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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 901631, 14 pages
http://dx.doi.org/10.1155/2011/901631
Research Article

Oscillation Properties for Second-Order Partial Differential Equations with Damping and Functional Arguments

Department of Mathematics, Qufu Normal University, Shandong, Qufu 273165, China

Received 19 September 2011; Accepted 21 October 2011

Academic Editor: NorioΒ Yoshida

Copyright Β© 2011 Run Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using an integral averaging method and generalized Riccati technique, by introducing a parameter 𝛽β‰₯1, we derive new oscillation criteria for second-order partial differential equations with damping. The results are of high degree of generality and sharper than most known ones.

1. Introduction

Consider the second-order partial delay differential equationπœ•ξ‚€πœ•πœ•π‘‘π‘Ÿ(𝑑)ξ‚πœ•π‘‘π‘’(π‘₯,𝑑)+𝑝(𝑑)πœ•π‘’(π‘₯,𝑑)πœ•π‘‘=π‘Ž(𝑑)Δ𝑒(π‘₯,𝑑)+π‘ ξ“π‘˜=1π‘Žπ‘˜ξ€·(𝑑)Δ𝑒π‘₯,π‘‘βˆ’πœŒπ‘˜ξ€Έβˆ’(𝑑)βˆ’π‘ž(π‘₯,𝑑)𝑓(𝑒(π‘₯,𝑑))π‘šξ“π‘—=1π‘žπ‘—(π‘₯,𝑑)𝑓𝑗𝑒π‘₯,π‘‘βˆ’πœŽπ‘—ξ€Έξ€Έ,(π‘₯,𝑑)βˆˆΞ©Γ—π‘…+≑𝐺,(1.1) where Ξ” is the Laplacian in 𝑅𝑁,𝑅+=[0,∞) and Ξ© is a bounded domain in 𝑅𝑁 with a piecewise smooth boundary πœ•Ξ©.

Throughout this paper, we assume that(H1)π‘Ÿ(𝑑)∈𝐢1(𝑅+,(0,∞)),𝑝(𝑑)∈𝐢(𝑅+,𝑅);(H2)π‘ž(π‘₯,𝑑),π‘žπ‘—(π‘₯,𝑑)∈𝐢(𝐺,𝑅+),π‘ž(𝑑)=minπ‘₯βˆˆπΊπ‘ž(π‘₯,𝑑),π‘žπ‘—(𝑑)=minπ‘₯βˆˆπΊπ‘žπ‘—(π‘₯,𝑑),π‘—βˆˆπΌπ‘š={1,2,…,π‘š};(H3)π‘Ž(𝑑),π‘Žπ‘˜(𝑑),πœŒπ‘˜(𝑑)∈𝐢(𝑅+,𝑅+),limπ‘‘β†’βˆž(π‘‘βˆ’πœŒπ‘˜(𝑑))=∞, and πœŽπ‘— are nonnegative constants, π‘—βˆˆπΌπ‘š, π‘˜βˆˆπΌπ‘ ={1,2,…,𝑠};(H4)𝑓(𝑒)∈𝐢1(𝑅,𝑅),𝑓𝑗(𝑒)∈𝐢(𝑅,𝑅) are convex in 𝑅+ with 𝑒𝑓𝑗(𝑒)>0,𝑒𝑓(𝑒)>0, and 𝑓′(𝑒)β‰₯πœ‡>0, (𝑒≠0).

We say that a continuous function 𝐻(𝑑,𝑠) belongs to the function class πœ”, denoted by π»βˆˆπœ”, if 𝐻∈𝐢(𝐷,𝑅+), where 𝐷={(𝑑,𝑠)βˆΆβˆ’βˆž<𝑠≀𝑑<+∞}, satisfy𝐻(𝑑,𝑑)=0,𝐻(𝑑,𝑠)>0,βˆ’βˆž<𝑠<𝑑<+∞.(1.2)

Furthermore, the continuous partial derivative πœ•π»/πœ•π‘† exists on 𝐷, and there is β„ŽβˆˆπΏloc(𝐷,𝑅), such thatπœ•π»βˆšπœ•π‘ =βˆ’β„Ž(𝑑,𝑠)𝐻(𝑑,𝑠).(1.3)

Various results on the oscillation for the partial functional differential equation have been obtained recently. We refer the reader to [1–3] for parabolic equations and to [4–11] for hyperbolic equations.

Recently, Li and Cui [12] studied the equation of the formπœ•ξƒ¬πœ•πœ•π‘‘π‘(𝑑)ξƒ©πœ•π‘‘π‘’(π‘₯,𝑑)+𝑙𝑖=1πœ†π‘–ξ€·(𝑑)𝑒π‘₯,π‘‘βˆ’πœπ‘–ξ€Έξƒͺξƒ­=π‘Ž(𝑑)Δ𝑒(π‘₯,𝑑)+π‘ ξ“π‘˜=1π‘Žπ‘˜ξ€·(𝑑)Δ𝑒π‘₯,π‘‘βˆ’πœŒπ‘˜ξ€Έβˆ’(𝑑)βˆ’π‘ž(𝑑)𝑒(π‘₯,𝑑)π‘šξ“π‘—=1π‘žπ‘—ξ€·(π‘₯,𝑑)𝑒π‘₯,π‘‘βˆ’πœŽπ‘—ξ€Έ,(π‘₯,𝑑)βˆˆΞ©Γ—π‘…+≑𝐺(1.4) with Robin boundary conditionπœ•π‘’(π‘₯,𝑑)πœ•π›Ύ+𝑔(π‘₯,𝑑)𝑒(π‘₯,𝑑)=0,(π‘₯,𝑑)βˆˆπœ•Ξ©Γ—π‘…+,(1.5) where 𝛾 is the unit exterior vector to πœ•Ξ© and 𝑔(π‘₯,𝑑) is a nonnegative continuous function on πœ•Ξ©Γ—π‘…+ and obtained the following result.

Theorem A (see [12, Theorem 2.2]). Suppose that π»βˆˆπœ”, let(C1)0<inf𝑠β‰₯𝑑0{liminfπ‘‘β†’βˆž(𝐻(𝑑,𝑠)/𝐻(𝑑,𝑑0))}β‰€βˆž, suppose that there exists some 𝑗0βˆˆπΌπ‘š and there exist two functions πœ™βˆˆπΆ1[𝑑0,∞),𝐴∈𝐢[𝑑0,∞) satisfying,(C2)limsupπ‘‘β†’βˆž(1/𝐻(𝑑,𝑑0∫))𝑑𝑑0𝑝(π‘ βˆ’πœŽπ‘—0)πœ™(𝑠)β„Ž2(𝑑,𝑠)𝑑𝑠<∞,(C3)βˆ«βˆžπ‘‘0(𝐴2+(𝑠)/𝑝(π‘ βˆ’πœŽπ‘—0)πœ™(𝑠))𝑑𝑠=∞, and for every 𝑑1β‰₯𝑑0,(C4)limsupπ‘‘β†’βˆž(1/𝐻(𝑑,𝑑1∫))𝑑𝑑1[𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’(1/4)πœ™(𝑠)𝑝(π‘ βˆ’πœŽπ‘—0)β„Ž2(𝑑,𝑠)]𝑑𝑠β‰₯𝐴(𝑑1), where βˆ«πœ™(𝑠)=exp{βˆ’2π‘ πœ™(πœ‰)π‘‘πœ‰},𝐴+(𝑠)=max{𝐴(𝑠),0}, and πœ“(𝑠)=πœ™(𝑠){𝛼𝑗0π‘žπ‘—0βˆ‘(𝑠)[1βˆ’π‘™π‘–=1πœ†π‘–(π‘ βˆ’πœŽπ‘—0)]+𝑝(π‘ βˆ’πœŽπ‘—0)πœ™2(𝑠)βˆ’[𝑝(π‘ βˆ’πœŽπ‘—0)πœ™(𝑠)]β€²}. Then every solution 𝑒(π‘₯,𝑑) of the problem (1.4), (1.5) is oscillatory in 𝐺.

In 2008, Rogovchenko and Tuncay [13] established new oscillation criteria for second-order nonlinear differential equations with damping termξ€·π‘Ÿ(𝑑)π‘₯ξ…žξ€Έ(𝑑)ξ…ž+𝑝(𝑑)π‘₯ξ…ž(𝑑)+π‘ž(𝑑)𝑓(π‘₯(𝑑))=0,(1.6) without an assumption that has been required in related results reported in the literature over the last two decades. Motivated by the ideas in [12, 13], by introducing a Parameter 𝛽β‰₯1, we will further improve Theorems A and derive new interval criteria for oscillation of (1.1). We suggest two different approaches which allow one to remove condition (C2) in Theorem A. A modified integral averaging technique enables one to simplify essentially the proofs of oscillation criteria.

2. Main Results

Theorem 2.1. Suppose that there exists a function π‘¦βˆˆπΆ1[𝑑0,∞) such that for some 𝛽β‰₯1 and for some π»βˆˆπœ”, limπ‘‘β†’βˆž1sup𝐻𝑑,𝑑0ξ€Έξ€œπ‘‘π‘‘0𝛽𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(𝑑,𝑠)𝑑𝑠=∞,𝑑β‰₯0,(2.1) where ξ‚΅ξ€œπ‘£(𝑑)=expβˆ’2π‘‘ξ‚΅πœ‡π‘¦(𝑠)βˆ’π‘(𝑠)ξ‚Άξ‚Ά,ξ€Ί2π‘Ÿ(𝑠)𝑑𝑠(2.2)πœ“(𝑑)=𝑣(𝑑)π‘ž(𝑑)+πœ‡π‘Ÿ(𝑑)𝑦2(𝑑)βˆ’π‘(𝑑)𝑦(𝑑)βˆ’(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξ€»,(2.3) then every solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Proof. Suppose to the contrary that there is a nonoscillatory solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) which has no zero on Ω×[𝑑0,∞) for some 𝑑0>0. Without loss of generality, we assume that 𝑒(π‘₯,𝑑)>0,𝑒(π‘₯,π‘‘βˆ’πœŒπ‘˜(𝑑))>0 and 𝑒(π‘₯,π‘‘βˆ’πœŽπ‘—)>0 in Ω×[𝑑1,∞),𝑑1β‰₯𝑑0,π‘˜βˆˆπΌπ‘ ,π‘—βˆˆπΌπ‘š. Integrating (1.1) with respect to π‘₯ over the domain Ξ©, we have π‘‘ξ‚΅π‘‘π‘‘π‘‘π‘Ÿ(𝑑)ξ€œπ‘‘π‘‘Ξ©ξ‚Άπ‘‘π‘’(π‘₯,𝑑)𝑑π‘₯+𝑝(𝑑)ξ€œπ‘‘π‘‘Ξ©ξ€œπ‘’(π‘₯,𝑑)𝑑π‘₯=π‘Ž(𝑑)ΩΔ𝑒(π‘₯,𝑑)𝑑π‘₯+π‘ ξ“π‘˜=1π‘Žπ‘˜(ξ€œπ‘‘)ΩΔ𝑒π‘₯,π‘‘βˆ’πœŒπ‘˜(ξ€Έβˆ’ξ€œπ‘‘)𝑑π‘₯Ξ©π‘ž(π‘₯,𝑑)𝑓(𝑒(π‘₯,𝑑))𝑑π‘₯βˆ’π‘šξ“π‘—=1ξ€œΞ©π‘žπ‘—(π‘₯,𝑑)𝑓𝑗𝑒π‘₯,π‘‘βˆ’πœŽπ‘—ξ€Έξ€Έπ‘‘π‘₯,𝑑β‰₯𝑑1.(2.4) From Green’s formula and the boundary condition (1.5), we have ξ€œΞ©ξ€œΞ”π‘’(π‘₯,𝑑)𝑑π‘₯=πœ•Ξ©πœ•π‘’(π‘₯,𝑑)ξ€œπœ•π›Ύπ‘‘π‘ =βˆ’πœ•Ξ©ξ€œπ‘”(π‘₯,𝑑)𝑒(π‘₯,𝑑)𝑑𝑠≀0,ΩΔ𝑒π‘₯,π‘‘βˆ’πœŒπ‘˜ξ€Έξ€œ(𝑑)𝑑π‘₯=πœ•Ξ©ξ€·πœ•π‘’π‘₯,π‘‘βˆ’πœŒπ‘˜ξ€Έ(𝑑)ξ€œπœ•π›Ύπ‘‘π‘ =βˆ’πœ•Ξ©π‘”ξ€·π‘₯,π‘‘βˆ’πœŒπ‘˜ξ€Έπ‘’ξ€·(𝑑)π‘₯,π‘‘βˆ’πœŒπ‘˜ξ€Έ(𝑑)𝑑𝑠≀0,𝑑β‰₯𝑑1,π‘˜βˆˆπΌπ‘ ,(2.5) where 𝑑𝑠 denotes the surface element on πœ•Ξ©. Moreover, from (H2), (H4) and Jensen’s inequality, we have ξ€œΞ©ξ€œπ‘ž(π‘₯,𝑑)𝑓(𝑒(π‘₯,𝑑))𝑑π‘₯β‰₯π‘ž(𝑑)Ξ©||Ξ©||ξ‚΅1𝑓(𝑒(π‘₯,𝑑))𝑑π‘₯β‰₯π‘ž(𝑑)𝑓||Ξ©||ξ€œΞ©ξ‚Ά,ξ€œπ‘’(π‘₯,𝑑)𝑑π‘₯Ξ©π‘žπ‘—(π‘₯,𝑑)𝑓𝑗𝑒π‘₯,π‘‘βˆ’πœŽπ‘—ξ€Έξ€Έπ‘‘π‘₯β‰₯π‘žπ‘—ξ€œ(𝑑)Ω𝑓𝑗𝑒π‘₯,π‘‘βˆ’πœŽπ‘—β‰₯||Ξ©||π‘žξ€Έξ€Έπ‘‘π‘₯𝑗(𝑑)𝑓𝑗1||Ξ©||ξ€œΞ©π‘’ξ€·π‘₯,π‘‘βˆ’πœŽπ‘—ξ€Έξ‚Άπ‘‘π‘₯,(2.6) where ∫|Ξ©|=Ω𝑑π‘₯.
Set 1π‘ˆ(𝑑)=||Ξ©||ξ€œΞ©π‘’(π‘₯,𝑑)𝑑π‘₯,𝑑β‰₯𝑑1.(2.7) In view of (2.5)–(2.7), (2.4) yields that ξ€·π‘Ÿ(𝑑)π‘ˆξ…žξ€Έ(𝑑)ξ…ž+𝑝(𝑑)π‘ˆξ…ž(𝑑)+π‘ž(𝑑)𝑓(π‘ˆ(𝑑))+π‘šξ“π‘—=1π‘žπ‘—(𝑑)π‘“π‘—ξ€·π‘ˆξ€·π‘‘βˆ’πœŽπ‘—ξ€Έξ€Έβ‰€0,𝑑β‰₯𝑑1.(2.8) Note that (H4), (2.8) yields that (π‘Ÿ(𝑑)π‘ˆβ€²(𝑑))β€²+𝑝(𝑑)π‘ˆβ€²(𝑑)+π‘ž(𝑑)𝑓(π‘ˆ(𝑑))≀0,𝑑β‰₯𝑑1.(2.9) Put ξ‚Έπ‘ˆπ‘€(𝑑)=𝑣(𝑑)π‘Ÿ(𝑑)ξ…ž(𝑑)𝑓(π‘ˆ(𝑑))+𝑦(𝑑),𝑑β‰₯𝑑1,(2.10) where 𝑣(𝑑) is given by (2.2), then π‘€ξ…žξ‚Έ(𝑑)=βˆ’2πœ‡π‘¦(𝑑)+𝑝(𝑑)ξ‚Ήξƒ¬ξ€·π‘Ÿ(𝑑)𝑀(𝑑)+𝑣(𝑑)π‘Ÿ(𝑑)π‘ˆξ…žξ€Έ(𝑑)ξ…žβˆ’ξ€·π‘ˆπ‘“(π‘ˆ(𝑑))π‘Ÿ(𝑑)𝑓′(π‘ˆ(𝑑))ξ…žξ€Έ(𝑑)2𝑓2(π‘ˆ(𝑑))+(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξƒ­β‰€ξ‚Έπ‘βˆ’2πœ‡π‘¦(𝑑)+(𝑑)ξ‚Ήξƒ¬ξ€·π‘Ÿ(𝑑)𝑀(𝑑)+𝑣(𝑑)π‘Ÿ(𝑑)π‘ˆξ…ž(𝑑)ξ…žβˆ’ξ€·π‘ˆπ‘“(π‘ˆ(𝑑))πœ‡π‘Ÿ(𝑑)ξ…ž(𝑑)2𝑓2(π‘ˆ(𝑑))+(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξƒ­β‰€ξ‚Έβˆ’2πœ‡π‘¦(𝑑)+𝑝(𝑑)ξ‚Ήξƒ¬π‘Ÿ(𝑑)𝑀(𝑑)βˆ’π‘£(𝑑)𝑝(𝑑)π‘ˆξ…ž(𝑑)ξ‚΅π‘ˆπ‘“(π‘ˆ(𝑑))+π‘ž(𝑑)+πœ‡π‘Ÿ(𝑑)ξ…ž(𝑑)𝑓(π‘ˆ(𝑑))2βˆ’(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξƒ­=ξ‚Έβˆ’2πœ‡π‘¦(𝑑)+𝑝(𝑑)ξ‚Ήξ‚Έξ‚΅π‘Ÿ(𝑑)𝑀(𝑑)βˆ’π‘£(𝑑)𝑝(𝑑)𝑀(𝑑)𝑣(𝑑)π‘Ÿ(𝑑)βˆ’π‘¦(𝑑)+π‘ž(𝑑)+πœ‡π‘Ÿ(𝑑)𝑀(𝑑)𝑣(𝑑)π‘Ÿ(𝑑)βˆ’π‘¦(𝑑)2βˆ’(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξƒ­=ξ‚Έπ‘βˆ’2πœ‡π‘¦(𝑑)+(𝑑)ξ‚Ήξ‚Έπ‘Ÿ(𝑑)𝑀(𝑑)βˆ’π‘£(𝑑)𝑝(𝑑)𝑀𝑣(𝑑)π‘Ÿ(𝑑)𝑀(𝑑)βˆ’π‘(𝑑)𝑦(𝑑)+π‘ž(𝑑)+πœ‡π‘Ÿ(𝑑)2(𝑑)𝑣2(𝑑)π‘Ÿ2(𝑑)βˆ’2πœ‡π‘€(𝑑)𝑦(𝑑)𝑣(𝑑)+πœ‡π‘Ÿ(𝑑)𝑦2(𝑑)βˆ’(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξ‚Ή=βˆ’2πœ‡π‘¦(𝑑)𝑀(𝑑)+𝑝(𝑑)π‘Ÿ(𝑑)𝑀(𝑑)βˆ’π‘(𝑑)ξ€Ίπ‘Ÿ(𝑑)𝑀(𝑑)βˆ’π‘£(𝑑)βˆ’π‘(𝑑)𝑦(𝑑)+π‘ž(𝑑)+πœ‡π‘Ÿ(𝑑)𝑦2(𝑑)βˆ’(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξ€»π‘€+2πœ‡π‘¦(𝑑)𝑀(𝑑)βˆ’πœ‡2(𝑑)𝑣(𝑑)π‘Ÿ(𝑑)=βˆ’π‘£(𝑑)βˆ’π‘(𝑑)𝑦(𝑑)+π‘ž(𝑑)+πœ‡π‘Ÿ(𝑑)𝑦2(𝑑)βˆ’(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξ€»π‘€βˆ’πœ‡2(𝑑)𝑀𝑣(𝑑)π‘Ÿ(𝑑)=βˆ’πœ“(𝑑)βˆ’πœ‡2(𝑑),𝑣(𝑑)π‘Ÿ(𝑑)(2.11) that is, πœ“(𝑑)β‰€βˆ’π‘€ξ…žπ‘€(𝑑)βˆ’πœ‡2(𝑑)𝑣(𝑑)π‘Ÿ(𝑑),(2.12) where πœ“(𝑑) is defined by (2.3). Multiplying (2.12) by 𝐻(𝑑,𝑠) and integrating from 𝑇 to 𝑑, we have, for some 𝛽β‰₯1 and for all 𝑑β‰₯𝑇β‰₯𝑑1, ξ€œπ‘‘π‘‡ξ€œπ»(𝑑,𝑠)πœ“(𝑠)π‘‘π‘ β‰€βˆ’π‘‘π‘‡π»(𝑑,𝑠)π‘€ξ…ž(ξ€œπ‘ )π‘‘π‘ βˆ’π‘‘π‘‡πœ‡π»(𝑑,𝑠)𝑀𝑣(𝑠)π‘Ÿ(𝑠)2(ξ€œπ‘ )𝑑𝑠=𝐻(𝑑,𝑇)𝑀(𝑇)βˆ’π‘‘π‘‡ξ‚΅βˆšβ„Ž(𝑑,𝑠)𝐻(𝑑,𝑠)𝑀(𝑠)+𝐻(𝑑,𝑠)πœ‡π‘€2(𝑠)π‘£ξ‚Άξ€œ(𝑠)π‘Ÿ(𝑠)𝑑𝑠=𝐻(𝑑,𝑇)𝑀(𝑇)βˆ’π‘‘π‘‡βŽ‘βŽ’βŽ’βŽ£ξƒŽπœ‡π»(𝑑,𝑠)ξƒŽπ›½π‘£(𝑠)π‘Ÿ(𝑠)𝑀(𝑠)+𝛽𝑣(𝑠)π‘Ÿ(𝑠)⎀βŽ₯βŽ₯⎦4πœ‡β„Ž(𝑑,𝑠)2+π›½π‘‘π‘ ξ€œ4πœ‡π‘‘π‘‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(ξ€œπ‘‘,𝑠)π‘‘π‘ βˆ’π‘‘π‘‡(π›½βˆ’1)πœ‡π›½π‘£(𝑠)π‘Ÿ(𝑠)𝐻(𝑑,𝑠)𝑀2(𝑠)𝑑𝑠.(2.13) Writing the latter inequality in the form ξ€œπ‘‘π‘‡ξ‚Έπ›½π»(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(ξ‚Ήξ€œπ‘‘,𝑠)𝑑𝑠≀𝐻(𝑑,𝑇)𝑀(𝑇)βˆ’π‘‘π‘‡βŽ‘βŽ’βŽ’βŽ£ξƒŽπœ‡π»(𝑑,𝑠)ξƒŽπ›½π‘£(𝑠)π‘Ÿ(𝑠)𝑀(𝑠)+𝛽𝑣(𝑠)π‘Ÿ(𝑠)⎀βŽ₯βŽ₯⎦4πœ‡β„Ž(𝑑,𝑠)2βˆ’ξ€œπ‘‘π‘ π‘‘π‘‡(π›½βˆ’1)πœ‡π›½π‘£(𝑠)π‘Ÿ(𝑠)𝐻(𝑑,𝑠)𝑀2(𝑠)𝑑𝑠.(2.14) Using the properties of 𝐻(𝑑,𝑠), we have ξ€œπ‘‘π‘‘1𝛽𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(𝑑,𝑠)𝑑𝑠≀𝐻𝑑,𝑑1ξ€Έ||𝑀𝑑1ξ€Έ||≀𝐻𝑑,𝑑0ξ€Έ||𝑀𝑑1ξ€Έ||,𝑑β‰₯𝑑1,(2.15) and for all 𝑑β‰₯𝑑1β‰₯𝑑0, ξ€œπ‘‘π‘‘0𝛽𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(𝑑,𝑠)𝑑𝑠≀𝐻𝑑,𝑑0ξ€Έξ‚Έξ€œπ‘‘1𝑑0||||||π‘€ξ€·π‘‘πœ“(𝑠)𝑑𝑠+1ξ€Έ||ξ‚Ή.(2.16) By (2.16), limπ‘‘β†’βˆž1sup𝐻𝑑,𝑑0ξ€Έξ€œπ‘‘π‘‘0𝛽𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(ξ‚Άξ€œπ‘‘,𝑠)𝑑𝑠≀𝑑1𝑑0||||||π‘€ξ€·π‘‘πœ“(𝑠)𝑑𝑠+1ξ€Έ||<∞,(2.17) which contradicts (2.1). This proves Theorem 2.1.

Consider a Kamenev-type function 𝐻(𝑑,𝑠) defined by 𝐻(𝑑,𝑠)=(π‘‘βˆ’π‘ )π‘›βˆ’1,(𝑑,𝑠)∈𝐷, where 𝑛>2 is an integer. Obviously, 𝐻 belongs to the class πœ”, and β„Ž(𝑑,𝑠)=(π‘›βˆ’1)(π‘‘βˆ’π‘ )(π‘›βˆ’3)/2,(𝑑,𝑠)∈𝐷. Then, we can get the following results.

Corollary 2.2. Suppose that there exists a function 𝑦(𝑑)∈𝐢1([𝑑0,∞);𝑅) such that for some integer 𝑛>2 and some 𝛽β‰₯1, limπ‘‘β†’βˆž1supπ‘‘π‘›βˆ’1ξ€œπ‘‘π‘‘0(π‘‘βˆ’π‘ )π‘›βˆ’3ξ‚Έ(π‘‘βˆ’π‘ )2πœ“(𝑠)βˆ’π›½(π‘›βˆ’1)2ξ‚Ή4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)𝑑𝑠=∞,(2.18) where 𝑣(𝑑) and πœ“(𝑑) are as defined in Theorem 2.1. Then every solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Theorem 2.3. Suppose that 0<inf𝑠β‰₯𝑑0limπ‘‘β†’βˆžinf𝐻(𝑑,𝑠)𝐻𝑑,𝑑0ξ€Έξƒͺβ‰€βˆž.(2.19) Assume that there exist functions π‘“βˆˆπΆ1([𝑑0,∞);𝑅) and πœ™βˆˆπΆ([𝑑0,∞);𝑅) such that, for all 𝑑β‰₯𝑇β‰₯𝑑0 and for some 𝛽>1, limπ‘‘β†’βˆž1supξ€œπ»(𝑑,𝑇)𝑑𝑇𝛽𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(𝑑,𝑠)𝑑𝑠β‰₯πœ™(𝑇),(2.20) where 𝑣(𝑑),πœ“(𝑑) are as defined in Theorem 2.1 and suppose further that limπ‘‘β†’βˆžξ€œsup𝑑𝑑0πœ™2+(𝑠)𝑣(𝑠)π‘Ÿ(𝑠)=∞,(2.21) where πœ™+(𝑑)=max(πœ™(𝑑),0). Then every solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Proof. Suppose to the contrary that there is a nonoscillatory solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) which has no zero on Ω×[𝑑1,∞) for some 𝑑1>𝑑0, without loss of generality, we assume that 𝑒(π‘₯,𝑑)>0,𝑒(π‘₯,π‘‘βˆ’πœŒπ‘˜(𝑑))>0 and 𝑒(π‘₯,π‘‘βˆ’πœŽπ‘—)>0 in Ω×[𝑑1,∞),𝑑β‰₯𝑑1β‰₯𝑑0,π‘˜βˆˆπΌπ‘ ,π‘—βˆˆπΌπ‘š.
As in the proof of Theorem 2.1, (2.14) holds for all 𝑑β‰₯𝑇β‰₯𝑑1, we have 1ξ€œπ»(𝑑,𝑇)𝑑𝑇𝛽𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(ξ‚Ή1𝑑,𝑠)𝑑𝑠≀𝑀(𝑇)βˆ’ξ€œπ»(𝑑,𝑇)π‘‘π‘‡βŽ‘βŽ’βŽ’βŽ£ξƒŽπœ‡π»(𝑑,𝑠)ξƒŽπ›½π‘£(𝑠)π‘Ÿ(𝑠)𝑀(𝑠)+𝛽𝑣(𝑠)π‘Ÿ(𝑠)⎀βŽ₯βŽ₯⎦4πœ‡β„Ž(𝑑,𝑠)2βˆ’1π‘‘π‘ ξ€œπ»(𝑑,𝑇)𝑑𝑇(π›½βˆ’1)πœ‡π›½π‘£(𝑠)π‘Ÿ(𝑠)𝐻(𝑑,𝑠)𝑀21(𝑠)𝑑𝑠≀𝑀(𝑇)βˆ’ξ€œπ»(𝑑,𝑇)𝑑𝑇(π›½βˆ’1)πœ‡π›½π‘£(𝑠)π‘Ÿ(𝑠)𝐻(𝑑,𝑠)𝑀2(𝑠)𝑑𝑠.(2.22) Therefore, for 𝑑>𝑇β‰₯𝑇0, limπ‘‘β†’βˆž1supξ€œπ»(𝑑,𝑇)𝑑𝑇𝛽𝐻(𝑑,𝑠)πœ“(𝑠)βˆ’4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2(𝑑,𝑠)𝑑𝑠≀𝑀(𝑇)βˆ’limπ‘‘β†’βˆž1infξ€œπ»(𝑑,𝑇)𝑑𝑇(π›½βˆ’1)πœ‡π›½π‘£(𝑠)π‘Ÿ(𝑠)𝐻(𝑑,𝑠)𝑀2(𝑠)𝑑𝑠.(2.23) It follows from (2.20) that 𝑀(𝑇)β‰₯πœ™(𝑇)+limπ‘‘β†’βˆž1infξ€œπ»(𝑑,𝑇)𝑑𝑇(π›½βˆ’1)πœ‡π›½π‘£(𝑠)π‘Ÿ(𝑠)𝐻(𝑑,𝑠)𝑀2(𝑠)𝑑𝑠(2.24) for all 𝑇β‰₯𝑑1 and for any 𝛽>1. Then, for all 𝑇β‰₯𝑑1, 𝑀(𝑇)β‰₯πœ™(𝑇),(2.25)limπ‘‘β†’βˆž1inf𝐻𝑑,𝑑1ξ€Έξ€œπ‘‘π‘‘1𝐻(𝑑,𝑠)𝑀𝑣(𝑠)π‘Ÿ(𝑠)2𝛽(𝑠)π‘‘π‘ β‰€ξ€·π‘€ξ€·π‘‘πœ‡(π›½βˆ’1)1ξ€Έξ€·π‘‘βˆ’πœ™1.ξ€Έξ€Έ(2.26) Now, we claim that ξ€œβˆžπ‘‘1𝑀2(𝑠)𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠<∞.(2.27) Suppose the contrary, that is, ξ€œβˆžπ‘‘1𝑀2(𝑠)𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠=∞.(2.28) By (2.19), there is a positive constant 𝑀1, satisfying inf𝑠β‰₯𝑑0limπ‘‘β†’βˆžinf𝐻(𝑑,𝑠)𝐻𝑑,𝑑0ξ€Έξƒͺ>𝑀1>0.(2.29) Let 𝑀 be any arbitrary positive number, then from (2.28) we get that there exists a 𝑇1>𝑑1 such that, for all 𝑑β‰₯𝑇1, ξ€œπ‘‘π‘‘1𝑀2(𝑠)𝑀𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠β‰₯𝑀1.(2.30) Using integration by parts, for all 𝑑β‰₯𝑇1, we get 1𝐻𝑑,𝑑1ξ€Έξ€œπ‘‘π‘‘1𝑀𝐻(𝑑,𝑠)2(𝑠)1𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠=𝐻𝑑,𝑑1ξ€Έξ€œπ‘‘π‘‘1ξ‚΅βˆ’πœ•π»(𝑑,𝑠)ξ€œπœ•π‘ ξ‚Άξ‚΅π‘ π‘‘1𝑀2(𝜏)ξ‚Άβ‰₯1𝑣(𝜏)π‘Ÿ(𝜏)π‘‘πœπ‘‘π‘ π»ξ€·π‘‘,𝑑1ξ€Έξ€œπ‘‘π‘‡1ξ‚΅βˆ’πœ•π»(𝑑,𝑠)ξ€œπœ•π‘ ξ‚Άξ‚΅π‘ π‘‘1𝑀2(𝜏)ξ‚Άβ‰₯𝑀𝑣(𝜏)π‘Ÿ(𝜏)π‘‘πœπ‘‘π‘ π‘€11𝐻𝑑,𝑑1ξ€Έξ€œπ‘‘π‘‡1ξ‚΅βˆ’πœ•π»(𝑑,𝑠)ξ‚Ά=π‘€πœ•π‘ π‘‘π‘ π‘€1𝐻𝑑,𝑇1𝐻𝑑,𝑑1ξ€Έ.(2.31) By (2.29), there exists a 𝑇2>𝑇1 such that, for all 𝑑β‰₯𝑇2, 𝐻𝑑,𝑇1𝐻𝑑,𝑑1ξ€Έβ‰₯𝑀1.(2.32) It follows from (2.31) that for all 𝑑β‰₯𝑇2, 1𝐻𝑑,𝑑1ξ€Έξ€œπ‘‘π‘‘1𝑀𝐻(𝑑,𝑠)2(𝑠)𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠β‰₯𝑀.(2.33) Since 𝑀 is an arbitrary positive constant, limπ‘‘β†’βˆž1inf𝐻𝑑,𝑑1ξ€Έξ€œπ‘‘π‘‘1𝑀𝐻(𝑑,𝑠)2(𝑠)𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠=∞,(2.34) which contradicts (2.26). Consequently, (2.27) holds. And from (2.25), we obtain ξ€œβˆžπ‘‘1πœ™2+(𝑠)ξ€œπ‘£(𝑠)π‘Ÿ(𝑠)π‘‘π‘ β‰€βˆžπ‘‘1𝑀2(𝑠)𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠<∞,(2.35) which contradicts (2.21). This completes the proof of Theorem 2.3.

Choosing 𝐻 as in Corollary 2.2, by Theorem 2.3, we can obtain the following corollary.

Corollary 2.4. Let 𝑣(𝑑) and πœ“(𝑑) be as in Theorem 2.1, assume further that there exist functions π‘“βˆˆπΆ1([𝑑0,∞);𝑅) and πœ™βˆˆπΆ([𝑑0,∞);𝑅) such that, for all 𝑇β‰₯𝑑0, for some integer 𝑛>2, and for some 𝛽>1, limπ‘‘β†’βˆž1supπ‘‘π‘›βˆ’1ξ€œπ‘‘π‘‡(π‘‘βˆ’π‘ )π‘›βˆ’3ξ‚Έ(π‘‘βˆ’π‘ )2πœ“(𝑠)βˆ’π›½(π‘›βˆ’1)2ξ‚Ή4πœ‡π‘£(𝑠)π‘Ÿ(𝑠)𝑑𝑠β‰₯πœ™(𝑇)(2.36) and (2.21) hold. Then every solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Theorem 2.5. Suppose that there exists a function π‘“βˆˆπΆ1([𝑑0,∞);𝑅) such that for some 𝛽β‰₯1 and for some π»βˆˆπœ”, limπ‘‘β†’βˆž1sup𝐻𝑑,𝑑0ξ€Έξ€œπ‘‘π‘‘0𝐻(𝑑,𝑠)π‘πœ“(𝑠)βˆ’2(𝑠)𝑣(𝑠)ξ‚Άβˆ’2πœ‡π‘Ÿ(𝑠)πœ‡π›½2𝑣(𝑠)π‘Ÿ(𝑠)β„Ž2ξ‚Ή(𝑑,𝑠)𝑑𝑠=∞,(2.37) where ξ‚΅ξ€œπ‘£(𝑑)=expβˆ’2πœ‡π‘‘ξ‚Ά,𝑦(𝑠)π‘‘π‘ πœ“(𝑑)=𝑣(𝑑)π‘ž(𝑑)+πœ‡π‘Ÿ(𝑑)𝑦2(𝑑)βˆ’π‘(𝑑)𝑦(𝑑)βˆ’(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξ€Έ.(2.38) Then every solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Proof. As in Theorem 2.1, without loss of generality, we assume that a nonoscillatory solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) satisfies 𝑒(π‘₯,𝑑)>0,𝑒(π‘₯,π‘‘βˆ’πœŒπ‘˜(𝑑))>0 and 𝑒(π‘₯,π‘‘βˆ’πœŽπ‘—)>0 in Ω×[𝑑1,∞), 𝑑1β‰₯𝑑0, π‘˜βˆˆπΌπ‘ ,π‘—βˆˆπΌπ‘š. Define a generalized Riccati transformation 𝑀(𝑑)=ξ‚Έπ‘ˆπ‘£(𝑑)π‘Ÿ(𝑑)ξ…ž(𝑑)𝑓(π‘ˆ(𝑑))+𝑦(𝑑),𝑑β‰₯𝑑1,(2.39) where 𝑣(𝑑) is given by (2.38). Then π‘€ξ…ž(𝑑)β‰€βˆ’2πœ‡π‘¦(𝑑)𝑀(𝑑)+𝑣(𝑑)βˆ’π‘ž(𝑑)+(π‘Ÿ(𝑑)𝑦(𝑑))ξ…žξ‚Έβˆ’π‘(𝑑)𝑀(𝑑)𝑣(𝑑)π‘Ÿ(𝑑)βˆ’π‘¦(𝑑)βˆ’πœ‡π‘Ÿ(𝑑)𝑀(𝑑)𝑣(𝑑)π‘Ÿ(𝑑)βˆ’π‘¦(𝑑)2ξƒ°=βˆ’πœ“(𝑑)βˆ’π‘(𝑑)π‘Ÿ(𝑑)1𝑀(𝑑)βˆ’πœ‡π‘Ÿ(𝑑)𝑣(𝑑)𝑀2(𝑑),𝑑β‰₯𝑑1.(2.40) Using an elementary inequality βˆ’π‘Žπ‘₯2π‘Ž+𝑏π‘₯β‰€βˆ’2π‘₯2+𝑏22π‘Ž,(2.41) for all π‘Ž>0 and for all 𝑏,π‘₯βˆˆπ‘…, we conclude from (2.40) that π‘πœ“(𝑑)βˆ’2(𝑑)𝑣(𝑑)2πœ‡π‘Ÿ(𝑑)β‰€βˆ’π‘€ξ…žπœ‡(𝑑)βˆ’2𝑣(𝑑)π‘Ÿ(𝑑)𝑀2(𝑑),𝑑β‰₯𝑑1.(2.42) Multiplying (2.42) by 𝐻(𝑑,𝑠) and integrating from 𝑇<𝑑, we obtain, for some 𝛽β‰₯1 and for all 𝑑β‰₯𝑇β‰₯𝑑1, ξ€œπ‘‘π‘‡ξ‚΅π»(𝑑,𝑠)π‘πœ“(𝑠)βˆ’2(𝑠)𝑣(𝑠)ξ‚Ά2πœ‡π‘Ÿ(𝑠)𝑑𝑠≀𝐻(𝑑,𝑇)ξ€œπ‘€(𝑇)βˆ’π‘‘π‘‡βˆšβ„Ž(𝑑,𝑠)𝐻(𝑑,𝑠)ξ€œπ‘€(𝑠)π‘‘π‘ βˆ’π‘‘π‘‡πœ‡π‘€2(𝑠)2𝑣(𝑠)π‘Ÿ(𝑠)𝑑𝑠≀𝐻(𝑑,𝑇)𝑀(𝑇)+πœ‡π›½2ξ€œπ‘‘π‘‡π‘£(𝑠)π‘Ÿ(𝑠)β„Ž2ξ€œ(𝑑,𝑠)π‘‘π‘ βˆ’πœ‡π‘‘π‘‡(π›½βˆ’1)𝐻(𝑑,𝑠)2𝛽𝑣(𝑠)π‘Ÿ(𝑠)𝑀2βˆ’πœ‡(𝑠)𝑑𝑠2ξ€œπ‘‘π‘‡ξƒ©ξƒŽπ»(𝑑,𝑠)𝛽𝑣(𝑠)π‘Ÿ(𝑠)βˆšπ‘€(𝑠)+𝛽ξƒͺ𝑣(𝑠)π‘Ÿ(𝑠)β„Ž(𝑑,𝑠)2𝑑𝑠.(2.43) Therefore, for all 𝑑β‰₯𝑇β‰₯𝑑1, we have ξ€œπ‘‘π‘‡ξ‚Έπ»(𝑑,𝑠)π‘πœ“(𝑠)βˆ’π»(𝑑,𝑠)2(𝑠)𝑣(𝑠)βˆ’2πœ‡π‘Ÿ(𝑠)πœ‡π›½2𝑣(𝑠)π‘Ÿ(𝑠)β„Ž2ξ‚Ή(𝑑,𝑠)𝑑𝑠≀𝐻(𝑑,𝑇)ξ€œπ‘€(𝑇)βˆ’πœ‡π‘‘π‘‡(π›½βˆ’1)𝐻(𝑑,𝑠)2𝛽𝑣(𝑠)π‘Ÿ(𝑠)𝑀2(βˆ’πœ‡π‘ )𝑑𝑠2ξ€œπ‘‘π‘‡ξƒ©ξƒŽπ»(𝑑,𝑠)𝛽𝑣(𝑠)π‘Ÿ(𝑠)βˆšπ‘€(𝑠)+𝛽𝑣ξƒͺ(𝑠)π‘Ÿ(𝑠)β„Ž(𝑑,𝑠)2𝑑𝑠.(2.44) Following the same lines as in the proof of Theorem 2.1, we have limπ‘‘β†’βˆž1sup𝐻𝑑,𝑑0ξ€Έξ€œπ‘‘π‘‘0𝐻(𝑑,𝑠)π‘πœ“(𝑠)βˆ’π»(𝑑,𝑠)2(𝑠)𝑣(𝑠)βˆ’2πœ‡π‘Ÿ(𝑠)πœ‡π›½2𝑣(𝑠)π‘Ÿ(𝑠)β„Ž2ξ‚Άβ‰€ξ€œ(𝑑,𝑠)𝑑𝑠𝑑1𝑑0||||||πœ“(𝑠)𝑑𝑠+𝑀𝑑1ξ€Έ||<∞(2.45) which contradicts the assumption (2.19).
This completes the proof.

Theorem 2.6. Let (2.19) holds. Assume that there exist functions π‘“βˆˆπΆ1([𝑑0,∞);𝑅) and πœ™βˆˆπΆ([𝑑0,∞),𝑅) such that, for all 𝑑β‰₯𝑑0, any 𝑇β‰₯𝑑0, and for some 𝛽>1, limπ‘‘β†’βˆž1supξ€œπ»(𝑑,𝑇)𝑑𝑇𝐻(𝑑,𝑠)π‘πœ“(𝑠)βˆ’2(𝑠)𝑣(𝑠)ξ‚Άβˆ’2πœ‡π‘Ÿ(𝑠)πœ‡π›½2𝑣(𝑠)π‘Ÿ(𝑠)β„Ž2ξ‚Ή(𝑑,𝑠)𝑑𝑠β‰₯πœ™(𝑇)(2.46) and (2.21) holds, where πœ“(𝑑),𝑣(𝑑) are defined as in Theorem 2.6 and πœ™+(𝑑)=max(πœ™(𝑑),0), then every solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Theorem 2.7. Let all assumptions of Theorem 2.6 be satisfied except that condition (2.46) be replaced by limπ‘‘β†’βˆž1infξ€œπ»(𝑑,𝑇)𝑑𝑇𝐻(𝑑,𝑠)π‘πœ“(𝑠)βˆ’2(𝑠)𝑣(𝑠)ξ‚Άβˆ’2πœ‡π‘Ÿ(𝑠)πœ‡π›½2𝑣(𝑠)π‘Ÿ(𝑠)β„Ž2ξ‚Ή(𝑑,𝑠)𝑑𝑠β‰₯πœ™(𝑇).(2.47) Then every solution 𝑒(π‘₯,𝑑) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Remark 2.8. By introducing the parameter 𝛽 in Theorem 2.3, we derive new oscillation criteria of the problem (1.1), (1.5) which are simpler than that in Theorem A; furthermore, modifications of the proofs through the refinement of the standard integral averaging method allowed us to shorten significantly the proofs of Theorem 2.3. We can also derive a number of oscillation criteria with the appropriate choice of the function 𝐻 and 𝜌, here, we omit the details.

3. Examples

Now, we consider these following examples.

Example 3.1. Consider the partial differential equation πœ•ξ‚ƒ1πœ•π‘‘π‘‘πœ•ξ‚€1πœ•π‘‘π‘’(π‘₯,𝑑)+2𝑒(π‘₯,π‘‘βˆ’πœ‹)+2cosπ‘‘πœ•π‘’(π‘₯,𝑑)1πœ•π‘‘=Δ𝑒(π‘₯,𝑑)+𝑑2ξ‚€3Δ𝑒π‘₯,π‘‘βˆ’2πœ‹ξ‚βˆ’ξ‚€2𝑑3+𝑑cos21𝑑+sin𝑑𝑓(𝑒(π‘₯,𝑑))βˆ’π‘‘2𝑓1(𝑒(π‘₯,π‘‘βˆ’πœ‹)),(π‘₯,𝑑)∈(0,πœ‹)Γ—(0,∞),(3.1) with the boundary condition 𝑒π‘₯(0,𝑑)=𝑒π‘₯(πœ‹,𝑑)=0,𝑑>0,(3.2) where 𝑓(𝑒)=𝑒3+𝑒,𝑓1(𝑒)=𝑒𝑒𝑒+𝑒.
Here, 𝑁=1,𝑙=1,𝑠=1,π‘š=1,πœ‡=1, π‘Ÿ(𝑑)=1/𝑑, 𝑝(𝑑)=2cos𝑑, π‘ž(π‘₯,𝑑)=π‘ž(𝑑)=(2/𝑑3+𝑑cos2𝑑+sin𝑑), π‘ž1(π‘₯,𝑑)=1/𝑑2, 𝑓(𝑒)=𝑓𝑗(𝑒)=𝑒,π‘Ž(𝑑)=1,π‘Ž1(𝑑)=1/𝑑2, 𝜌1(𝑑)=(3/2)πœ‹,𝜎1=πœ‹,𝜏1=πœ‹.
Let 1𝑦(𝑑)=𝑑+𝑑cos𝑑,(3.3) then 𝑣(𝑑)=𝑑2,πœ“(𝑑)=π‘‘βˆ’1.(3.4) Let 𝑛=3, for any 𝛽β‰₯1, limπ‘‘β†’βˆž1sup𝑑2ξ€œπ‘‘1(π‘‘βˆ’π‘ )2π‘ βˆ’1βˆ’π›½π‘ 21𝑠𝑑𝑠=limπ‘‘β†’βˆž1sup𝑑2ξ€œπ‘‘1ξ€Ί(π‘‘βˆ’π‘ )2π‘ βˆ’1ξ€»βˆ’π›½π‘ π‘‘π‘ =∞.(3.5) Therefore, Corollary 2.2 holds, then every solution 𝑒(π‘₯,𝑑) of the problem (3.1), (3.2) oscillates in (0,πœ‹)Γ—(0,∞).

Example 3.2. Consider the partial differential equation πœ•ξ‚Έξ‚€1πœ•π‘‘1+2𝑑3(2+sin𝑑)πœ•π‘’(π‘₯,𝑑)ξ‚Ή+3πœ•π‘‘π‘‘ξ‚€11+2𝑑3(2+sin𝑑)πœ•π‘’(π‘₯,𝑑)ξ‚€3πœ•π‘‘=3Δ𝑒(π‘₯,𝑑)+(2βˆ’cos𝑑)Δ𝑒π‘₯,π‘‘βˆ’2πœ‹ξ‚βˆ’π‘‘βˆ’3ξ€·ξ€·1βˆ’π‘‘3+2𝑑2ξ€Έξ€Έβˆ’6𝑑sin𝑑+12𝑑𝑓(𝑒(π‘₯,𝑑))βˆ’(2𝑑+sin𝑑)𝑓1(𝑒(π‘₯,π‘‘βˆ’πœ‹))βˆ’2𝑓2ξ‚€π‘’ξ‚€πœ‹π‘₯,π‘‘βˆ’2,(π‘₯,𝑑)∈(0,πœ‹)Γ—(0,∞)(3.6) with the boundary condition (3.2), where 𝑓(𝑒)=𝑒5+𝑒,𝑓1(𝑒)=𝑒sin2𝑒,𝑓2(𝑒)=𝑒3cos2𝑒.
Here 𝑁=1,𝑠=1,π‘š=2,πœ‡=1, π‘Ÿ(𝑑)=(1+1/2𝑑3)(2+sin𝑑), 𝑝(𝑑)=(3/𝑑)(1+1/2𝑑3)(2+sin𝑑), π‘ž(𝑑)=π‘‘βˆ’3[(1βˆ’π‘‘3+2𝑑2βˆ’6𝑑)sin𝑑+12𝑑], π‘Ž(𝑑)=3,π‘Ž1(𝑑)=2βˆ’cos𝑑,π‘ž1(π‘₯,𝑑)=2+sin𝑑,π‘ž2(π‘₯,𝑑)=2, 𝜌1(𝑑)=(3/2)πœ‹, 𝜎1=πœ‹,𝜎2=πœ‹/2.
Let 𝑦(𝑑)=0, then 𝑣(𝑑)=𝑑3 and πœ“(𝑑)=𝑣(𝑑)π‘ž(𝑑)=(1βˆ’π‘‘3+2𝑑2βˆ’6𝑑)sin𝑑+12𝑑.
Choose 𝛽=2, 𝑛=3, a straightforward computation yields limsupπ‘‘β†’βˆž1𝑑2ξ€œπ‘‘π‘‡ξ€Ί(π‘‘βˆ’π‘ )2ξ€·ξ€·1βˆ’π‘ 3+2𝑠2ξ€Έξ€Έβˆ’ξ€·βˆ’6𝑠sin𝑠+12𝑠2𝑠3ξ€Έ(ξ€»+12+sin𝑠)𝑑𝑠=16βˆ’π‘‡3cos𝑇+𝑇2(2cosπ‘‡βˆ’6+3sin𝑇)βˆ’4𝑇sinπ‘‡βˆ’3cos𝑇=πœ™(𝑇).(3.7) Let πœ™+(𝑑)=max(πœ™(𝑑),0). It is not difficult to see that limsupπ‘‘β†’βˆžξ€œπ‘‘1πœ™2+(𝑠)𝑠3ξ€Έ+(1/2)(2+sin𝑠)𝑑𝑠β‰₯limsupπ‘‘β†’βˆžξ€œπ‘‘1πœ™2+(𝑠)3𝑠3ξ€Έ+(1/2)𝑑𝑠=∞.(3.8) By Corollary 2.4, we obtain that every solution of problem (3.6), (3.2) oscillates in (0,πœ‹)Γ—(0,∞).
Note that in this example, limsupπ‘‘β†’βˆž1𝑑2ξ€œπ‘‘14𝑠3+12+(2+sin𝑠)𝑑𝑠=∞,(3.9) so the condition (C2) would not have been satisfied with the same choices of 𝑣(𝑑).

Acknowledgments

This research was supported by National Science Foundation of China (11171178), the fund of subject for doctor of ministry of education (20103705110003), and the Natural Science Foundations of Shandong Province of China (ZR2009AM011 and ZR2009AL015).

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