Oscillation for Certain Nonlinear Neutral Partial Differential Equations

Abstract

We present some new oscillation criteria for second-order neutral partial functional differential equations of the form (𝜕/𝜕𝑡){𝑝(𝑡)(𝜕/𝜕𝑡)[𝑢(𝑥,𝑡)+∑^𝑙_𝑖=1𝜆_𝑖(𝑡)𝑢(𝑥,𝑡−𝜏_𝑖)]}=𝑎(𝑡)Δ𝑢(𝑥,𝑡)+∑^𝑠_𝑘=1𝑎_𝑘(𝑡)Δ𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))−𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))−∑^𝑚_𝑗=1𝑞_𝑗(𝑥,𝑡)𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗)), (𝑥,𝑡)∈Ω×𝑅⁺≡𝐺, where Ω is a bounded domain in the Euclidean 𝑁-space 𝑅^𝑁 with a piecewise smooth boundary 𝜕Ω and Δ is the Laplacian in 𝑅^𝑁. Our results improve some known results and show that the oscillation of some second-order linear ordinary differential equations implies the oscillation of relevant nonlinear neutral partial functional differential equations.

1. Introduction

In this paper, we consider the oscillatory behavior of solutions to the neutral partial functional differential equation𝜕𝜕𝜕𝑡{𝑝(𝑡)𝜕𝑡[𝑢(𝑥,𝑡)+^𝑙_𝑖=1𝜆_𝑖(𝑡)𝑢(𝑥,𝑡−𝜏_𝑖)]}=𝑎(𝑡)Δ𝑢(𝑥,𝑡)+^𝑠_𝑘=1𝑎_𝑘(𝑡)Δ𝑢(𝑥,𝑡−𝜌_𝑘−(𝑡))−𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))_𝑚𝑗=1𝑞_𝑗(𝑥,𝑡)𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗)),(𝑥,𝑡)∈Ω×𝑅⁺≡𝐺, (1.1) with the boundary condition𝜕𝑢(𝑥,𝑡)𝜕𝜈+𝑔(𝑥,𝑡)𝑢(𝑥,𝑡)=0,(𝑥,𝑡)∈𝜕Ω×𝑅⁺≡𝐺 (1.2) or𝑢(𝑥,𝑡)=0,(𝑥,𝑡)∈𝜕Ω×𝑅⁺≡𝐺, (1.3) where Δ is the Laplacian in Euclidean N-space 𝑅^𝑁,𝑅⁺∶=(0,+∞), Ω is a bounded domain in 𝑅^𝑁 with a piecewise smooth boundary 𝜕Ω,𝜈 denotes the unit exterior normal vector to 𝜕Ω, and 𝑔(𝑥,𝑡) is a nonnegative continuous function on 𝜕Ω×𝑅⁺.

Throughout this paper we assume that the following conditions hold:(𝐶₁)𝑝∈𝐶¹(𝑅⁺,𝑅⁺),∫^∞_𝑡0(1/𝑝(𝑠))𝑑𝑠=∞,𝑡₀>0;(𝐶₂)𝜆_𝑖∈𝐶²(𝑅⁺,𝑅⁺),0≤∑^𝑙_𝑖=1𝜆_𝑖≤1, and the numbers 𝜏_𝑖 are nonnegative real constants for 𝑖∈𝐼_𝑙={1,2,…,𝑙};(𝐶₃)𝑞,𝑞_𝑗∈𝐶(𝐺,𝑅⁺),𝑞(𝑡)=min_𝑥∈Ω𝑞(𝑥,𝑡), and 𝑞_𝑗(𝑡)=min_𝑥∈Ω𝑞_𝑗(𝑥,𝑡),𝑗∈𝐼_𝑚={1,2,…,𝑚};(𝐶₄)𝑎,𝑎_𝑘∈𝐶(𝑅⁺,𝑅⁺),𝜌_𝑘∈𝐶(𝑅⁺,𝑅⁺),lim_𝑡→∞(𝑡−𝜌_𝑘(𝑡))=∞, 𝑘∈𝐼_𝑠={1,2,…,𝑠}, and 𝜎_𝑗(𝑗∈𝐼_𝑚) are nonnegative constants;(𝐶₅)𝑓,𝑓_𝑗∈𝐶(𝑅,𝑅) are convex in 𝑅⁺ with 𝑓(𝑢)/𝑢≥𝛼>0, 𝑓_𝑗(𝑢)/𝑢≥𝛼_𝑗>0 for 𝑢≠0, where 𝛼 and 𝛼_𝑗 are positive constants for 𝑗∈𝐼_𝑚.

We refer to these five conditions collectively as condition (C).

A function 𝑢∈𝐶²(𝐺)∪𝐶¹(𝐺) is called a solution of the problem (1.1), (1.2) (or (1.1), (1.3)), if it satisfies (1.1) in the domain 𝐺 and the corresponding boundary condition. A solution 𝑢 of the problem (1.1), (1.2) (or (1.1), (1.3)) is called oscillatory in the domain 𝐺 if for each positive number 𝑏 there exists a point (𝑥₀,𝑡₀)∈Ω×[𝑏,∞) such that 𝑢(𝑥₀,𝑡₀)=0.

The theory of partial differential equations with deviating arguments has received much attention (see [1]). We mention here [1–7] concerning oscillatory properties of solutions to some parabolic equations and some hyperbolic equations with deviating arguments.

By considering the function 𝐻(𝑡,𝑠), in 1999 Li and Cui [4] obtained some oscillation criteria for solutions of the problems (1.1), (1.2) and (1.1), (1.3). One of the theorems in [4] is as follows.

Theorem 1.1. Set 𝐷={(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀}. Let 𝐻∈(𝐷;𝑅) satisfy the following conditions:(i)𝐻(𝑡,𝑡)=0 for 𝑡≥𝑡₀;𝐻(𝑡,𝑠)>0 for 𝑡≥𝑠≥𝑡₀;(ii)𝐻 has a continuous and nonpositive partial derivative on 𝐷 with respect to the second variable.(iii)ℎ∶𝐷→𝑅 is a continuous function with −𝜕𝜕𝑠𝐻(𝑡,𝑠)=ℎ(𝑡,𝑠)𝐻(𝑡,𝑠)∀(𝑡,𝑠)∈𝐷. (1.4)If there exists a function 𝜙∈𝐶¹[𝑡₀,∞) and there exists some 𝑗₀∈𝐼_𝑚 such that limsup_𝑡→∞1𝐻(𝑡,𝑡₀)^𝑡_𝑡01[𝐻(𝑡,𝑠)𝜓(𝑠)−4Φ(𝑠)𝑝(𝑠−𝜎_𝑗0)ℎ²(𝑡,𝑠)]𝑑𝑠=∞, (1.5) where Φ(𝑠)=𝑒^{−2∫𝑠𝜙(𝜉)𝑑𝜉} and 𝜓(𝑡)=Φ(𝑡){𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]+𝑝(𝑡−𝜎_𝑗0)𝜙²(𝑡)−[𝑝(𝑡−𝜎_𝑗0)𝜙(𝑡)]^′}, (1.6) then every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺.

In this paper, we shall establish some new oscillation results for solutions of the problems (1.1), (1.2) and (1.1), (1.3). Our results are extensive version of Theorem 1.1. Meanwhile, our results show that the oscillation of some second-order linear ordinary differential equations implies the oscillation of relevant nonlinear second-order neutral partial functional differential (1.1), thus we can obtain some new oscillation theorems for (1.1), which do not need the condition of the integrals of the coefficient.

2. Main Results

Theorem 2.1. Let condition (C) hold, and 𝜙∈𝐶¹[𝑡₀,∞)(𝑡₀>0). Assume that there exists 𝑗₀∈𝐼_𝑚 such that the inequality 𝑊^′𝑊(𝑡)+𝜓(𝑡)+²(𝑡)𝑝(𝑡−𝜎_𝑗0)Φ(𝑡)≤0 (2.1) has no eventually positive solution, where Φ(𝑠)=exp{−2∫^𝑠𝜙(𝜉)𝑑𝜉} and 𝜓(𝑡)=Φ(𝑡){𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]+𝑝(𝑡−𝜎_𝑗0)𝜙²(𝑡)−[𝑝(𝑡−𝜎_𝑗0)𝜙(𝑡)]^′}, (2.2) then every solution u(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺.

Proof. Suppose to the contrary that there is a nonoscillatory solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) which has no zero in Ω×[𝑡₀,∞) for some 𝑡₀>0. Without loss of generality we may assume that 𝑢(𝑥,𝑡)>0,𝑢(𝑥,𝑡−𝜏_𝑖)>0,𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))>0, and 𝑢(𝑥,𝑡−𝜎_𝑗)>0 in Ω×[𝑡₁,∞),𝑡₁≥𝑡₀,𝑖∈𝐼_𝑙,𝑘∈𝐼_𝑠,𝑗∈𝐼_𝑚.
Integrating (1.1) with respect to 𝑥 over the domain Ω, we have 𝑑𝑑𝑑𝑡{𝑝(𝑡)[𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥+^𝑙_𝑖=1𝜆_𝑖(𝑡)_Ω𝑢(𝑥,𝑡−𝜏_𝑖)𝑑𝑥]}=𝑎(𝑡)_ΩΔ𝑢(𝑥,𝑡)𝑑𝑥+^𝑠_𝑘=1𝑎_𝑘(𝑡)_ΩΔ𝑢(𝑥,𝑡−𝜌_𝑘−(𝑡))𝑑𝑥_Ω𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))𝑑𝑥−^𝑚_𝑗=1Ω𝑞_𝑗(𝑥,𝑡)𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗))𝑑𝑥,𝑡≥𝑡₁. (2.3)
From Green's formula and boundary condition (1.2), it follows that _ΩΔ𝑢(𝑥,𝑡)𝑑𝑥=_𝜕Ω𝜕𝑢(𝑥,𝑡)𝜕𝜈𝑑𝑆=−_𝜕Ω𝑔(𝑥,𝑡)𝑢(𝑥,𝑡)𝑑𝑆≤0,𝑡≥𝑡₁,_ΩΔ𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))𝑑𝑥=_𝜕Ω𝜕𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))𝜕𝜈𝑑𝑆=−_𝜕Ω𝑔(𝑥,𝑡−𝜌_𝑘(𝑡))𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))𝑑𝑆≤0,𝑡≥𝑡₁,𝑘∈𝐼_𝑠, (2.4)(2.5) where 𝑑𝑆 is the surface element on 𝜕Ω. Moreover, from (𝐶₃),(𝐶₅), and Jensen's inequality it follows that _Ω𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))𝑑𝑥≥𝑞(𝑡)_Ω𝑓(𝑢(𝑥,𝑡))𝑑𝑥≥𝑞(𝑡)_𝜕Ω𝑑𝑥𝑓(_Ω𝑢(𝑥,𝑡)𝑑𝑥(_Ω𝑑𝑥)⁻¹),𝑡≥𝑡₁,_Ω𝑞_𝑗(𝑥,𝑡)𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗))𝑑𝑥≥𝑞_𝑗(𝑡)_Ω𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗))𝑑𝑥≥𝑞_𝑗(𝑡)_𝜕Ω𝑑𝑥𝑓_𝑗(_Ω𝑢(𝑥,𝑡−𝜎_𝑗)𝑑𝑥(_Ω𝑑𝑥)⁻¹),𝑡≥𝑡₁. (2.6)(2.7) Set 𝑉₁(𝑡)=_Ω𝑢(𝑥,𝑡)𝑑𝑥(_Ω𝑑𝑥)⁻¹,𝑡≥𝑡₁. (2.8)
In view of (2.4)–(2.8), (2.3) yields 𝑑𝑑𝑑𝑡{𝑝(𝑡)𝑑𝑡[𝑉₁(𝑡)+^𝑙_𝑖=1𝜆_𝑖(𝑡)𝑉₁(𝑡−𝜏_𝑖)]}+𝑞(𝑡)𝑓(𝑉₁(𝑡))+^𝑚_𝑗=1𝑞_𝑗(𝑡)𝑓_𝑗(𝑉₁(𝑡−𝜎_𝑗))≤0,𝑡≥𝑡₁. (2.9) Let 𝑍(𝑡)=𝑉₁(𝑡)+∑^𝑙_𝑖=1𝜆_𝑖(𝑡)𝑉₁(𝑡−𝜏_𝑖). We have 𝑍(𝑡)>0 and [𝑝(𝑡)𝑍^′(𝑡)]^′<0 for 𝑡≥𝑡₁. Hence 𝑝(𝑡)𝑍^′(𝑡) is a decreasing function in the interval [𝑡₁,∞). We can claim that 𝑝(𝑡)𝑍^′(𝑡)>0 for 𝑡≥𝑡₁. In fact, if 𝑝(𝑡)𝑍^′(𝑡)≤0 for 𝑡≥𝑡₁, then there exists a 𝑇≥𝑡₁ such that 𝑝(𝑇)𝑍^′(𝑇)<0. This implies that 𝑍^′(𝑡)≤𝑝(𝑇)𝑍^′(𝑇)𝑝(𝑡)for𝑡≥𝑇,𝑍(𝑡)−𝑍(𝑇)≤𝑝(𝑇)𝑍^′(𝑇)^𝑡_𝑇1𝑝(𝑠)𝑑𝑠,𝑡≥𝑇. (2.10)
Therefore lim_𝑡→∞𝑍(𝑡)=−∞, which contradicts the fact that 𝑍(𝑡)>0.
From(2.9), for the 𝑗₀ in (2.1) we obtain [𝑝(𝑡)𝑍^′(𝑡)]^′+𝑞_𝑗0(𝑡)𝑓_𝑗0(𝑉₁(𝑡−𝜎_𝑗0))≤0,𝑡≥𝑡₁. (2.11)
Noting condition (𝐶₅), from (2.11) we have [𝑝(𝑡)𝑍^′(𝑡)]^′+𝛼_𝑗0𝑞_𝑗0(𝑡)𝑉₁(𝑡−𝜎_𝑗0)≤0,𝑡≥𝑡₁ (2.12) or [𝑝(𝑡)𝑍^′(𝑡)]^′+𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]𝑍(𝑡−𝜎_𝑗0)≤0,𝑡≥𝑡₁. (2.13) Let 𝑊(𝑡)=Φ(𝑡)[𝑝(𝑡)𝑍^′(𝑡)𝑍(𝑡−𝜎_𝑗0)+𝑝(𝑡−𝜎_𝑗0)𝜙(𝑡)]; (2.14) we have 𝑊^′(𝑡)≤−2𝜙(𝑡)𝑊(𝑡)+Φ(𝑡){−𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0−)]𝑝(𝑡)𝑍^′(𝑡)𝑍^′(𝑡−𝜎_𝑗0)𝑍²(𝑡−𝜎_𝑗0)+[𝑝(𝑡−𝜎_𝑗0)𝜙(𝑡)]^′}. (2.15) Using the fact that 𝑝(𝑡)𝑍^′(𝑡) is decreasing, we get 𝑝(𝑡)𝑍^′(𝑡)≤𝑝(𝑡−𝜎_𝑗0)𝑍^′(𝑡−𝜎_𝑗0),for𝑡≥𝑡₁. (2.16) Thus 𝑊^′(𝑡)≤−2𝜙(𝑡)𝑊(𝑡)+Φ(𝑡){−𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0−1)]𝑝(𝑡−𝜎_𝑗0)(𝑝(𝑡)𝑍^′(𝑡)𝑍(𝑡−𝜎_𝑗0))²+[𝑝(𝑡−𝜎_𝑗0)𝜙(𝑡)]^′}=−2𝜙(𝑡)𝑊(𝑡)+Φ(𝑡)×{−𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0−1)]𝑝(𝑡−𝜎_𝑗0)(𝑊(𝑡)Φ(𝑡)−𝑝(𝑡−𝜎_𝑗0)𝜙(𝑡))²+[𝑝(𝑡−𝜎_𝑗0)𝜙(𝑡)]^′}𝑊=−𝜓(𝑡)−²(𝑡)𝑝(𝑡−𝜎_𝑗0,)Φ(𝑡) (2.17) that is, 𝑊(𝑡) is a positive solution of (2.1), which contradicts the assumption. This completes the proof of Theorem 2.1.

In order to study oscillation of the problem (1.1) and (1.3), the following fact will be used (see [2]). The smallest eigenvalue 𝜂₀ of the Dirichlet problem Δ𝑢(𝑥)+𝜂𝑢(𝑥)=0,inΩ,𝑢(𝑥)=0,on𝜕Ω (2.18) is positive, and the corresponding eigenfunction 𝜑(𝑥) is positive in Ω.

Theorem 2.2. Let all conditions in Theorem 2.1 hold, then every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Proof. Suppose to the contrary that there is a nonoscillatory solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) which has no zero in Ω×[𝑡₀,∞) for some 𝑡₀>0. Without loss of generality, we may assume that 𝑢(𝑥,𝑡)>0,𝑢(𝑥,𝑡−𝜏_𝑖)>0,𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))>0, and 𝑢(𝑥,𝑡−𝜎_𝑗)>0 in Ω×[𝑡₁,∞),𝑡₁≥𝑡₀,𝑖∈𝐼_𝑙,𝑘∈𝐼_𝑠,𝑗∈𝐼_𝑚.
Multiplying both sides of (1.1) by 𝜑(𝑥)>0 and integrating (1.1) with respect to 𝑥 over the domain Ω, we have 𝑑𝑑𝑑𝑡{𝑝(𝑡)[𝑑𝑡_Ω𝑢(𝑥,𝑡)𝜑(𝑥)𝑑𝑥+^𝑙_𝑖=1𝜆_𝑖(𝑡)_Ω𝑢(𝑥,𝑡−𝜏_𝑖)𝜑(𝑥)𝑑𝑥]}=𝑎(𝑡)_ΩΔ𝑢(𝑥,𝑡)𝜑(𝑥)𝑑𝑥+^𝑠_𝑘=1𝑎_𝑘(𝑡)_ΩΔ𝑢(𝑥,𝑡−𝜌_𝑘−(𝑡))𝜑(𝑥)𝑑𝑥_Ω𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))𝜑(𝑥)𝑑𝑥−^𝑚_𝑗=1Ω𝑞_𝑗(𝑥,𝑡)𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗)𝜑(𝑥))𝑑𝑥,𝑡≥𝑡₁. (2.19)
From Green's formula and boundary condition (1.3), it follows that _ΩΔ𝑢(𝑥,𝑡)𝜑(𝑥)𝑑𝑥=_Ω𝑢(𝑥,𝑡)Δ𝜑(𝑥)𝑑𝑥=−𝛽_0Ω𝑢(𝑥,𝑡)𝜑(𝑥)𝑑𝑥≤0,𝑡≥𝑡₁,_ΩΔ𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))𝜑(𝑥)𝑑𝑥=_Ω𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))Δ𝜑(𝑥)𝑑𝑥=−𝛽_0Ω𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))𝜑(𝑥)𝑑𝑥≤0,𝑡≥𝑡₁,𝑘∈𝐼_𝑠. (2.20)(2.21) Moreover, from (𝐶₃) and (𝐶₅) by Jensen's inequality it follows that _Ω𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))𝜑(𝑥)d𝑥≥𝑞(𝑡)_Ω𝑓(𝑢(𝑥,𝑡))𝜑(𝑥)𝑑𝑥≥𝑞(𝑡)_Ω𝜑(𝑥)𝑑𝑥𝑓(_Ω𝑢(𝑥,𝑡)𝜑(𝑥)𝑑𝑥(_Ω𝜑(𝑥)𝑑𝑥)⁻¹),𝑡≥𝑡₁,_Ω𝑞_𝑗(𝑥,𝑡)𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗))𝜑(𝑥)𝑑𝑥≥𝑞_𝑗(𝑡)_Ω𝑓_𝑗(𝑢(𝑥,𝑡−𝜎_𝑗))𝜑(𝑥)𝑑𝑥≥𝑞_𝑗(𝑡)_Ω𝜑(𝑥)𝑑𝑥𝑓_𝑗(_Ω𝑢(𝑥,𝑡−𝜎_𝑗)𝜑(𝑥)𝑑𝑥(_Ω𝜑(𝑥)𝑑𝑥)⁻¹),𝑡≥𝑡₁. (2.22)(2.23) Set 𝑉₂(𝑡)=_Ω𝑢(𝑥,𝑡)𝜑(𝑥)𝑑𝑥(_Ω𝜑(𝑥)𝑑𝑥)⁻¹,𝑡≥𝑡₁. (2.24)
In view of (2.20)–(2.24), (2.19) yields 𝑑𝑑𝑑𝑡{𝑝(𝑡)𝑑𝑡[𝑉₂(𝑡)+^𝑙_𝑖=1𝜆_𝑖(𝑡)𝑉₂(𝑡−𝜏_𝑖)]}+𝑞(𝑡)𝑓(𝑉₂(𝑡))+^𝑚_𝑗=1𝑞_𝑗(𝑡)𝑓_𝑗(𝑉₂(𝑡−𝜎_𝑗))≤0,𝑡≥𝑡₁. (2.25) Let 𝑍(𝑡)=𝑉₂(𝑡)+∑^𝑙_𝑖=1𝜆_𝑖(𝑡)𝑉₂(𝑡−𝜏_𝑖); the remainder of the proof is similar to that of Theorem 2.1, so we omit it.

Theorem 2.3. Let the condition (C) hold, and 𝜙∈𝐶¹[𝑡₀,∞),𝐹∈𝐶([𝑡₀,∞),𝑅). Suppose that there exists 𝑗₀∈𝐼_𝑚 such that limsup_𝑡→∞^𝑡_𝑡01[𝜓(𝑠)−4𝑝(𝑠−𝜎_𝑗0)Φ(𝑠)𝐹²(𝑠)]𝑒^{∫𝑠𝑡0𝐹(𝜏)𝑑𝜏}𝑑𝑠=∞, (2.26) where Φ(𝑠)=𝑒^{−2∫𝑠𝜙(𝜏)𝑑𝜏} and 𝜓(𝑠) is defined as in (2.2). Then(I)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺;(II)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Proof. (I) From Theorem 2.1, we only need to prove that (2.1) has no eventually positive solution. Suppose to the contrary that there is a solution 𝑤(𝑡) of system (2.1) which has no zero in [𝑡₀,∞) for some 𝑡₀>0. Without loss of generality we may assume that 𝑤0𝑥02029(𝑡)>0 in [𝑡₁,∞),𝑡₁≥𝑡₀. Hence for all 𝑡≥𝑡₁, we have by (2.1) 𝑤^′1(𝑡)≤−[𝜓(𝑡)−4𝑝(𝑡−𝜎_𝑗0)Φ(𝑡)𝐹²𝑤(𝑡)]−[²(𝑡)𝑝(𝑡−𝜎_𝑗0+1)Φ(𝑡)4𝑝(𝑡−𝜎_𝑗0)Φ(𝑡)𝐹²(𝑡)], (2.27) that is, 𝑤^′1(𝑡)+𝐹(𝑡)𝑤(𝑡)≤−[𝜓(𝑡)−4𝑝(𝑡−𝜎_𝑗0)Φ(𝑡)𝐹²(𝑡)],𝑤(𝑡)𝑒^{∫𝑡𝑡0𝐹(𝜏)𝑑𝜏}−𝑤(𝑇)𝑒^{∫𝑇𝑡0𝐹(𝜏)𝑑𝜏}≤−^𝑡_𝑇1[𝜓(𝑠)−4𝑝(𝑠−𝜎_𝑗0)Φ(𝑠)𝐹²(𝑠)]𝑒^{∫𝑠𝑡0𝐹(𝜏)𝑑𝜏}𝑑𝑠. (2.28) Hence ^𝑡_𝑇1[𝜓(𝑠)−4𝑝(𝑠−𝜎_𝑗0)Φ(𝑠)𝐹²(𝑠)]𝑒^{∫𝑠𝑡0𝐹(𝜏)𝑑𝜏}𝑑𝑠≤𝑤(𝑇)𝑒^{∫𝑇𝑡0𝐹(𝜏)𝑑𝜏}−𝑤(𝑡)𝑒^{∫𝑡𝑡0𝐹(𝜏)𝑑𝜏}. (2.29) In view of 𝑤(𝑡)≥0, we get limsup_𝑡→∞^𝑡_𝑡01[𝜓(𝑠)−4𝑝(𝑠−𝜎_𝑗0)Φ(𝑠)𝐹²(𝑠)]𝑒^{∫𝑠𝑡0𝐹(𝜏)𝑑𝜏}𝑑𝑠≤𝑤(𝑇)𝑒^{∫𝑇𝑡0𝐹(𝜏)𝑑𝜏}, (2.30) which contradicts assumption (2.26). Hence, (2.1) has no eventually positive solution. By Theorem 2.1, every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺.
(II) According to Theorem 2.2, the remainder of the proof is similar to that of the proof of part (I), so we omit the details. The proof of Theorem 2.3 is complete.

Set 𝐷={(𝑡,𝑠)∶𝑡≥𝑠≥𝑡₀}. Let 𝐻∈𝐶(𝐷,𝑅) satisfy the following conditions:(i)𝐻(𝑡,𝑡)=0, for 𝑡≥𝑡₀,𝐻(𝑡,𝑠)>0 for 𝑡>𝑠≥𝑡₀;(ii)𝐻 has a continuous and nonpositive partial derivative on 𝐷 with respect to the second variable;(iii)ℎ∶𝐷→𝑅 is a continuous function with −𝜕𝜕𝑠𝐻(𝑡,𝑠)=ℎ(𝑡,𝑠)𝐻(𝑡,𝑠),∀(𝑡,𝑠)∈𝐷. (2.31)

Taking 𝐹(𝑠)=(𝜕𝐻(𝑡,𝑠)/𝜕𝑠)/𝐻(𝑡,𝑠), we have the following Philo's type theorem in [8].

Theorem 2.4. Let the condition (C) hold, and 𝜙∈𝐶¹[𝑡₀,∞). Suppose that there exists 𝑗₀∈𝐼_𝑚 such that limsup_𝑡→∞1𝐻(𝑡,𝑡₀)^𝑡_𝑡01[𝐻(𝑡,𝑠)𝜓(𝑠)−4Φ(𝑠)𝑝(𝑠−𝜎_𝑗0)ℎ²(𝑡,𝑠)]𝑑𝑠=∞, (2.32) where Φ(𝑠)=𝑒^{−2∫𝑠𝜙(𝜏)𝑑𝜏} and Φ(𝑠) is defined as in (2.2). Then(I)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺;(II)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Remark 2.5. We can establish a lot of oscillation criteria from Theorem 2.3 if we choose differential 𝜙 and 𝐹. For example, taking 𝜙=0,𝐹=0, Theorem 2.3 reduces to a Grammatikopoulos's type criteria in [9].

Next we present another oscillation theorem.

Theorem 2.6. Let the condition (C) hold. Suppose that there exists 𝑗₀∈𝐼_𝑚 such that the following ordinary differential equation 𝑦^′′+𝑄(𝑡)𝑦(𝑡)=0 (2.33) is oscillatory, where 1𝑄(𝑡)=𝑝(𝑡−𝜎_𝑗0){𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]+[𝑝^′(𝑡−𝜎_𝑗0)]²4𝑝(𝑡−𝜎_𝑗0)−𝑝^′′(𝑡−𝜎_𝑗0)2}, (2.34) then(I) every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺;(II) every solution 𝑢(x,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Proof. Let 𝑦(𝑡) be a nonoscillatory solution of (2.33) Without loss of generality, we assume that 𝑦(𝑡)>0,𝑡≥𝑇₀≥𝑡₀. Similar to the proof Theorem 2.3, we can get 𝑤^′(𝑡)≤−𝑄(𝑡)−𝑤²(𝑡),for𝑡≥𝑇₀, (2.35) where 𝑄(𝑡) is defined as in (2.34). In fact, taking 𝜙(𝑡)=(𝑝^′(𝑡−𝜎_𝑗0))/(2𝑝(𝑡−𝜎_𝑗0)) in Theorem 2.3, we obtain (2.35) from (2.1).
Therefore, from (2.35), by using Theorem 7.2 in [10, Chap. XI], we see that (2.33) is nonoscillatory. This contradicts the fact that (2.33) is oscillatory. The proof of Theorem 2.6 is complete.

Corollary 2.7. Let the condition (C) hold. Suppose that there exists 𝑗₀∈𝐼_𝑚 such that ∞≥lim_𝑟→∞𝑡²1𝑝(𝑡−𝜎_𝑗0){𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]+[𝑝^′(𝑡−𝜎_𝑗0)]²4𝑝(𝑡−𝜎_𝑗0)−𝑝^′′(𝑡−𝜎_𝑗0)2}>14, (2.36) then(I) every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺;(II) every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Proof. From Theorem 2.5 and Theorem 7.1 in [10, Chap.XI], it is easy to see that the result of Corollary 2.7 is true.

Corollary 2.8. Let the condition (C) hold. Suppose that there exists 𝑗₀∈𝐼_𝑚 such that ∞≥liminf_𝑟→∞𝑡^∞_𝑡1𝑝(𝑡−𝜎_𝑗0)×{𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]+[𝑝^′(𝑡−𝜎_𝑗0)]²4𝑝(𝑡−𝜎_𝑗0)−𝑝^′′(𝑡−𝜎_𝑗0)2>1}𝑑𝑡4, (2.37) then(I)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺;(II)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Corollary 2.9. Let condition (C) hold. If there exist 𝑇>𝑡₀,𝛼>3−2√2, and 𝑗₀∈𝐼_𝑚 such that for every 𝑛∈𝑁, ^2𝑛+1𝑇_2𝑛𝑇1𝑝(𝑡−𝜎_𝑗0){𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]+[𝑝^′(𝑡−𝜎_𝑗0)]²4𝑝(𝑡−𝜎_𝑗0)−𝑝^′′(𝑡−𝜎_𝑗0)2𝛼}𝑑𝑡>2^𝑛𝑇, (2.38) then(I)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺;(II)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Corollary 2.10. Let the condition (C) hold, 𝜆>1, and 𝛼₀=(√𝜆−1)². If there exist T>𝑡₀,𝛼>𝛼₀, and 𝑗₀∈𝐼_𝑚 such that for every 𝑛∈𝑁, ^{𝜆𝑛+1𝑇}_𝜆𝑛𝑇1𝑝(𝑡−𝜎_𝑗0){𝛼_𝑗0𝑞_𝑗0(𝑡)[1−^𝑙_𝑖=1𝜆_𝑖(𝑡−𝜎_𝑗0)]+[𝑝^′(𝑡−𝜎_𝑗0)]²4𝑝(𝑡−𝜎_𝑗0)−𝑝^′′(𝑡−𝜎_𝑗0)2𝛼}𝑑𝑡>(𝜆−1)𝜆^𝑛𝑇, (2.39) then(I)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.2) is oscillatory in 𝐺;(II)every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.3) is oscillatory in 𝐺.

Remark 2.11. Corollaries 2.8–2.10 are easy to be proved by Theorem 2.6 of this paper, Theorems A and 2 of Huang [11], or Theorem 2 of Wong [12]. Corollaries 2.9 and 2.10 are different from the most known ones in the sense that they are based on the information only on a sequence of intervals such as [2^𝑛𝑇,2^𝑛+1𝑇], rather than on the whole half-line [t₀,∞).

Example 2.12. Let constants 𝑐>0 and 𝜇>0. Consider the partial differential equation 𝜕{1𝜕𝑡𝜕𝑡+𝜋+13𝜕𝑡[𝑢(𝑥,𝑡)+=1𝑡+2𝜋𝑢(𝑥,𝑡−2𝜋)]}1𝑡+𝜋+1Δ𝑢(𝑥,𝑡)+[(𝑡+𝜋+1)²+6(𝑡+𝜋+1)(𝑡+2𝜋)²+3(𝑡+𝜋+1)²(𝑡+2𝜋)]Δ𝑢(𝑥,𝑡−3𝜋2)6+[(𝑡+𝜋+1)(𝑡+2𝜋)³+3(𝑡+𝜋+1)²(𝑡+2𝜋)²3]Δ𝑢(𝑥,𝑡−𝜋)−[+(𝑡+𝜋+1)(𝑡+2𝜋)𝑡+𝜋𝜇𝑡+𝜋−3ln²𝑐(𝑡+1)]𝑢(𝑥,𝑡)[1+1+𝑢²]−(𝑥,𝑡)𝑡+𝜋𝜇𝑡+𝜋−3ln²(𝑡+1)𝑢(𝑥,𝑡−𝜋),(𝑥,𝑡)∈(0,𝜋)×𝑅⁺≡𝐺, (2.40) with the boundary condition 𝑢(0,𝑡)=𝑢(𝜋,𝑡)=0,𝑡≥0. (2.41)
A straightforward verification shows that the functions 𝑞₁(𝑡)=((𝑡+𝜋)/(𝑡+𝜋−3))(𝜇/ln²(𝑡+1)),𝜆₁(𝑡−𝜎₁)=𝜆₁(𝑡−𝜋)=(3/(𝑡+𝜋)), and 𝑝(𝑡−𝜎₁)=𝑝(𝑡−𝜋)=(1/(𝑡+1)). By simple computation, for constant 𝜇>0 and for each 𝑡≥0, we have 𝜇𝑄(𝑡)=(𝑡+1){ln²+1(𝑡+1)4(𝑡+1)³−1(𝑡+1)³}=𝜇(𝑡+1)ln²−3(𝑡+1)4(𝑡+1)². (2.42) Then, for constant 𝜇>0, lim_𝑡→∞𝑡²𝑄(𝑡)=lim_𝑡→∞[𝜇𝑡²(𝑡+1)ln²−(𝑡+1)3𝑡²4(𝑡+1)²1]=∞>4. (2.43) Hence, by Corollary 2.7, (2.40) is oscillatory if 𝜇>0. For example, if 𝑐=0,𝑢(𝑥,𝑡)=sin𝑥cos𝑡 is such a solution. However, criteria in [1–6] fail to imply this fact and in [7] fail to apply to (2.40) when 0<𝜇≤1. In addition, those criteria are quite difficult to apply to get oscillation of all solutions of problem (2.40), (2.41) for 𝑐>0.

Acknowledgments

The project was supported by NSF of Guangdong Province (no. 1015160150100003) and NNSF of China (no. 10971231).