((1-h)\tilde{z}(\sigma_j(t)) - \tilde{k}\big)}\\ &\quad - r(t)\tilde{z}'(t)\frac{\varphi_j'\big((1-h) \tilde{z}(\sigma_j(t)) - \tilde{k}\big) (1-h) \tilde{z}'(\sigma_j(t)) \sigma_j'(t)}{\varphi_j^2 \big((1-h) \tilde{z}(\sigma_j(t)) - \tilde{k}\big)} \\ &\leq - q_j(t)\frac{\varphi_j(y(\sigma_j(t)))}{\varphi_j \big((1-h) \tilde{z}(\sigma_j(t)) - \tilde{k}\big)} - \frac{(1-h) \sigma \varphi_j'\big((1-h)\tilde{z}(\sigma_j(t)) - \tilde{k}\big)}{r(\sigma_j(t))} \tilde{w}^2(t). \end{split} \end{equation} It follow from \eqref{d2} and (H4) that \begin{equation}\label{d4} \varphi_j'\left((1-h) \tilde{z}(\sigma_j(t)) - \tilde{k}\right) \ge \varphi_j'(\tilde{k}) \equiv K, \quad t \ge t_3. \end{equation} Combining \eqref{d3} with \eqref{d4} yields \begin{equation}\label{d5} \tilde{w}'(t) + \frac{1}{2}\frac{1}{P_K(t)} \tilde{w}^2(t) \le -q_j(t), \quad t \ge t_3. \end{equation} Therefore, $\tilde{w}(t)$ is a solution of \eqref{re1}. This contradicts the hypothesis and completes the proof. \end{proof} \begin{theorem} \label{thm5} Assume that {\rm (C1) (H4)}--{\rm (H7)} (or that {\rm (H4)--(H6), (H8)}) hold. If for each $T >0$ and some $K > 0$, there exist $(H_1,H_2)\in \mathbb{H}$, $\phi(t)\in C^{1}((0,\infty);(0,\infty ))$ and $a,b,c \in {\mathbb{R}}$ such that $T \le a 0, \end{split} \end{equation} where $$ \lambda_1(s,t) = \frac{\phi'(s)}{\phi(s)} + h_1(s,t), \quad \lambda _2(t,s) = \frac{\phi'(s)}{\phi (s)} - h_2(t,s). $$ Then \eqref{3} has no eventually positive solutions. \end{theorem} \begin{proof} Suppose that $y(t)$ is a positive solution of \eqref{3} on $[t_0,\infty )$ for some $t_0 >0$. Proceeding as in the proof of Theorem \ref{thm3}, multiplying \eqref{c5} or \eqref{d5} by $H_2(t,s)$ and integrating over $[c,t]$ for $t \in [c,b)$, we have \begin{align*} &\int_{c}^{t}H_2(t,s)q_j(s)\phi(s)ds \\ &\leq -\int_{c}^{t}H_2(t,s)w'(s)\phi(s)ds - \frac{1}{2}\int_{c}^{t}H_2(t,s)\frac{1}{P_{K}(s)}w^{2}(s)\phi(s)ds \\ &\leq H_2(t,c)w(c)\phi(c) + \frac{1}{2}\int_{c}^{t}H_2(t,s)P_{K}(s)\lambda_2^{2}(t,s)\phi(s)ds \\ & \quad - \frac{1}{2}\int_{c}^{t}H_2(t,s) \{w(s)/\sqrt{P_{K}(s)} - \lambda_2(t,s) \sqrt{P_{K}(s)}\}^{2}\phi(s)ds, \end{align*} and so $$ \frac{1}{H_2(t,c)}\int_{c}^{t}H_2(t,s)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda _2^{2}(t,s)\} \phi(s)ds \le w(c)\phi (c). $$ Letting $t \to b^{-}$ in the last inequality, we obtain \begin{equation}\label{e1b} \frac{1}{H_2(b,c)}\int_{c}^{b}H_2(b,s)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda_2^{2}(b,s)\} \phi(s)ds \le w(c)\phi(c). \end{equation} On the other hand, multiplying \eqref{c5} by $H_1(s,t)$, integrating over $[t,c]$ for $t \in (a,c]$ and letting $t \to a^{+}$, we obtain \begin{equation}\label{e2b} \frac{1}{H_1(c,a)}\int_{a}^{c}H_1(s,a)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda_1^{2}(s,a)\} \phi(s) ds \le -w(c)\phi(c). \end{equation} Adding \eqref{e1} and \eqref{e2}, we obtain \begin{align*} & \frac{1}{H_1(c,a)}\int_{a}^{c}H_1(s,a)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda_1^{2}(s,a)\} \phi(s) ds \\ & + \frac{1}{H_2(b,c)}\int_{c}^{b}H_2(b,s)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda _2^{2}(b,s)\} \phi(s) ds \le 0, \end{align*} which is contrary to \eqref{ec1}. Pick up a sequence $\{T_{i}\} \subset [t_0,\infty)$ such that $T_{i} \to \infty$ as $i \to \infty$. By the assumptions, for each $i \in {\mathbb{N}}$, there exists $a_{i},\ b_{i},\ c_{i}\in [0,\infty )$ such that $T_{i} \leq a_{i} < c_{i} < b_{i}$, and \eqref{ec1} holds with $a,\ b,\ c$ replaced by $a_{i},\ b_{i},\ c_{i}$, respectively. Therefore, every solution $y(t)$ of \eqref{3} has at least one zero $t_{i} \in (a_{i},b_{i})$. The case when \eqref{d5} follows by a similar arguments. This is a contradiction and the proof is complete. \end{proof} \begin{theorem} \label{thm6} Assume {\rm (C1), (H4)--(H7)} (or {\rm (H4)--(H6), (H8)}). If for each $T >0$ and some $K >0$, there exist functions $(H_1,H_2)\in \mathbb{H}$, $\phi (t) \in C^{1}((0,\infty);(0,\infty))$, such that \begin{equation}\label{fc1} \limsup_{t \to \infty}\int_{T}^{t}H_1(s,T)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda_1^{2}(s,T)\} \phi(s)ds > 0 \end{equation} and \begin{equation}\label{fc2} \limsup_{t \to \infty}\int_{T}^{t}H_2(t,s)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda_2^{2}(t,s)\} \phi(s)ds > 0, \end{equation} then \eqref{3} has no eventually positive solutions. \end{theorem} \begin{proof} For any $T\geq t_{0}$, let $a=T$ and choose $T=a$ in \eqref{ec1}. Then there exists $c>a$ such that \begin{equation}\label{f1} \int_{a}^{c}H_1(s,a)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda_1^{2}(s,a)\} \phi(s)ds > 0. \end{equation} Next, choose $T=c$ in \eqref{fc2}. Then there exists $b>c$ such that \begin{equation}\label{f2} \int_{c}^{b}H_2(b,s)\{q_j(s) - \frac{1}{2}P_{K}(s)\lambda_2^{2}(b,s)\} \phi(s)ds > 0. \end{equation} Combining \eqref{f1} and \eqref{f2}, we obtain \eqref{ec1}. By Theorem \ref{thm5}, the proof is complete. \end{proof} \subsection{Case: (C2) is satisfied} We use the following notation: \begin{gather*} \rho_{*}(t) = \min_{1 \le i \le l}\rho_i(t), \quad \pi(t) = \int_{t}^{\infty}\frac{1}{r(s)}ds, \\ A(t) = 1 - \sum_{i=1}^{l}h_i(t) - \log{\frac{\pi(\rho_{*}(t))}{\pi(t)}}, \quad [\delta(t)]_{\pm} = \max\{0, \pm\delta(t)\}. \end{gather*} \begin{theorem} \label{thm7} Assume that{\rm (C2), (H4)--(H7)} hold. If the Riccati inequality \begin{equation}\label{re2} z'_i(t) + \frac{1}{2}\frac{1}{P_i(t)}z^{2}_i(t) \leq -Q_i(t) \ (i=1,2) \end{equation} has no solution on $[T,\infty)$ for all large $T$, then \eqref{3} has no eventually positive solutions, where \begin{gather*} P_1(t) = P_{K}(t), \quad P_2(t) = \frac{r(t)}{2\varphi'_j(c_1 \pi(t))}, \\ Q_1(t) = q_j(t), \ Q_2(t) = q_j(t)\frac{\varphi_j \big([c_1 A(\sigma_j(t)) \pi(\rho_{*}(\sigma_j(t)))]_+ \big)} {\tilde{K}}. \end{gather*} \end{theorem} \begin{proof} Suppose that $y(t)$ is a positive solution of \eqref{3} on $[t_0,\infty)$ for some $t_0 >0$. Proceeding as in the proof of Theorem \ref{thm3}, we obtain the inequality \eqref{c1}. Thus we see that $z'(t) \ge 0$, $z(t) \ge 0$ or $z'(t) < 0$, $z(t) \ge 0$, $t \ge t_1$ for some $t_1 \ge t_0$. \noindent\textbf{Case 1.} $z'(t) \ge 0$, $z(t) \ge 0$ for $t \ge t_1$. The proof of this case is similar as Theorem \ref{thm3}, and so we omit it. \noindent\textbf{Case 2.} $z'(t) < 0$, $z(t) \ge 0$ for $t \ge t_1$. Then there exists a constant $k_1 >0$ such that $z(t) \le k_1$, $t \ge t_2$ for some $t_2 \ge t_1$. Consequently we have \begin{equation}\label{g1} \varphi_j(z(t)) \le \varphi_j(k_1) \equiv \tilde{K}, \quad t \ge t_2. \end{equation} If we define $$ w_2(t) = \frac{r(t) z'(t)}{\varphi_j(z(t))}, $$ then \begin{equation} \label{g2} \begin{split} w'_2(t) &= \frac{(r(t) z'(t))'}{\varphi_j(z(t))} - r(t)z'(t)\frac{\varphi'_j(z(t)) z'(t)}{\varphi_{j}^{2}(z(t))} \\ &\leq -q_j(t)\frac{\varphi_j(y(\sigma_j(t)))}{\varphi_j(z(t))} - \frac{\varphi'_j(z(t))}{r(t)}w_2^{2}(t), \quad t \ge t_2. \end{split} \end{equation} Using \cite[Lemma 5.2]{t1}, we see that $z(t) \ge c_1 \pi(t)$, $t \ge t_3$ for some $t_3 \ge t_2$, and that \begin{equation}\label{g3} \varphi'_j(z(t)) \ge \varphi'_j(c_1 \pi(t)), \quad t \ge t_3. \end{equation} By \cite[Theorem 3.2]{t1}, we show that $$ y(t) \ge c_1 A(t) \pi(\rho_*(t)), \quad t \ge t_3, $$ and that \begin{equation}\label{g4} \varphi_j(y(\sigma_j(t))) \ge \varphi_j([c_1 A(\sigma_j(t)) \pi(\rho_*(\sigma_j(t)))]_{+}), \quad t \ge t_3. \end{equation} Combining \eqref{g1}--\eqref{g4}, we can derive the inequality $$ w'_2(t) + \frac{1}{2}\frac{1}{P_2(t)}w_2^{2}(t) \le -Q_2(t), \quad t \ge t_3. $$ Therefore, $w_2(t)$ is a solution of \eqref{re2}. This contradicts the hypothesis and completes the proof. \end{proof} \begin{theorem} \label{thm8} Assume that {\rm (C2), (H4)--(H6), (H8)} hold. If the Riccati inequality \begin{equation}\label{re3} z'_i(t)+\frac{1}{2}\frac{1}{P_i(t)}z^{2}_i(t) \leq -\tilde{Q}_i(t) \quad (i=1,2) \end{equation} has no solution on $[T,\infty)$ for all large $T$, then \eqref{3} has no eventually positive solutions, where $$ \tilde{Q}_1(t) = q_j(t), \quad \tilde{Q}_2(t)= q_j(t)\frac{\varphi_j\Big([c_1 A(\sigma_j(t)) \pi(\rho_{*}(\sigma_j(t))) + \tilde{\theta}(\sigma_j(t))]_+\Big)} {\tilde{K}}. $$ \end{theorem} \begin{proof} Suppose that $y(t)$ is a positive solution of \eqref{3} on $[t_0,\infty)$ for some $t_0 >0$. Proceeding as in the proof of Theorem \ref{thm4}, we see that $\tilde{z}'(t) \ge 0$, $\tilde{z}(t) \ge 0$ or $\tilde{z}'(t) < 0$, $\tilde{z}(t) \ge 0$, $t \ge t_1$ for some $t_1 \ge t_0$. \noindent\textbf{Case 1.} $\tilde{z}'(t) \ge 0$, $\tilde{z} \ge 0$. Then it can be treated similarly as in the proof of Theorem \ref{thm4}. \noindent\textbf{Case 2.} $\tilde{z}'(t) < 0$, $\tilde{z}(t) \ge 0$. By Tanaka \cite[Theorem 3.2]{t1}, we obtain $$ y(\sigma_j(t)) \ge [c_1 A(\sigma_j(t)) \pi(\rho_{*}(\sigma_j(t))) + \tilde{\theta}(\sigma_j(t))]_{+}, \quad t \ge t_2. $$ Setting $\tilde{w}_2(t) = w_2(t)$, it obvious that $$ \tilde{w}'_2(t) \le -q_j(t) \frac{\varphi_j(y(\sigma_j(t)))}{\varphi_j(z(t))} - \frac{\varphi'_j(z(t))}{r(t)}\tilde{w}_2^{2}(t), \quad t \ge t_2. $$ Substituting \eqref{g1} and \eqref{g3} into this inequality yields $$ \tilde{w}'_2(t) + \frac{1}{2}\frac{1}{P_2(t)}\tilde{w}_2^{2}(t) \le -q_j(t)\frac{\varphi_j(y(\sigma_j(t)))}{\tilde{K}}. $$ It is clear that $\tilde{w}_2(t)$ is a solution of \eqref{re3}. This contradicts the hypothesis and completes the proof. \end{proof} \begin{theorem} \label{thm9} Assume that {\rm (C2), (H4)--(H7)} hold. If for each $T >0$ and some $K > 0$, $\tilde{K} >0$ there exist $(H_1,H_2)\in \mathbb{H}$, $\phi(t)\in C^{1}((0,\infty);(0,\infty ))$ and $a,b,c \in {\mathbb{R}}$ such that $T \le a 0 \end{split} \end{equation} hold, then \eqref{3} has no eventually positive solutions. \end{theorem} \begin{theorem} \label{thm10} Assume that {\rm (C2), (H4)--(H7)} hold. If for each $T >0$ and some $K >0$, $\tilde{K} >0$, there exist functions $(H_1,H_2)\in \mathbb{H}$, $\phi (t) \in C^{1}((0,\infty);(0,\infty))$, such that \eqref{fc1}, \eqref{fc2} and \begin{equation}\label{jc1} \limsup_{t \to \infty}\int_{T}^{t}H_1(s,T)\{Q_2(s) - \frac{1}{2}P_2(s)\lambda_1^{2}(s,T)\} \phi(s)ds > 0 \end{equation} and \begin{equation}\label{jc2} \limsup_{t \to \infty}\int_{T}^{t}H_2(t,s)\{Q_2(s) - \frac{1}{2}P_2(s)\lambda_2^{2}(t,s)\} \phi(s)ds > 0, \end{equation} then \eqref{3} has no eventually positive solutions. \end{theorem} \begin{theorem} \label{thm11} Assume that {\rm (C2), (H4)--(H6), (H8)} hold. If for each $T >0$ and some $K > 0$, $\tilde{K} > 0$, there exist $(H_1,H_2)\in \mathbb{H}$, $\phi(t)\in C^{1}((0,\infty);(0,\infty ))$ and $a,b,c \in {\mathbb{R}}$ such that $T \le a 0 \end{split} \end{equation} hold, then \eqref{3} has no eventually positive solutions. \end{theorem} \begin{theorem} \label{thm12} Assume that {\rm (C2), (H4)--(H6), (H8)} hold. If for each $T >0$ and some $K > 0$, $\tilde{K} >0$, there exist functions $(H_1,H_2)\in \mathbb{H}$, $\phi (t) \in C^{1}((0,\infty);(0,\infty))$, such that \eqref{fc1}, \eqref{fc2} and \begin{equation}\label{lc1} \limsup_{t \to \infty}\int_{T}^{t}H_1(s,T)\{\tilde{Q}_2(s) - \frac{1}{2}P_2(s)\lambda_1^{2}(s,T)\} \phi(s)ds > 0 \end{equation} and \begin{equation}\label{lc2} \limsup_{t \to \infty}\int_{T}^{t}H_2(t,s)\{\tilde{Q}_2(s) - \frac{1}{2}P_2(s)\lambda_2^{2}(t,s)\} \phi(s)ds > 0, \end{equation} then \eqref{3} has no eventually positive solutions. \end{theorem} \section{Oscillation criteria for \eqref{eE}} In this section, by combining the results of Sections 2 and 3, we establish sufficient conditions for oscillation of solutions to \eqref{eE}. \begin{itemize} \item[(H9)] There exists $T \le a < b \le \tilde{a} < \tilde{b}$ such that \[ G(t) \ [\text{resp. } \tilde{G}(t)] = \begin{cases} \le 0, & t \in [a,b], \\ \ge 0, & t \in [\tilde{a},\tilde{b}] \end{cases} \] for each $T \ge 0$; \item[(H10)] there exists an oscillatory function $\Theta(t)$ such that $$ \Big(r(t) \Theta'(t)\Big)' = G(t) \ [\text{resp}. \tilde{G}(t)], \quad \lim_{t \to \infty}\tilde{\Theta}(t)=0, $$ where $$ \tilde{\Theta}(t) = \Theta(t) - \sum_{i=1}^{l}h_i(t)\Theta(\rho_i(t)). $$ \end{itemize} Using the Riccati inequality, we derive sufficient conditions for every solution of hyperbolic equation \eqref{eE} to be oscillatory. We are going to use the following lemma which is due to Usami \cite{u1}. \begin{lemma} \label{lem1} If there exists a function $\phi(t) \in C^{1}([T_0,\infty );(0,\infty ))$ such that \begin{gather*} \int_{T_1}^{\infty} \Big(\frac{\bar{p}(t)|\phi'(t)|^{\beta }}{\phi(t)}\Big)^{1/(\beta - 1)} dt < \infty, \quad \int_{T_1}^{\infty}\frac{1}{\bar{p}(t)(\phi(t))^{\beta -1}}dt = \infty, \\ \int_{T_1}^{\infty}\phi (t)\bar{q}(t)dt = \infty \end{gather*} for some $T_1 \ge T_0$, then the Riccati inequality $$ x'(t) + \frac{1}{\beta }\frac{1}{\bar{p}(t)}|x(t)|^{\beta } \le -\bar{q}(t) $$ has no solution on $[T,\infty)$ for all large $T$, where $\beta >1$, $\bar{p}(t) \in C([T_0,\infty );(0,\infty ))$ and $\bar{q}(t)\in C([T_{0},\infty );{\mathbb{R}})$, \end{lemma} \subsection{Oscillation results by Riccati inequality for case (C1)} Combining Theorems \ref{thm1}--\ref{thm4} and Lemma \ref{lem1}, we obtain the following theorem. \begin{theorem} \label{thm13} Assume that {\rm (C1), (H1)--(H6), (H9)} (or {\rm (H1)--(H6), (H10)}) and that if \[ \int_{T_1}^{\infty}\Big(\frac{P_K(t) \phi'(t)^2}{\phi(t)}\Big) dt < \infty , \quad \int_{T_1}^{\infty}\frac{1}{P_K(t)\phi(t)}dt = \infty, \quad \int_{T_1}^{\infty}\phi(t)q_j(t)dt = \infty, \] then every solution $u(x,t)$ of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega $. \end{theorem} \begin{example} \label{exa1} \rm Consider the equation \begin{gather} \label{ex11} \begin{split} & \frac{\partial}{\partial t}\Big(e^{-2t} \frac{\partial}{\partial t} \Big(u(x,t) + \frac{1}{2}u(x,t-\pi)\Big) -e^{-3t}\Delta u(x,t) \\ &- \frac{1}{2}e^{-2t}\Delta u(x,t-2\pi) - \left(e^{-t}+e^{-2t}\right)\Delta u\big(x,t-\frac{3}{2}\pi\big) + e^{-t}u\big(x,t-\frac{\pi}{2}\big)\\ & = e^{-3t}\sin{x}\sin{t}, \ (0,\pi) \times (0,\infty), \end{split}\\ u(0,t) = u(\pi,t) = 0, \quad t > 0. \label{ex12} \end{gather} \end{example} Here $l=m=1$, $k=2$, $r(t)=e^{-2t}$, $h_1(t)=1/2$, $\rho_1(t)=t-\pi$, $q_1(x,t)=e^{-t}$, $\sigma_1(t)=t-\pi/2$ and $f(x,t)=e^{-3t}\sin{x}\sin{t}$. It is easy to see that $\Phi(x)=\sin{x}$ and $$ G(t) = F(t) = \frac{\pi}{4}e^{-3t}\sin{t}, \quad \tilde{\Theta}(t) = \frac{\pi}{16}\big(1+\frac{1}{2}e^{\pi}\big)e^{-t}\cos{t}. $$ Then $\int^{\infty}e^{-t}dt < \infty$; hence, \cite[Corollary 2.1]{t1} is not applicable to this problem. Taking $\phi(t) = e^t$, we find \begin{gather*} \int^{\infty}\Big(\frac{P_K(t) \phi'(t)^2}{\phi(t)}\Big)dt = \int^{\infty}\Big(\frac{e^{-2t+\pi} \cdot e^{2t}}{e^t}\Big)dt < \infty, \\ \int^{\infty}\Big(\frac{1}{P_K(t)\phi(t)}\Big)dt = \int^{\infty}\Big(\frac{1}{e^{-2t+\pi} \cdot e^t}\Big)dt = \infty, \\ \int^{\infty}\phi(t)q_1(t)dt = \int^{\infty}\left(e^{t} \cdot e^{-t}\right)dt = \infty. \end{gather*} It follows from Theorem \ref{thm13} that every solution $u$ of \eqref{ex11}, \eqref{ex12} is oscillatory in $(0,\pi) \times (0,\infty)$. For example, $u=\sin{x}\sin{t}$ is such a solution. \subsection{Interval oscillation results for case (C1)} Combining Theorems \ref{thm1}, \ref{thm2}, \ref{thm5}, and \ref{thm6}, we have the following theorems. \begin{theorem} \label{thm14} Assume that {\rm (C1), (H1)--(H6), (H9)} hold. If for each $T >0$ and some $K >0$, there exist functions $(H_1,H_2)\in \mathbb{H}$, $\phi (t) \in C^{1}((0,\infty);(0,\infty))$ and $a,b,c,\tilde{a},\tilde{b},\tilde{c} \in {\mathbb{R}}$ such that $T \le a 0 \end{align*} hold, then every solution $u(x,t)$ of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega $. \end{theorem} \begin{theorem} \label{thm15} Assume that {\rm (C1), (H1), (H6), (H10)} hold. If for each $T >0$ and some $K >0$, there exist functions $(H_1,H_2)\in \mathbb{H}$, $\phi (t) \in C^{1}((0,\infty);(0,\infty))$ and $a,b,c\in {\mathbb{R}}$ such that $T \le a 0$, the conditions \eqref{fc1} and \eqref{fc2} hold, then every solution of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega$. \end{theorem} \begin{example} \label{exa2} \rm Consider the problem \begin{gather} \label{ex21} \begin{split} & \frac{\partial^2}{\partial t^2}\Big(u(x,t) + \frac{1}{2}u(x,t-\pi)\Big) - \Delta u(x,t) - 5t^{-2}\Delta u(x,t-2\pi) + 5t^{-2}u(x,t-\pi) \\ &= \frac{1}{2}\sin{x}\sin{t}, \quad (0,\pi) \times (0,\infty), \end{split} \\ u(0,t) = u(\pi,t) = 0, \quad t > 0. \label{ex22} \end{gather} Here $l=k=m=1$, $r(t)=1$, $h_1(t)=1/2$, $\rho_1(t)=t-\pi$, $q_1(x,t)=5t^{-2}$, $\sigma_1(t)=t-\pi$ and $f(x,t)=\frac{1}{2}\sin{x}\sin{t}$. \end{example} It is easy to verify that $\Phi(x)=\sin{x}$ and $$ G(t) = F(t) = \frac{\pi}{8}\sin{t} \quad \text{and} \quad \tilde{\Theta}(t) = -\frac{3}{16}\pi\sin{t}. $$ Since $$ \int^{\infty}5t^{-2}[\frac{1}{2} \pm \frac{3}{16}\pi\sin{t}]_{+}dt < \infty, $$ Then \cite[Theorem 2.1]{t1} does not apply; however, by choosing $\phi(t)=t^2$ and $H_1(s,t)=H_2(t,s)=(t-s)^2$, $$ \limsup_{t \to \infty}\int_{T}^{t}(s-T)^2 \{5s^{-2} - \frac{1}{2}\frac{1}{2}\frac{4T^2}{s^2 (s-T)^2}\}s^2 ds > 0 $$ and $$ \limsup_{t \to \infty}\int_{T}^{t}(t-s)^2 \{5s^{-2} - \frac{1}{2}\frac{1}{2}\frac{4(t-2s)^2}{s^2 (t-s)^2}\}s^2 ds > 0 $$ hold. Therefore, Theorem \ref{thm16} implies that every solution $u$ of the problem \eqref{ex21}, \eqref{ex22} is oscillatory in $(0,\pi) \times (0,\infty)$. In fact, one such solution is $u=\sin{x}\sin{t}$. \subsection{Oscillation results by Riccati inequality for case (C2)} Combining Theorems \ref{thm1}, \ref{thm2}, and \ref{thm7}, we have the following theorem. \begin{theorem} \label{thm17} Assume that {\rm (C2), (H1)--(H6), (H9)} hold. If for $i=1,2$, \begin{equation} \label{th15c} \int_{T_1}^{\infty}\Big( \frac{P_i(t) \phi'(t)^2}{\phi(t)}\Big) dt < \infty , \quad \int_{T_1}^{\infty}\frac{1}{P_i(t)\phi(t)}dt = \infty , \quad \int_{T_1}^{\infty}\phi(t)Q_i(t)dt = \infty, \end{equation} then every solution of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega $. \end{theorem} \begin{example} \label{exa3} \rm Consider the equation \begin{gather} \label{ex31} \begin{split} &\frac{\partial}{\partial t}\Big(e^{1/8} \frac{\partial}{\partial t} \Big(u(x,t) + \frac{1}{2}u(x,t-\pi)\Big)\Big) \\ & - \frac{1}{2}e^{1/8}\Delta u(x,t) - \frac{1}{16}e^{1/8}\Delta u\big(x,t-\frac{\pi}{2}\big) + e^{2t}u(x,t-2\pi)\\ &= e^{2t}\sin{x}\sin{t}, \quad (0,\pi) \times (0,\infty), \end{split}\\ u(0,t) = u(\pi,t) = 0, \quad t > 0. \label{ex32} \end{gather} \end{example} Here $l=k=m=1$, $r(t)=e^{t/8}$, $h_1(t)=1/2$, $\rho_1(t)=t-\pi$, $q_1(x,t)=e^{2t}$, $\sigma_1(t)=t-2\pi$ and $f(x,t)=e^{2t}\sin{x}\sin{t}$. It is easy to see that $\Phi(x)=\sin{x}$ and \begin{gather*} \int^{\infty}\Big(\frac{P_1(t) \phi'(t)^2}{\phi(t)}\Big)dt = \int^{\infty}\Big(\frac{e^{\frac{1}{8}(t-2\pi)} \cdot e^{-2t}}{e^{-t}}\Big)dt < \infty, \\ \int^{\infty}\Big(\frac{P_2(t) \phi'(t)^2}{\phi(t)}\Big)dt = \int^{\infty}\Big(\frac{\frac{1}{2}e^{\frac{1}{8}t} \cdot e^{-2t}}{e^{-t}}\Big)dt < \infty, \\ \int^{\infty}\frac{1}{P_1(t)\phi(t)}dt = \int^{\infty}\frac{1}{\big(e^{\frac{1}{8}(t-2\pi)} \cdot e^{-t}\big)}dt = \infty, \\ \int^{\infty}\frac{1}{P_2(t)\phi(t)}dt = \int^{\infty}\frac{1}{\big(\frac{1}{2}e^{\frac{1}{8}t} \cdot e^{-t}\big)}dt = \infty, \\ \int^{\infty}\phi(t)Q_1(t)dt = \int^{\infty} (e^{-t} \cdot e^{2t}) = \infty, \\ \int^{\infty}\phi(t)Q_2(t)dt = \int^{\infty}e^{-t} \cdot e^{2t}\big[c\big(\frac{1}{2} - \frac{\pi}{8}\big) \cdot 8e^{-\frac{1}{8}(t-3\pi)}\big]_{+}dt = \infty, \end{gather*} where $\phi(t)=e^{-t}$. Therefore it follows from Theorem \ref{thm17} that every solution $u$ of problem \eqref{ex31}, \eqref{ex32} is oscillatory in $(0,\pi) \times (0,\infty)$. For example $u=\sin{x}\sin{t}$ is such a solution. Combining Theorems \ref{thm1}, \ref{thm2}, and \ref{thm8}, we have the following result. \begin{theorem} \label{thm18} Assume {\rm (C1), (H1)--(H6), (H10)}. If \eqref{th15c} and $$ \int_{T_1}^{\infty}\phi(t)\tilde{Q}_i(t)dt = \infty \quad (i=1,2) $$ hold, then every solution $u(x,t)$ of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega $, where $$ \tilde{Q}_2(t) = q_j(t) \frac{1}{\tilde{K}} \varphi_j \Big(\big[c_1 A(\sigma_j(t)) \pi(\rho_{*}(\sigma_j(t))) + \tilde{\Theta}(\sigma_j(t))\big]_+\Big). $$ \end{theorem} \subsection{Interval oscillation results for case (C2)} Combining Theorems \ref{thm1}, \ref{thm2}, \ref{thm9}, and \ref{thm10}, we have the following result. \begin{theorem} \label{thm19.} Assume that {\rm (C2), (H1)--(H6), (H9)} hold. If for each $T >0$ and some $K >0$, $\tilde{K} >0$, there exist functions $(H_1,H_2)\in \mathbb{H}$, $\phi (t) \in C^{1}((0,\infty);(0,\infty ))$ and $a,b,c,\tilde{a},\tilde{b},\tilde{c} \in {\mathbb{R}}$ such that $T \le a 0 \end{align*} and \begin{align*} & \frac{1}{H_1(\tilde{c},\tilde{a})}\int_{\tilde{a}}^{\tilde{c}} H_1(s,\tilde{a})\{Q_2(s) - \frac{1}{2}P_2(s)\lambda_1^{2}(s,\tilde{a})\}\phi(s)ds \\ & + \frac{1}{H_2(\tilde{b},\tilde{c})}\int_{\tilde{c}}^{\tilde{b}} H_2(\tilde{b},s)\{Q_2(s) - \frac{1}{2}P_2(s)\lambda_2^{2}(\tilde{b},s)\} \phi(s)ds > 0 \end{align*} hold, then every solution of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega $. \end{theorem} \begin{theorem} \label{thm20} Assume {\rm (C2), (H1)--(H4), (H9)}. Also assume that for some functions $(H_1,H_2) \in \mathbb{H}$, each $T \ge 0$ and some $K > 0$, $\tilde{K} >0$. If \eqref{fc1}, \eqref{fc2}, \eqref{jc1}, and \eqref{jc2} hold, then every solution of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega$. \end{theorem} Combining Theorems \ref{thm1}, \ref{thm2}, \ref{thm11}, and \ref{thm12}, we have the following result. \begin{theorem} \label{thm21} Assume that {\rm (C2), (H1)--(H6), (H10)} hold. If for each $T >0$ and some $K > 0$, $\tilde{K} > 0$, there exist functions $(H_1,H_2) \in \mathbb{H}$, $\phi(t) \in C^{1}((0,\infty );(0,\infty ))$ such that \eqref{ec1} and \eqref{kc1} hold, then every solution of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega $. \end{theorem} \begin{theorem} \label{thm22} Assume {\rm (C2), (H1)--(H4), (H10)}. Also assume that some functions $(H_1,H_2) \in \mathbb{H}$ for each $T \ge 0$ and some $K > 0$, $\tilde{K} > 0$. If \eqref{fc1}, \eqref{fc2}, \eqref{lc1}, \eqref{lc2} hold, then every solution of \eqref{eE}, \eqref{eB1} (or \eqref{eE}, \eqref{eB2}) is oscillatory in $\Omega$. \end{theorem} \begin{example} \label{exa44} \rm Consider the equation \begin{gather} \label{ex41} \begin{split} & \frac{\partial}{\partial t}\Big(t^3 \frac{\partial}{\partial t} \Big(u(x,t) + \frac{1}{2}u(x,t-\pi)\Big)\Big) \\ & - \frac{t^3}{2} \Delta u(x,t) - \big(t + \frac{3}{2}t^2\big)\Delta u\big(x,t-\frac{\pi}{2}\big) + u(x,t-2\pi) \\ &= (\sin{t} - t\cos{t})\sin{x}, \quad (0,\pi) \times (T_0,\infty), \end{split} \\ u(0,t) = u(\pi,t) = 0, \quad t > T_0 = \pi/(1-e^{-1/4}). \label{ex42} \end{gather} \end{example} Here $l=k=m=1$, $r(t)=t^3$, $h_1(t)=1/2$, $\rho_1(t)=t-\pi$, $q_1(x,t)=1$, $\sigma_1(t)=t-2\pi$ and $f(x,t)=(\sin{t} - t\cos{t})\sin{x}$. An easy computation shows that $\Phi(x)=\sin{x}$ and \[ \pi(t) = \frac{1}{2}t^{-2}, \quad \tilde{\Theta}(t)=\frac{\pi}{4} \Big(t^{-2} + \frac{1}{2}(t-\pi)^{-2}\Big)\cos{t}, \quad A(t) = \frac{1}{2} + 2\log\big(\frac{t-\pi}{t}\big) > 0. \] Since $$ \int^{\infty}\big(\frac{1}{2}t^{-2}\big) [cA(t-2\pi)\pi(t-3\pi) \pm \Theta(t-2\pi)]_{+}dt < \infty, $$ Note that \cite[Theorem 3.2]{t1} is not applicable to this problem. However, we see from $\phi(t)=t^3$ and $H_1(s,t)=H_2(t,s)=(t-s)^3$ that \begin{gather*} \limsup_{t \to \infty}\int_{T}^{t}(s-T)^3 \{1 - \frac{1}{2}(s-2\pi)^3 \frac{9T^2}{s^2 (s-T)^2}\}s^3 ds > 0, \\ \limsup_{t \to \infty}\int_{T}^{t}(s-t)^3 \{1 - \frac{1}{2}(s-2\pi)^3 \frac{9(t-2s)^2}{s^2 (s-t)^2}\}s^3 ds > 0, \\ \limsup_{t \to \infty}\int_{T}^{t}(s-T)^3 \{1 - \frac{1}{2} \frac{s^3}{2}\frac{9T^2}{s^2 (s-T)^2}\}s^3 ds > 0, \\ \limsup_{t \to \infty}\int_{T}^{t}(t-s)^3 \{[cA(t-2\pi)\pi(t-3\pi) \pm \tilde{\Theta}(t-2\pi)]_{+} - \frac{1}{2}\frac{s^3}{2}\frac{9(t-2s)^2}{s^2 (s-t)^2}\}s^3 ds > 0. \end{gather*} Therefore, Theorem \ref{thm22} implies that every solution $u$ of the problem \eqref{ex41}, \eqref{ex42} is oscillatory in $(0,\pi) \times (T_0,\infty)$. In fact, one such solution is $u=\sin{x}\sin{t}$. \begin{thebibliography}{99} \bibitem{c1} S. Cui, Z. Xu; Interval oscillation theorems for second order nonlinear partial delay differential equations, \emph{Differ. Equ. 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