# Oscillation for Certain Nonlinear Neutral Partial Differential Equations

Authors: Quanwen Lin; Rongkun Zhuang
Journal: International Journal of Differential Equations
 (2010)

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## 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)+∑i=1lλi(t)u(x,t-τi)]}=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t-ρk(t))-q(x,t)f(u(x,t))-∑j=1mqj(x,t)fj(u(x,t-σj)), (x,t)∈Ω×R+≡G, where Ω is a bounded domain in the Euclidean N-space RN with a piecewise smooth boundary ∂Ω and Δ is the Laplacian in RN. 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.


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## Body

## 1. Introduction

In this paper, we consider the oscillatory behavior of solutions to the neutral partial functional differential equation(1.1)∂∂t{p(t)∂∂t[u(x,t)+∑i=1lλi(t)u(x,t-τi)]}=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t-ρk(t))-q(x,t)f(u(x,t))-∑j=1mqj(x,t)fj(u(x,t-σj)),(x,t)∈Ω×R+≡G,
with the boundary condition(1.2)∂u(x,t)∂ν+g(x,t)u(x,t)=0,(x,t)∈∂Ω×R+≡G
or(1.3)u(x,t)=0,(x,t)∈∂Ω×R+≡G,
where Δ is the Laplacian in Euclidean N-space RN,R+:=(0,+∞), Ω is a bounded domain in RN 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)
λi∈C2(R+,R+),0≤∑i=1lλi≤1, and the numbers τi are nonnegative real constants for i∈Il={1,2,…,l};(C3)
q,qj∈C(G¯,R+),q(t)=minx∈Ω¯q(x,t), and qj(t)=minx∈Ω¯qj(x,t),j∈Im={1,2,…,m};(C4)
a,ak∈C(R+,R+),ρk∈C(R+,R+),limt→∞(t-ρk(t))=∞, k∈Is={1,2,…,s}, and σj(j∈Im) are nonnegative constants;(C5)
f,fj∈C(R,R) are convex in R+ with f(u)/u≥α>0, fj(u)/u≥αj>0 for u≠0, where α and αj are positive constants for j∈Im.We refer to these five conditions collectively as condition (C).A functionu∈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 functionH(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.
SetD={(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
(1.4)-∂∂sH(t,s)=h(t,s)H(t,s)∀(t,s)∈D.If there exists a function ϕ∈C1[t0,∞) and there exists some j0∈Im such that
(1.5)lim supt→∞1H(t,t0)∫t0t[H(t,s)ψ(s)-14Φ(s)p(s-σj0)h2(t,s)]ds=∞,
where Φ(s)=e-2∫sϕ(ξ)dξ and
(1.6)ψ(t)=Φ(t){αj0qj0(t)[1-∑i=1lλi(t-σj0)]+p(t-σj0)ϕ2(t)-[p(t-σj0)ϕ(t)]′},
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, andϕ∈C1[t0,∞)(t0>0). Assume that there exists j0∈Im such that the inequality
(2.1)W′(t)+ψ(t)+W2(t)p(t-σj0)Φ(t)≤0
has no eventually positive solution, where Φ(s)=exp{-2∫sϕ(ξ)dξ} and
(2.2)ψ(t)=Φ(t){αj0qj0(t)[1-∑i=1lλi(t-σj0)]+p(t-σj0)ϕ2(t)-[p(t-σj0)ϕ(t)]′},
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 solutionu(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-τi)>0,u(x,t-ρk(t))>0, and u(x,t-σj)>0 in Ω×[t1,∞),t1≥t0,i∈Il,k∈Is,j∈Im.
Integrating (1.1) with respect to x over the domain Ω, we have
(2.3)ddt{p(t)ddt[∫Ωu(x,t)dx+∑i=1lλi(t)∫Ωu(x,t-τi)dx]}=a(t)∫ΩΔu(x,t)dx+∑k=1sak(t)∫ΩΔu(x,t-ρk(t))dx-∫Ωq(x,t)f(u(x,t))dx-∑j=1m∫Ωqj(x,t)fj(u(x,t-σj))dx,t≥t1.
From Green's formula and boundary condition (1.2), it follows that
(2.4)∫ΩΔu(x,t)dx=∫∂Ω∂u(x,t)∂νdS=-∫∂Ωg(x,t)u(x,t)dS≤0,t≥t1,(2.5)∫ΩΔu(x,t-ρk(t))dx=∫∂Ω∂u(x,t-ρk(t))∂νdS=-∫∂Ωg(x,t-ρk(t))u(x,t-ρk(t))dS≤0,t≥t1,k∈Is,
where dS is the surface element on ∂Ω. Moreover, from (C3),(C5), and Jensen's inequality it follows that
(2.6)∫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,(2.7)∫Ωqj(x,t)fj(u(x,t-σj))dx≥qj(t)∫Ωfj(u(x,t-σj))dx≥qj(t)∫∂Ωdxfj(∫Ωu(x,t-σj)dx(∫Ωdx)-1),t≥t1.
Set
(2.8)V1(t)=∫Ωu(x,t)dx(∫Ωdx)-1,t≥t1.
In view of (2.4)–(2.8), (2.3) yields
(2.9)ddt{p(t)ddt[V1(t)+∑i=1lλi(t)V1(t-τi)]}+q(t)f(V1(t))+∑j=1mqj(t)fj(V1(t-σj))≤0,t≥t1.
Let Z(t)=V1(t)+∑i=1lλi(t)V1(t-τi). 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
(2.10)Z′(t)≤p(T)Z′(T)p(t)fort≥T,Z(t)-Z(T)≤p(T)Z′(T)∫Tt1p(s)ds,t≥T.
Thereforelimt→∞Z(t)=-∞, which contradicts the fact that Z(t)>0.
From(2.9), for the j0 in (2.1) we obtain
(2.11)[p(t)Z′(t)]′+qj0(t)fj0(V1(t-σj0))≤0,t≥t1.
Noting condition(C5), from (2.11) we have
(2.12)[p(t)Z′(t)]′+αj0qj0(t)V1(t-σj0)≤0,t≥t1
or
(2.13)[p(t)Z′(t)]′+αj0qj0(t)[1-∑i=1lλi(t-σj0)]Z(t-σj0)≤0,t≥t1.
Let
(2.14)W(t)=Φ(t)[p(t)Z′(t)Z(t-σj0)+p(t-σj0)ϕ(t)];
we have
(2.15)W′(t)≤-2ϕ(t)W(t)+Φ(t){-αj0qj0(t)[1-∑i=1lλi(t-σj0)]-p(t)Z′(t)Z′(t-σj0)Z2(t-σj0)+[p(t-σj0)ϕ(t)]′[1-∑i=1lλi(t-σj0)]}.
Using the fact that p(t)Z′(t) is decreasing, we get
(2.16)p(t)Z′(t)≤p(t-σj0)Z′(t-σj0),fort≥t1.
Thus
(2.17)W′(t)≤-2ϕ(t)W(t)+Φ(t){-αj0qj0(t)[1-∑i=1lλi(t-σj0)]-1p(t-σj0)(p(t)Z′(t)Z(t-σj0))2+[p(t-σj0)ϕ(t)]′[1-∑i=1lλi(t-σj0)]}=-2ϕ(t)W(t)+Φ(t)×{-αj0qj0(t)[1-∑i=1lλi(t-σj0)]-1p(t-σj0)(W(t)Φ(t)-p(t-σj0)ϕ(t))2+[p(t-σj0)ϕ(t)]′(W(t)Φ(t)-p(t-σj0)ϕ(t))2}=-ψ(t)-W2(t)p(t-σj0)Φ(t),
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
(2.18)Δu(x)+ηu(x)=0,inΩ,u(x)=0,on∂Ω
is positive, and the corresponding eigenfunction φ(x) is positive in Ω.Theorem 2.2.
Let all conditions in Theorem2.1 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 solutionu(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-τi)>0,u(x,t-ρk(t))>0, and u(x,t-σj)>0 in Ω×[t1,∞),t1≥t0,i∈Il,k∈Is,j∈Im.
Multiplying both sides of (1.1) by φ(x)>0 and integrating (1.1) with respect to x over the domain Ω, we have
(2.19)ddt{p(t)ddt[∫Ωu(x,t)φ(x)dx+∑i=1lλi(t)∫Ωu(x,t-τi)φ(x)dx]}=a(t)∫ΩΔu(x,t)φ(x)dx+∑k=1sak(t)∫ΩΔu(x,t-ρk(t))φ(x)dx-∫Ωq(x,t)f(u(x,t))φ(x)dx-∑j=1m∫Ωqj(x,t)fj(u(x,t-σj)φ(x))dx,t≥t1.
From Green's formula and boundary condition (1.3), it follows that
(2.20)∫ΩΔu(x,t)φ(x)dx=∫Ωu(x,t)Δφ(x)dx=-β0∫Ωu(x,t)φ(x)dx≤0,t≥t1,(2.21)∫ΩΔu(x,t-ρk(t))φ(x)dx=∫Ωu(x,t-ρk(t))Δφ(x)dx=-β0∫Ωu(x,t-ρk(t))φ(x)dx≤0,t≥t1,k∈Is.
Moreover, from (C3) and (C5) by Jensen's inequality it follows that
(2.22)∫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,(2.23)∫Ωqj(x,t)fj(u(x,t-σj))φ(x)dx≥qj(t)∫Ωfj(u(x,t-σj))φ(x)dx≥qj(t)∫Ωφ(x)dxfj((∫Ωφ(x)dx)-1∫Ωu(x,t-σj)φ(x)dx(∫Ωφ(x)dx)-1),t≥t1.
Set
(2.24)V2(t)=∫Ωu(x,t)φ(x)dx(∫Ωφ(x)dx)-1,t≥t1.
In view of (2.20)–(2.24), (2.19) yields
(2.25)ddt{p(t)ddt[V2(t)+∑i=1lλi(t)V2(t-τi)]}+q(t)f(V2(t))+∑j=1mqj(t)fj(V2(t-σj))≤0,t≥t1.
Let Z(t)=V2(t)+∑i=1lλi(t)V2(t-τi); 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, andϕ∈C1[t0,∞),F∈C([t0,∞),R). Suppose that there exists j0∈Im such that
(2.26)lim supt→∞∫t0t[ψ(s)-14p(s-σj0)Φ(s)F2(s)]e∫t0sF(τ)dτds=∞,
where Φ(s)=e-2∫sϕ(τ)dτ and ψ(s) is defined as in (2.2). Then(I)
every solutionu(x,t) of the problem (1.1), (1.2) is oscillatory in G;(II)
every solutionu(x,t) of the problem (1.1), (1.3) is oscillatory in G.Proof.
(I) From Theorem2.1, 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 w(t)>0 in [t1,∞),t1≥t0. Hence for all t≥t1, we have by (2.1)
(2.27)w′(t)≤-[ψ(t)-14p(t-σj0)Φ(t)F2(t)]-[w2(t)p(t-σj0)Φ(t)+14p(t-σj0)Φ(t)F2(t)],
that is,
(2.28)w′(t)+F(t)w(t)≤-[ψ(t)-14p(t-σj0)Φ(t)F2(t)],w(t)e∫t0tF(τ)dτ-w(T)e∫t0TF(τ)dτ≤-∫Tt[ψ(s)-14p(s-σj0)Φ(s)F2(s)]e∫t0sF(τ)dτds.
Hence
(2.29)∫Tt[ψ(s)-14p(s-σj0)Φ(s)F2(s)]e∫t0sF(τ)dτds≤w(T)e∫t0TF(τ)dτ-w(t)e∫t0tF(τ)dτ.
In view of w(t)≥0, we get
(2.30)lim supt→∞∫t0t[ψ(s)-14p(s-σj0)Φ(s)F2(s)]e∫t0sF(τ)dτds≤w(T)e∫t0TF(τ)dτ,
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 Theorem2.2, 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.SetD={(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
(2.31)-∂∂sH(t,s)=h(t,s)H(t,s),∀(t,s)∈D.TakingF(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, andϕ∈C1[t0,∞). Suppose that there exists j0∈Im such that
(2.32)lim supt→∞1H(t,t0)∫t0t[H(t,s)ψ(s)-14Φ(s)p(s-σj0)h2(t,s)]ds=∞,
where Φ(s)=e-2∫sϕ(τ)dτ and Φ(s) is defined as in (2.2). Then(I)
every solutionu(x,t) of the problem (1.1), (1.2) is oscillatory in G;(II)
every solutionu(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 Theorem2.3 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 existsj0∈Im such that the following ordinary differential equation
(2.33)y′′+Q(t)y(t)=0
is oscillatory, where
(2.34)Q(t)=1p(t-σj0){αj0qj0(t)[1-∑i=1lλi(t-σj0)]+[p′(t-σj0)]24p(t-σj0)-p′′(t-σj0)2},
then(I)
every solutionu(x,t) of the problem (1.1), (1.2) is oscillatory in G;(II)
every solutionu(x,t) of the problem (1.1), (1.3) is oscillatory in G.Proof.
Lety(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
(2.35)w′(t)≤-Q(t)-w2(t),fort≥T0,
where Q(t) is defined as in (2.34). In fact, taking ϕ(t)=(p′(t-σj0))/(2p(t-σj0)) 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 existsj0∈Im such that
(2.36)∞≥limr→∞t21p(t-σj0){αj0qj0(t)[1-∑i=1lλi(t-σj0)]+[p′(t-σj0)]24p(t-σj0)-p′′(t-σj0)2}>14,
then(I)
every solutionu(x,t) of the problem (1.1), (1.2) is oscillatory in G;(II)
every solutionu(x,t) of the problem (1.1), (1.3) is oscillatory in G.Proof.
From Theorem2.5 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 existsj0∈Im such that
(2.37)∞≥liminfr→∞t∫t∞1p(t-σj0)×{αj0qj0(t)[1-∑i=1lλi(t-σj0)]+[p′(t-σj0)]24p(t-σj0)-p′′(t-σj0)2}dt>14,
then(I)
every solutionu(x,t) of the problem (1.1), (1.2) is oscillatory in G;(II)
every solutionu(x,t) of the problem (1.1), (1.3) is oscillatory in G.Corollary 2.9.
Let condition (C) hold. If there existT>t0,α>3-22, and j0∈Im such that for every n∈N,
(2.38)∫2nT2n+1T1p(t-σj0){αj0qj0(t)[1-∑i=1lλi(t-σj0)]+[p′(t-σj0)]24p(t-σj0)-p′′(t-σj0)2}dt>α2nT,
then(I)
every solutionu(x,t) of the problem (1.1), (1.2) is oscillatory in G;(II)
every solutionu(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∈Im such that for every n∈N,
(2.39)∫λnTλn+1T1p(t-σj0){αj0qj0(t)[1-∑i=1lλi(t-σj0)]+[p′(t-σj0)]24p(t-σj0)-p′′(t-σj0)2}dt>α(λ-1)λnT,
then(I)
every solutionu(x,t) of the problem (1.1), (1.2) is oscillatory in G;(II)
every solutionu(x,t) of the problem (1.1), (1.3) is oscillatory in G.Remark 2.11.
Corollaries2.8–2.10 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 [2nT,2n+1T], rather than on the whole half-line [t0,∞).Example 2.12.
Let constantsc>0 and μ>0. Consider the partial differential equation
(2.40)∂∂t{1t+π+1∂∂t[u(x,t)+3t+2πu(x,t-2π)]}=1t+π+1Δu(x,t)+[1(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)2]Δu(x,t-π)-[3(t+π+1)(t+2π)+t+πt+π-3μln2(t+1)]u(x,t)[1+c1+u2(x,t)]-t+πt+π-3μln2(t+1)u(x,t-π),(x,t)∈(0,π)×R+≡G,
with the boundary condition
(2.41)u(0,t)=u(π,t)=0,t≥0.
A straightforward verification shows that the functionsq1(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
(2.42)Q(t)=(t+1){μln2(t+1)+14(t+1)3-1(t+1)3}=μ(t+1)ln2(t+1)-34(t+1)2.
Then, for constant μ>0,
(2.43)limt→∞t2Q(t)=limt→∞[μt2(t+1)ln2(t+1)-3t24(t+1)2]=∞>14.
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.



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