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

Abstract

Using an integral averaging method and generalized Riccati technique, by introducing a parameter 𝛽≥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 equation𝜕𝜕𝑡𝑟(𝑡)𝜕𝜕𝑡𝑢(𝑥,𝑡)+𝑝(𝑡)𝜕𝑢𝜕(𝑥𝑡,𝑡)=𝑎(𝑡)Δ𝑢(𝑥,𝑡)+^𝑠_𝑘=1−𝑎_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,(1.1) where Δ is the Laplacian in 𝑅^𝑁,𝑅₊=[0,∞) and Ω is a bounded domain in 𝑅^𝑁 with a piecewise smooth boundary 𝜕Ω.

Throughout this paper, we assume that(H₁)𝑟(𝑡)∈𝐶¹(𝑅₊,(0,∞)),𝑝(𝑡)∈𝐶(𝑅₊,𝑅);(H₂)𝑞(𝑥,𝑡),𝑞_𝑗(𝑥,𝑡)∈𝐶(𝐺,𝑅₊),𝑞(𝑡)=min_𝑥∈𝐺𝑞(𝑥,𝑡),𝑞_𝑗(𝑡)=min_𝑥∈𝐺𝑞_𝑗(𝑥,𝑡),𝑗∈𝐼_𝑚={1,2,…,𝑚};(H₃)𝑎(𝑡),𝑎_𝑘(𝑡),𝜌_𝑘(𝑡)∈𝐶(𝑅₊,𝑅₊),lim_𝑡→∞(𝑡−𝜌_𝑘(𝑡))=∞, and 𝜎_𝑗 are nonnegative constants, 𝑗∈𝐼_𝑚, 𝑘∈𝐼_𝑠={1,2,…,𝑠};(H₄)𝑓(𝑢)∈𝐶¹(𝑅,𝑅),𝑓_𝑗(𝑢)∈𝐶(𝑅,𝑅) are convex in 𝑅₊ with 𝑢𝑓_𝑗(𝑢)>0,𝑢𝑓(𝑢)>0, and 𝑓′(𝑢)≥𝜇>0, (𝑢≠0).

We say that a continuous function 𝐻(𝑡,𝑠) belongs to the function class 𝜔, denoted by 𝐻∈𝜔, if 𝐻∈𝐶(𝐷,𝑅₊), where 𝐷={(𝑡,𝑠)∶−∞<𝑠≤𝑡<+∞}, satisfy𝐻(𝑡,𝑡)=0,𝐻(𝑡,𝑠)>0,−∞<𝑠<𝑡<+∞.(1.2)

Furthermore, the continuous partial derivative 𝜕𝐻/𝜕𝑆 exists on 𝐷, and there is ℎ∈𝐿_loc(𝐷,𝑅), such that𝜕𝐻𝜕𝑠=−ℎ(𝑡,𝑠)√𝐻(𝑡,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 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=1−𝑎_𝑘(𝑡)Δ𝑢𝑥,𝑡−𝜌_𝑘(𝑡)−𝑞(𝑡)𝑢(𝑥,𝑡)_𝑚_𝑗=1𝑞_𝑗(𝑥,𝑡)𝑢𝑥,𝑡−𝜎_𝑗,(𝑥,𝑡)∈Ω×𝑅₊≡𝐺(1.4) with Robin boundary condition𝜕𝑢(𝑥,𝑡)𝜕𝛾+𝑔(𝑥,𝑡)𝑢(𝑥,𝑡)=0,(𝑥,𝑡)∈𝜕Ω×𝑅⁺,(1.5) where 𝛾 is the unit exterior vector to 𝜕Ω and 𝑔(𝑥,𝑡) is a nonnegative continuous function on 𝜕Ω×𝑅₊ and obtained the following result.

Theorem A (see [12, Theorem 2.2]). Suppose that 𝐻∈𝜔, let(C₁)0<inf_𝑠≥𝑡0{liminf_𝑡→∞(𝐻(𝑡,𝑠)/𝐻(𝑡,𝑡₀))}≤∞, suppose that there exists some 𝑗₀∈𝐼_𝑚 and there exist two functions 𝜙∈𝐶¹[𝑡₀,∞),𝐴∈𝐶[𝑡₀,∞) satisfying,(C₂)limsup_𝑡→∞(1/𝐻(𝑡,𝑡₀))∫^𝑡𝑡₀𝑝(𝑠−𝜎_𝑗0)𝜙(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠<∞,(C₃)∫^∞𝑡₀(𝐴²⁺(𝑠)/𝑝(𝑠−𝜎_𝑗0)𝜙(𝑠))𝑑𝑠=∞, and for every 𝑡₁≥𝑡₀,(C₄)limsup_𝑡→∞(1/𝐻(𝑡,𝑡₁))∫^𝑡𝑡₁[𝐻(𝑡,𝑠)𝜓(𝑠)−(1/4)𝜙(𝑠)𝑝(𝑠−𝜎_𝑗0)ℎ²(𝑡,𝑠)]𝑑𝑠≥𝐴(𝑡₁), where 𝜙(𝑠)=exp{−2∫^𝑠𝜙(𝜉)𝑑𝜉},𝐴₊(𝑠)=max{𝐴(𝑠),0}, and 𝜓(𝑠)=𝜙(𝑠){𝛼_𝑗0𝑞_𝑗0(𝑠)[1−∑^𝑙𝑖=1𝜆_𝑖(𝑠−𝜎_𝑗0)]+𝑝(𝑠−𝜎_𝑗0)𝜙²(𝑠)−[𝑝(𝑠−𝜎_𝑗0)𝜙(𝑠)]′}. Then every solution 𝑢(𝑥,𝑡) 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 term𝑟(𝑡)𝑥^(𝑡)^+𝑝(𝑡)𝑥^(𝑡)+𝑞(𝑡)𝑓(𝑥(𝑡))=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 𝛽≥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 𝑦∈𝐶¹[𝑡₀,∞) such that for some 𝛽≥1 and for some 𝐻∈𝜔, lim_𝑡→∞sup1𝐻𝑡,𝑡₀^𝑡𝑡₀𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠=∞,𝑡≥0,(2.1) where 𝑣(𝑡)=exp−2^𝑡𝜓(𝑡)=𝑣(𝑡)𝑞(𝑡)+𝜇𝑟(𝑡)𝑦₂𝜇𝑦(𝑠)−2𝑝𝑟((𝑠𝑠))𝑑𝑠,(2.2)(𝑡)−𝑝(𝑡)𝑦(𝑡)−(𝑟(𝑡)𝑦(𝑡))^,(2.3) then every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Proof. Suppose to the contrary that there is a nonoscillatory solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.5) which has no zero on Ω×[𝑡₀,∞) for some 𝑡₀>0. Without loss of generality, we assume that 𝑢(𝑥,𝑡)>0,𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))>0 and 𝑢(𝑥,𝑡−𝜎_𝑗)>0 in Ω×[𝑡₁,∞),𝑡₁≥𝑡₀,𝑘∈𝐼_𝑠,𝑗∈𝐼_𝑚. Integrating (1.1) with respect to 𝑥 over the domain Ω, we have 𝑑𝑑𝑡𝑟(𝑡)𝑑𝑑𝑡_Ω𝑢(𝑥,𝑡)𝑑𝑥+𝑝(𝑡)𝑑𝑑𝑡_Ω=𝑎(𝑡)_Ω𝑢(𝑥,𝑡)𝑑𝑥Δ𝑢(𝑥,𝑡)𝑑𝑥+^𝑠_𝑘=1−𝑎_𝑘(𝑡)_ΩΔ𝑢𝑥,𝑡−𝜌_𝑘(𝑡)𝑑𝑥_Ω𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))𝑑𝑥−^𝑚_𝑗=1_Ω𝑞_𝑗(𝑥,𝑡)𝑓_𝑗𝑢𝑥,𝑡−𝜎_𝑗𝑑𝑥,𝑡≥𝑡₁.(2.4) From Green’s formula and the boundary condition (1.5), we have _ΩΔ𝑢(𝑥,𝑡)𝑑𝑥=_𝜕Ω𝜕𝑢(𝑥,𝑡)𝜕𝛾𝑑𝑠=−_𝜕Ω𝑔(𝑥,𝑡)𝑢(𝑥,𝑡)𝑑𝑠≤0,_ΩΔ𝑢𝑥,𝑡−𝜌_𝑘(𝑡)𝑑𝑥=_𝜕Ω𝜕𝑢𝑥,𝑡−𝜌_𝑘(𝑡)=−_𝜕Ω𝜕𝛾𝑑𝑠𝑔𝑥,𝑡−𝜌_𝑘(𝑡)𝑢𝑥,𝑡−𝜌_𝑘(𝑡)𝑑𝑠≤0,𝑡≥𝑡₁,𝑘∈𝐼_𝑠,(2.5) where 𝑑𝑠 denotes the surface element on 𝜕Ω. Moreover, from (H₂), (H₄) and Jensen’s inequality, we have _Ω𝑞(𝑥,𝑡)𝑓(𝑢(𝑥,𝑡))𝑑𝑥≥𝑞(𝑡)_Ω𝑓(𝑢(𝑥,𝑡))𝑑𝑥≥||Ω||𝑞(𝑡)𝑓1||Ω||_Ω𝑢(𝑥,𝑡)𝑑𝑥_Ω,𝑞_𝑗(𝑥,𝑡)𝑓_𝑗𝑢𝑥,𝑡−𝜎_𝑗𝑑𝑥≥≥𝑞||Ω_𝑗(||𝑡𝑞)_Ω𝑓_𝑗𝑢𝑥,𝑡−𝜎_𝑗𝑑𝑥_𝑗(𝑡)𝑓_𝑗||Ω1||_Ω𝑢𝑥,𝑡−𝜎_𝑗𝑑𝑥,(2.6) where |Ω|=∫_Ω𝑑𝑥.
Set 𝑈(𝑡)=1||Ω||_Ω𝑢(𝑥,𝑡)𝑑𝑥,𝑡≥𝑡₁.(2.7) In view of (2.5)–(2.7), (2.4) yields that 𝑟(𝑡)𝑈^(𝑡)^+𝑝(𝑡)𝑈^(𝑡)+𝑞(𝑡)𝑓(𝑈(𝑡))+^𝑚_𝑗=1𝑞_𝑗(𝑡)𝑓_𝑗𝑈𝑡−𝜎_𝑗≤0,𝑡≥𝑡₁.(2.8) Note that (H₄), (2.8) yields that (𝑟(𝑡)𝑈′(𝑡))′+𝑝(𝑡)𝑈′(𝑡)+𝑞(𝑡)𝑓(𝑈(𝑡))≤0,𝑡≥𝑡₁.(2.9) Put 𝑤(𝑡)=𝑣(𝑡)𝑟(𝑡)𝑈^(𝑡)𝑓(𝑈(𝑡))+𝑦(𝑡),𝑡≥𝑡¹,(2.10) where 𝑣(𝑡) is given by (2.2), then 𝑤^(𝑡)=−2𝜇𝑦(𝑡)+𝑝(𝑡)𝑟(𝑡)𝑤(𝑡)+𝑣(𝑡)𝑟(𝑡)𝑈^(𝑡)^𝑓(𝑈(𝑡))−𝑟(𝑡)𝑓′(𝑈(𝑡))𝑈^(𝑡)²≤−2𝜇𝑦(𝑡)+𝑝(𝑡)𝑓²(𝑈(𝑡))+(𝑟(𝑡)𝑦(𝑡))^𝑟(𝑡)𝑤(𝑡)+𝑣(𝑡)𝑟(𝑡)𝑈^(𝑡)^𝑓(𝑈(𝑡))−𝜇𝑟(𝑡)𝑈^(𝑡)²≤−2𝜇𝑦(𝑡)+𝑝(𝑡)𝑓²(𝑈(𝑡))+(𝑟(𝑡)𝑦(𝑡))^𝑟(𝑡)𝑤(𝑡)−𝑣(𝑡)𝑝(𝑡)𝑈^(𝑡)𝑓(𝑈(𝑡))+𝑞(𝑡)+𝜇𝑟(𝑡)𝑈^(𝑡)𝑓(𝑈(𝑡))²−(𝑟(𝑡)𝑦(𝑡))^=−2𝜇𝑦(𝑡)+𝑝(𝑡)𝑟(𝑡)𝑤(𝑡)−+𝑣𝜇(𝑟𝑡)(𝑡)𝑝(𝑡)𝑤(𝑣𝑡)(𝑤𝑡)(𝑟𝑡)(𝑡)−𝑦(𝑡)+𝑞(𝑡)𝑣(𝑡)𝑟(𝑡)−𝑦(𝑡)²−(𝑟(𝑡)𝑦(𝑡))^=−2𝜇𝑦(𝑡)+𝑝(𝑡)−𝑣(𝑡)𝑝(𝑡𝑟)(𝑡)𝑤(𝑡)+𝑣𝜇(𝑡𝑟)(𝑟𝑡()𝑡)𝑤𝑤(₂𝑡)−𝑝(𝑡)𝑦(𝑡)+𝑞(𝑡)(𝑡)𝑣²(𝑡)𝑟²(𝑡)−2𝜇𝑤(𝑣𝑡)(𝑡𝑦)(𝑡)+𝜇𝑟(𝑡)𝑦²(𝑡)−(𝑟(𝑡)𝑦(𝑡))^=−2𝜇𝑦(𝑡)𝑤(𝑡)+𝑝(𝑡)𝑟(𝑡)𝑤(𝑡)−𝑝𝑟((𝑡𝑡))𝑤(𝑡)−𝑣(𝑡)−𝑝(𝑡)𝑦(𝑡)+𝑞(𝑡)+𝜇𝑟(𝑡)𝑦²+2𝜇𝑦(𝑡)𝑤(𝑡)−𝜇𝑤₂(𝑡)−(𝑟(𝑡)𝑦(𝑡))^(𝑡)=−𝑣(𝑡)−𝑝(𝑡)𝑦(𝑡)+𝑣(𝑞𝑡)(𝑡𝑟)(+𝑡)𝜇𝑟(𝑡)𝑦₂(𝑡)−(𝑟(𝑡)𝑦(𝑡))^−𝜇𝑤²(𝑡)=−𝜓(𝑡)−𝜇𝑤₂𝑣(𝑡)𝑟(𝑡)(𝑡)𝑣(𝑡)𝑟(𝑡),(2.11) that is, 𝜓(𝑡)≤−𝑤^(𝑡)−𝜇𝑤²(𝑡)𝑣(𝑡)𝑟(𝑡),(2.12) where 𝜓(𝑡) is defined by (2.3). Multiplying (2.12) by 𝐻(𝑡,𝑠) and integrating from 𝑇 to 𝑡, we have, for some 𝛽≥1 and for all 𝑡≥𝑇≥𝑡₁, ^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)𝑑𝑠≤−^𝑡𝑇𝐻(𝑡,𝑠)𝑤^(𝑠)𝑑𝑠−^𝑡𝑇𝐻(𝑡,𝑠)𝜇𝑣(𝑠)𝑟(𝑠)𝑤²=𝐻(𝑡,𝑇)𝑤(𝑇)−_𝑡𝑇(𝑠)𝑑𝑠ℎ(𝑡,𝑠)√𝐻(𝑡,𝑠)𝑤(𝑠)+𝐻(𝑡,𝑠)𝜇𝑤²(𝑠)=𝐻(𝑡,𝑇)𝑤(𝑇)−_𝑡𝑇𝑣(𝑠)𝑟(𝑠)𝑑𝑠⎡⎢⎢⎣𝜇𝐻(𝑡,𝑠)𝛽𝑣(𝑠)𝑟(𝑠)𝑤(𝑠)+𝛽𝑣(𝑠)𝑟(𝑠)4𝜇ℎ(𝑡,𝑠)⎤⎥⎥⎦²+𝛽𝑑𝑠4𝜇^𝑡𝑇𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠−^𝑡𝑇(𝛽−1)𝜇𝛽𝑣(𝑠)𝑟(𝑠)𝐻(𝑡,𝑠)𝑤²(𝑠)𝑑𝑠.(2.13) Writing the latter inequality in the form ^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²≤𝐻(𝑡,𝑇)𝑤(𝑇)−_𝑡𝑇(𝑡,𝑠)𝑑𝑠⎡⎢⎢⎣𝜇𝐻(𝑡,𝑠)𝛽𝑣(𝑠)𝑟(𝑠)𝑤(𝑠)+𝛽𝑣(𝑠)𝑟(𝑠)4𝜇ℎ(𝑡,𝑠)⎤⎥⎥⎦²−𝑑𝑠_𝑡𝑇(𝛽−1)𝜇𝛽𝑣(𝑠)𝑟(𝑠)𝐻(𝑡,𝑠)𝑤²(𝑠)𝑑𝑠.(2.14) Using the properties of 𝐻(𝑡,𝑠), we have ^𝑡𝑡₁𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠≤𝐻𝑡,𝑡₁||𝑤𝑡₁||≤𝐻𝑡,𝑡₀||𝑤𝑡₁||,𝑡≥𝑡₁,(2.15) and for all 𝑡≥𝑡₁≥𝑡₀, ^𝑡𝑡₀𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠≤𝐻𝑡,𝑡₀^𝑡1_𝑡0||𝜓(𝑠)||𝑑𝑠+||𝑤𝑡₁||.(2.16) By (2.16), lim_𝑡→∞sup1𝐻𝑡,𝑡₀^𝑡𝑡₀𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠≤^𝑡1_𝑡0||𝜓(𝑠)||𝑑𝑠+||𝑤𝑡₁||<∞,(2.17) which contradicts (2.1). This proves Theorem 2.1.

Consider a Kamenev-type function 𝐻(𝑡,𝑠) defined by 𝐻(𝑡,𝑠)=(𝑡−𝑠)^𝑛−1,(𝑡,𝑠)∈𝐷, where 𝑛>2 is an integer. Obviously, 𝐻 belongs to the class 𝜔, and ℎ(𝑡,𝑠)=(𝑛−1)(𝑡−𝑠)^(𝑛−3)/2,(𝑡,𝑠)∈𝐷. Then, we can get the following results.

Corollary 2.2. Suppose that there exists a function 𝑦(𝑡)∈𝐶¹([𝑡₀,∞);𝑅) such that for some integer 𝑛>2 and some 𝛽≥1, lim_𝑡→∞sup1𝑡^𝑛−1^𝑡𝑡₀(𝑡−𝑠)^𝑛−3(𝑡−𝑠)²𝜓(𝑠)−𝛽(𝑛−1)²4𝜇𝑣(𝑠)𝑟(𝑠)𝑑𝑠=∞,(2.18) where 𝑣(𝑡) and 𝜓(𝑡) are as defined in Theorem 2.1. Then every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Theorem 2.3. Suppose that 0<inf_𝑠≥𝑡0lim^𝑡→∞inf𝐻(𝑡,𝑠)𝐻𝑡,𝑡₀≤∞.(2.19) Assume that there exist functions 𝑓∈𝐶¹([𝑡₀,∞);𝑅) and 𝜙∈𝐶([𝑡₀,∞);𝑅) such that, for all 𝑡≥𝑇≥𝑡₀ and for some 𝛽>1, lim_𝑡→∞sup1𝐻(𝑡,𝑇)^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠≥𝜙(𝑇),(2.20) where 𝑣(𝑡),𝜓(𝑡) are as defined in Theorem 2.1 and suppose further that lim_𝑡→∞sup^𝑡𝑡₀𝜙²⁺(𝑠)𝑣(𝑠)𝑟(𝑠)=∞,(2.21) where 𝜙₊(𝑡)=max(𝜙(𝑡),0). Then every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Proof. Suppose to the contrary that there is a nonoscillatory solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.5) which has no zero on Ω×[𝑡₁,∞) for some 𝑡₁>𝑡₀, without loss of generality, we assume that 𝑢(𝑥,𝑡)>0,𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))>0 and 𝑢(𝑥,𝑡−𝜎_𝑗)>0 in Ω×[𝑡₁,∞),𝑡≥𝑡₁≥𝑡₀,𝑘∈𝐼_𝑠,𝑗∈𝐼_𝑚.
As in the proof of Theorem 2.1, (2.14) holds for all 𝑡≥𝑇≥𝑡₁, we have 1𝐻(𝑡,𝑇)^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²≤𝑤(𝑇)−1(𝑡,𝑠)𝑑𝑠𝐻(𝑡,𝑇)^𝑡𝑇⎡⎢⎢⎣𝜇𝐻(𝑡,𝑠)𝛽𝑣(𝑠)𝑟(𝑠)𝑤(𝑠)+𝛽𝑣(𝑠)𝑟(𝑠)4𝜇ℎ(𝑡,𝑠)⎤⎥⎥⎦²−1𝑑𝑠𝐻(𝑡,𝑇)^𝑡𝑇(𝛽−1)𝜇𝛽𝑣(𝑠)𝑟(𝑠)𝐻(𝑡,𝑠)𝑤²≤𝑤(𝑇)−1(𝑠)𝑑𝑠𝐻(𝑡,𝑇)^𝑡𝑇(𝛽−1)𝜇𝛽𝑣(𝑠)𝑟(𝑠)𝐻(𝑡,𝑠)𝑤²(𝑠)𝑑𝑠.(2.22) Therefore, for 𝑡>𝑇≥𝑇₀, lim_𝑡→∞sup1𝐻(𝑡,𝑇)^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−4𝛽𝜇𝑣(𝑠)𝑟(𝑠)ℎ²≤𝑤(𝑇)−lim_𝑡→∞(𝑡,𝑠)𝑑𝑠inf1𝐻(𝑡,𝑇)^𝑡𝑇(𝛽−1)𝜇𝛽𝑣(𝑠)𝑟(𝑠)𝐻(𝑡,𝑠)𝑤²(𝑠)𝑑𝑠.(2.23) It follows from (2.20) that 𝑤(𝑇)≥𝜙(𝑇)+lim_𝑡→∞inf1𝐻(𝑡,𝑇)^𝑡𝑇(𝛽−1)𝜇𝛽𝑣(𝑠)𝑟(𝑠)𝐻(𝑡,𝑠)𝑤²(𝑠)𝑑𝑠(2.24) for all 𝑇≥𝑡₁ and for any 𝛽>1. Then, for all 𝑇≥𝑡₁, lim_𝑡→∞𝑤(𝑇)≥𝜙(𝑇),(2.25)inf1𝐻𝑡,𝑡₁^𝑡𝑡₁𝐻(𝑡,𝑠)𝑣(𝑠)𝑟(𝑠)𝑤²(𝑠)𝑑𝑠≤𝛽𝜇(𝛽−1)𝑤𝑡¹−𝜙𝑡₁.(2.26) Now, we claim that ^∞𝑡₁𝑤²(𝑠)𝑣(𝑠)𝑟(𝑠)𝑑𝑠<∞.(2.27) Suppose the contrary, that is, ^∞𝑡₁𝑤²(𝑠)𝑣(𝑠)𝑟(𝑠)𝑑𝑠=∞.(2.28) By (2.19), there is a positive constant 𝑀₁, satisfying inf_𝑠≥𝑡0lim^𝑡→∞inf𝐻(𝑡,𝑠)𝐻𝑡,𝑡₀>𝑀¹>0.(2.29) Let 𝑀 be any arbitrary positive number, then from (2.28) we get that there exists a 𝑇₁>𝑡₁ such that, for all 𝑡≥𝑇₁, ^𝑡𝑡₁𝑤²(𝑠)𝑣(𝑠)𝑟(𝑠)𝑑𝑠≥𝑀𝑀₁.(2.30) Using integration by parts, for all 𝑡≥𝑇₁, we get 1𝐻𝑡,𝑡₁^𝑡𝑡₁𝐻(𝑡,𝑠)𝑤²(𝑠)𝑣(𝑠)𝑟(𝑠)𝑑𝑠=𝐻1𝑡,𝑡₁^𝑡𝑡₁−𝜕𝐻𝜕(𝑠𝑡,𝑠)^𝑠𝑡₁𝑤²(𝜏)≥1𝑣(𝜏)𝑟(𝜏)𝑑𝜏𝑑𝑠𝐻𝑡,𝑡₁^𝑡𝑇₁−𝜕𝐻𝜕(𝑠𝑡,𝑠)^𝑠𝑡₁𝑤²(𝜏)≥𝑀𝑣(𝜏)𝑟(𝜏)𝑑𝜏𝑑𝑠𝑀₁1=𝑀𝐻𝑡,𝑡₁^𝑡𝑇₁−𝜕𝐻𝜕(𝑠𝑡,𝑠)𝑑𝑠𝑀₁𝐻𝑡,𝑇₁𝐻𝑡,𝑡₁.(2.31) By (2.29), there exists a 𝑇₂>𝑇₁ such that, for all 𝑡≥𝑇₂, 𝐻𝑡,𝑇₁𝐻𝑡,𝑡₁≥𝑀¹.(2.32) It follows from (2.31) that for all 𝑡≥𝑇₂, 1𝐻𝑡,𝑡₁^𝑡𝑡₁𝐻(𝑡,𝑠)𝑤²(𝑠)𝑣(𝑠)𝑟(𝑠)𝑑𝑠≥𝑀.(2.33) Since 𝑀 is an arbitrary positive constant, lim_𝑡→∞inf1𝐻𝑡,𝑡₁^𝑡𝑡₁𝐻(𝑡,𝑠)𝑤²(𝑠)𝑣(𝑠)𝑟(𝑠)𝑑𝑠=∞,(2.34) which contradicts (2.26). Consequently, (2.27) holds. And from (2.25), we obtain ^∞𝑡₁𝜙²⁺(𝑠)𝑣(𝑠)𝑟(𝑠)𝑑𝑠≤^∞𝑡₁𝑤²(𝑠)𝑣(𝑠)𝑟(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(𝑡) and 𝜓(𝑡) be as in Theorem 2.1, assume further that there exist functions 𝑓∈𝐶¹([𝑡₀,∞);𝑅) and 𝜙∈𝐶([𝑡₀,∞);𝑅) such that, for all 𝑇≥𝑡₀, for some integer 𝑛>2, and for some 𝛽>1, lim_𝑡→∞sup1𝑡^𝑛−1^𝑡𝑇(𝑡−𝑠)^𝑛−3(𝑡−𝑠)²𝜓(𝑠)−𝛽(𝑛−1)²4𝜇𝑣(𝑠)𝑟(𝑠)𝑑𝑠≥𝜙(𝑇)(2.36) and (2.21) hold. Then every solution 𝑢(𝑥,𝑡) of the problem (1.1), (1.5) is oscillatory in 𝐺.

Theorem 2.5. Suppose that there exists a function 𝑓∈𝐶¹([𝑡₀,∞);𝑅) such that for some 𝛽≥1 and for some 𝐻∈𝜔, lim_𝑡→∞sup1𝐻𝑡,𝑡₀^𝑡𝑡₀𝐻(𝑡,𝑠)𝜓(𝑠)−𝑝²(𝑠)𝑣(𝑠)2𝜇𝑟(𝑠)−𝜇2𝛽𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠=∞,(2.37) where 𝑣(𝑡)=exp−2𝜇^𝑡𝑦(𝑠)𝑑𝑠,𝜓(𝑡)=𝑣(𝑡)𝑞(𝑡)+𝜇𝑟(𝑡)𝑦²(𝑡)−𝑝(𝑡)𝑦(𝑡)−(𝑟(𝑡)𝑦(𝑡))^.(2.38) Then every solution 𝑢(𝑥,𝑡) 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 𝑢(𝑥,𝑡) of the problem (1.1), (1.5) satisfies 𝑢(𝑥,𝑡)>0,𝑢(𝑥,𝑡−𝜌_𝑘(𝑡))>0 and 𝑢(𝑥,𝑡−𝜎_𝑗)>0 in Ω×[𝑡₁,∞), 𝑡₁≥𝑡₀, 𝑘∈𝐼_𝑠,𝑗∈𝐼_𝑚. Define a generalized Riccati transformation 𝑤(𝑡)=𝑣(𝑡)𝑟(𝑡)𝑈^(𝑡)𝑓(𝑈(𝑡))+𝑦(𝑡),𝑡≥𝑡¹,(2.39) where 𝑣(𝑡) is given by (2.38). Then 𝑤^(𝑡)≤−2𝜇𝑦(𝑡)𝑤(𝑡)+𝑣(𝑡)−𝑞(𝑡)+(𝑟(𝑡)𝑦(𝑡))^−𝑝(𝑡)𝑤(𝑡)𝑣(𝑡)𝑟(𝑡)−𝑦(𝑡)−𝜇𝑟(𝑡)𝑤(𝑡)=−𝑣(𝑡)𝑟(𝑡)−𝑦(𝑡)²𝜓(𝑡)−𝑝(𝑡)𝑟(𝑡)𝑤(𝑡)−𝜇1𝑟(𝑡)𝑣(𝑡)𝑤²(𝑡),𝑡≥𝑡₁.(2.40) Using an elementary inequality −𝑎𝑥²+𝑏𝑥≤−𝑎2𝑥²+𝑏²2𝑎,(2.41) for all 𝑎>0 and for all 𝑏,𝑥∈𝑅, we conclude from (2.40) that 𝜓(𝑡)−𝑝²(𝑡)𝑣(𝑡)2𝜇𝑟(𝑡)≤−𝑤^(𝑡)−𝜇2𝑣(𝑡)𝑟(𝑡)𝑤²(𝑡),𝑡≥𝑡₁.(2.42) Multiplying (2.42) by 𝐻(𝑡,𝑠) and integrating from 𝑇<𝑡, we obtain, for some 𝛽≥1 and for all 𝑡≥𝑇≥𝑡₁, ^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−𝑝²(𝑠)𝑣(𝑠)2𝜇𝑟(𝑠)𝑑𝑠≤𝐻(𝑡,𝑇)𝑤(𝑇)−^𝑡𝑇ℎ(𝑡,𝑠)√𝐻(𝑡,𝑠)𝑤(𝑠)𝑑𝑠−^𝑡𝑇𝜇𝑤²(𝑠)≤𝐻(𝑡,𝑇)2𝑣(𝑠)𝑟(𝑠)𝑑𝑠𝑤(𝑇)+𝜇𝛽2^𝑡𝑇−𝜇_𝑡𝑇𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠(𝛽−1)𝐻(𝑡,𝑠)2𝛽𝑣(𝑠)𝑟(𝑠)−𝜇𝑤²(𝑠)𝑑𝑠2^𝑡𝑇𝐻(𝑡,𝑠)𝛽𝑣(𝑠)𝑟(𝑠)𝑤(𝑠)+√𝛽𝑣(𝑠)𝑟(𝑠)ℎ(𝑡,𝑠)²𝑑𝑠.(2.43) Therefore, for all 𝑡≥𝑇≥𝑡₁, we have ^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−𝐻(𝑡,𝑠)𝑝²(𝑠)𝑣(𝑠)2𝜇𝑟(𝑠)−𝜇2𝛽≤𝐻(𝑡,𝑇)𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠𝑤(𝑇)−𝜇^𝑡𝑇(𝛽−1)𝐻(𝑡,𝑠)2𝛽𝑣(𝑠)𝑟(𝑠)−𝜇𝑤²(𝑠)𝑑𝑠2^𝑡𝑇𝐻(𝑡,𝑠)𝛽𝑣(𝑠)𝑟(𝑠)𝑤(𝑠)+√𝛽𝑣(𝑠)𝑟(𝑠)ℎ(𝑡,𝑠)²𝑑𝑠.(2.44) Following the same lines as in the proof of Theorem 2.1, we have lim_𝑡→∞sup1𝐻𝑡,𝑡₀^𝑡𝑡₀𝐻(𝑡,𝑠)𝜓(𝑠)−𝐻(𝑡,𝑠)𝑝²(𝑠)𝑣(𝑠)2𝜇𝑟(𝑠)−𝜇2𝛽≤𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠_𝑡1𝑡0||𝜓(𝑠)||𝑑𝑠+||𝑤𝑡₁||<∞(2.45) which contradicts the assumption (2.19).
This completes the proof.

Theorem 2.6. Let (2.19) holds. Assume that there exist functions 𝑓∈𝐶¹([𝑡₀,∞);𝑅) and 𝜙∈𝐶([𝑡₀,∞),𝑅) such that, for all 𝑡≥𝑡₀, any 𝑇≥𝑡₀, and for some 𝛽>1, lim_𝑡→∞sup1𝐻(𝑡,𝑇)^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−𝑝²(𝑠)𝑣(𝑠)2𝜇𝑟(𝑠)−𝜇2𝛽𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠≥𝜙(𝑇)(2.46) and (2.21) holds, where 𝜓(𝑡),𝑣(𝑡) are defined as in Theorem 2.6 and 𝜙₊(𝑡)=max(𝜙(𝑡),0), then every solution 𝑢(𝑥,𝑡) 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 lim_𝑡→∞inf1𝐻(𝑡,𝑇)^𝑡𝑇𝐻(𝑡,𝑠)𝜓(𝑠)−𝑝²(𝑠)𝑣(𝑠)2𝜇𝑟(𝑠)−𝜇2𝛽𝑣(𝑠)𝑟(𝑠)ℎ²(𝑡,𝑠)𝑑𝑠≥𝜙(𝑇).(2.47) Then every solution 𝑢(𝑥,𝑡) 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 𝜕𝜕𝑡1𝑡=𝜕𝜕Δ𝑡𝑢𝑢(𝑥(𝑥,𝑡,)𝑡)++112𝑢(𝑥,𝑡−𝜋)+2cos𝑡𝜕𝑢𝜕(𝑥𝑡,𝑡)𝑡²Δ𝑢𝑥,𝑡−3−22𝜋𝑡³+𝑡cos²𝑡+sin𝑡𝑓(𝑢(𝑥,𝑡))−1𝑡²𝑓₁(𝑢(𝑥,𝑡−𝜋)),(𝑥,𝑡)∈(0,𝜋)×(0,∞),(3.1) with the boundary condition 𝑢_𝑥(0,𝑡)=𝑢_𝑥(𝜋,𝑡)=0,𝑡>0,(3.2) where 𝑓(𝑢)=𝑢³+𝑢,𝑓₁(𝑢)=𝑢𝑒^𝑢+𝑢.
Here, 𝑁=1,𝑙=1,𝑠=1,𝑚=1,𝜇=1, 𝑟(𝑡)=1/𝑡, 𝑝(𝑡)=2cos𝑡, 𝑞(𝑥,𝑡)=𝑞(𝑡)=(2/𝑡³+𝑡cos²𝑡+sin𝑡), 𝑞₁(𝑥,𝑡)=1/𝑡², 𝑓(𝑢)=𝑓_𝑗(𝑢)=𝑢,𝑎(𝑡)=1,𝑎₁(𝑡)=1/𝑡², 𝜌₁(𝑡)=(3/2)𝜋,𝜎₁=𝜋,𝜏₁=𝜋.
Let 𝑦(𝑡)=1𝑡+𝑡cos𝑡,(3.3) then 𝑣(𝑡)=𝑡²,𝜓(𝑡)=𝑡⁻¹.(3.4) Let 𝑛=3, for any 𝛽≥1, lim_𝑡→∞sup1𝑡²^𝑡1(𝑡−𝑠)²𝑠⁻¹−𝛽𝑠²1𝑠𝑑𝑠=lim^𝑡→∞sup1𝑡²^𝑡1(𝑡−𝑠)²𝑠⁻¹−𝛽𝑠𝑑𝑠=∞.(3.5) Therefore, Corollary 2.2 holds, then every solution 𝑢(𝑥,𝑡) of the problem (3.1), (3.2) oscillates in (0,𝜋)×(0,∞).

Example 3.2. Consider the partial differential equation 𝜕𝜕𝑡=13+Δ2𝑢1𝑡(³𝑥,𝑡()2++(s2in−𝑡)co𝜕s𝑢𝑡𝜕()𝑥𝑡Δ,𝑡𝑢)𝑥+,𝑡3𝑡−31+21𝑡³(2+sin𝑡)𝜕𝑢𝜕(𝑥𝑡,𝑡)2𝜋−𝑡⁻³1−𝑡³−(2𝑡+sin𝑡)𝑓₁+2𝑡²−6𝑡sin𝑡+12𝑡𝑓(𝑢(𝑥,𝑡))(𝑢(𝑥,𝑡−𝜋))−2𝑓₂𝑢𝑥,𝑡−𝜋2,(𝑥,𝑡)∈(0,𝜋)×(0,∞)(3.6) with the boundary condition (3.2), where 𝑓(𝑢)=𝑢⁵+𝑢,𝑓₁(𝑢)=𝑢sin²𝑢,𝑓₂(𝑢)=𝑢³cos²𝑢.
Here 𝑁=1,𝑠=1,𝑚=2,𝜇=1, 𝑟(𝑡)=(1+1/2𝑡³)(2+sin𝑡), 𝑝(𝑡)=(3/𝑡)(1+1/2𝑡³)(2+sin𝑡), 𝑞(𝑡)=𝑡⁻³[(1−𝑡³+2𝑡²−6𝑡)sin𝑡+12𝑡], 𝑎(𝑡)=3,𝑎₁(𝑡)=2−cos𝑡,𝑞₁(𝑥,𝑡)=2+sin𝑡,𝑞₂(𝑥,𝑡)=2, 𝜌₁(𝑡)=(3/2)𝜋, 𝜎₁=𝜋,𝜎₂=𝜋/2.
Let 𝑦(𝑡)=0, then 𝑣(𝑡)=𝑡³ and 𝜓(𝑡)=𝑣(𝑡)𝑞(𝑡)=(1−𝑡³+2𝑡²−6𝑡)sin𝑡+12𝑡.
Choose 𝛽=2, 𝑛=3, a straightforward computation yields limsup_𝑡→∞1𝑡²^𝑡𝑇=16−𝑇(₃𝑡−𝑠)²1−𝑠³+2𝑠²−6𝑠sin𝑠+12𝑠−2𝑠³+1(2+sin𝑠)𝑑𝑠cos𝑇+𝑇²(2cos𝑇−6+3sin𝑇)−4𝑇sin𝑇−3cos𝑇=𝜙(𝑇).(3.7) Let 𝜙₊(𝑡)=max(𝜙(𝑡),0). It is not difficult to see that limsup_𝑡→∞^𝑡1𝜙²⁺(𝑠)𝑠³+(1/2)(2+sin𝑠)𝑑𝑠≥limsup^𝑡→∞^𝑡1𝜙²⁺(𝑠)3𝑠³+(1/2)𝑑𝑠=∞.(3.8) By Corollary 2.4, we obtain that every solution of problem (3.6), (3.2) oscillates in (0,𝜋)×(0,∞).
Note that in this example, limsup_𝑡→∞1𝑡²^𝑡14𝑠³+12+(2+sin𝑠)𝑑𝑠=∞,(3.9) so the condition (C₂) would not have been satisfied with the same choices of 𝑣(𝑡).

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).