Oscillation and asymptotic behavior for a class of delay parabolic differential

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1 Applied Mthemtics Letters 19 (2006) Oscilltion nd symptotic behvior for clss of dely prbolic differentil Qisheng Wng,b,,ZigenOuyng,Jiding Lio School of Mthemtics nd Physicl Science, Nnhu University, Hengyng Hunn, , PR Chin b School of Mthemtics nd Computing Science, Xingtn University, Xingtn Hunn, , PR Chin Received 14 My 2004; received in revised form 19 August 2005; ccepted 27 September 2005 Abstrct Some comprtive theorems re given for the oscilltion nd symptotic behvior for clss of high order dely prbolic differentil equtions of the form t n (t) u c(x, t, u) q(x, t,ξ)f (u(x, g 1 (t,ξ)),...,u(x, g l (t,ξ))) dσ(ξ) = 0, (x, t) R G, where n is n odd integer, is bounded domin in R m with smooth boundry,nd is the Lplcin opertion with three boundry vlue conditions. Our results extend some of those of P. Wng, Oscilltory criteri of nonliner hyperbolic equtions with continuous deviting rguments, Appl. Mth. Comput. 106 (1999), ] substntilly. c 2005 Elsevier Ltd. All rights reserved. Keywords: Oscilltion; High order; Dely prbolic differentil eqution; Asymptoticlly positive solution; Asymptotic behvior 1. Introduction Consider the following odd-order dely prbolic differentil eqution: t n (t) u c(x, t, u) q(x, t,ξ)f (u(x, g 1 (t,ξ)),...,u(x, g l (t,ξ))) dσ(ξ) = 0, (x, t) G, (1.1) where n is n odd integer, τ is positive constnt, R =0, );wewill ssume throughout this work tht Supported by the Nturl Science Foundtion of Chin under Grnt No Corresponding uthor t: School of Mthemtics nd Physicl Science, Nnhu University, Hengyng Hunn, , PR Chin. E-mil ddress: wngqisheng001@yhoo.com.cn (Q. Wng) /$ - see front mtter c 2005 Elsevier Ltd. All rights reserved. doi: /j.ml

2 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) (H 1 ) (t) C(R, R ), g i (t,ξ) C(R, b], R), g i (t,ξ) t,ξ, b] nd lim t g i (t,ξ) =, for i = 1,...,l. (H 2 ) q(x, t,ξ) C( R, b], R ), Q(t,ξ) = min x {q(x, t,ξ)}, f C(R m R, b], R), u i f 0 when ech u i > 0, i = 1,...,l, f is convex, f = f ( u 1,..., u l ), in which u i = u(x, g i (t,ξ)), for i = 1,...,l. (H 3 ) p(t) C(R, R ), nd lim t p(t) = p < 1. (H 4 ) c(x, t, u) C( R R, R); h 1 (t)ϕ 1 (ξ) c(x, t,ξ) h 2 (t)ϕ 2 (ξ) for ξ > 0, in which h 1 (t) C(R, R ), ϕ 1 (ξ) C(, b], R), ϕ 1 (ξ) is positive nd convex function in (0, ), nd c(x, t, ξ) = c(x, t,ξ),ϕ 1 ( ξ) = ϕ 1 (ξ). We consider the following boundry conditions: u(x, t) = 0, (x, t) R, (B 1 ) u(x, t) N = 0, (x, t) R, (B 2 ) u(x, t) νu = 0, (x, t) R, (B 3 ) N where N is the unit exteriorvectornormlto,ndν(x, t) is nonnegtivecontinuousfunctionon R. Recently, mny uthors 1 8] hve studied oscilltions for the solutions of prbolic differentil equtions; they obtined some comprtive theorems for the oscilltions of these equtions. In more recent times, Wng 10] hs investigted the following nonliner hyperbolic equtions: 2 (u(x, t) p(t)u(x, t τ)) t 2 (t) u c(x, t, u) q(x, t,ξ)u(x, g(t,ξ))dσ(ξ) = 0, (x, t) G, (1.2) where (t), g(t), c(x, t, u), p(t) re defined s bove. Some comprtive oscilltion for the solutions of the boundry vlue problem (1.2) (B 3 ) were obtined. But few uthors 9] hve studied oscilltion of the high order prbolic differentil eqution. In this work, we investigte Eq. (1.1) with the boundry conditions (B 1 ), (B 2 ) nd (B 3 ).Thiswork is orgnized s follows: in Section 2, we discuss the comprtive oscilltion for the solutions of Eq. (1.1) with boundry conditions (B 1 ), (B 2 ) nd (B 3 ),ndsomecomprtive results will be obtined; in Section 3, wewill investigte the symptotic behvior of non-oscilltorysolutions of (1.1),nd we shll give one exmple to explin the oscilltion nd symptotic behvior of Eq. (1.1) with the bove boundry conditions. We first recll some definitions s follows. Definition 1.1. Afunction u(x, t) C 2 () C n (R ) is sid to be solution to the problem (1.1) (B i ) (i = 1, 2, 3) if it stisfies (1.1) in the domin G nd stisfies the boundry condition (B i ) (i = 1, 2, 3). Definition 1.2. The solution u(x, t) of problem (1.1) is sid to be oscilltory in the domin G if for ny positive number µ, thereexists point (x 1, t 1 ) µ, ) such tht the equlity u(x 1, t 1 ) = 0 holds. If every solution of Eq. (1.1) is oscilltory, then Eq. (1.1) is clled oscilltory. Definition 1.3. A function u(x, t) is clled eventully positive (negtive) if there exists number T 0 such tht u(x, t) >0 (< 0) for every (x, t) T, ). Remrk. If solution is non-oscilltory, then it is eventully positive or eventully negtive.

3 760 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) Comprtive oscilltion of (1.1) with the boundry vlue conditions (B 1 ), (B 2 ), (B 3 ) Forconvenience, we consider the following Dirichlet boundry vlue problem in the domin : u αu = 0, in (x, t) G, u = 0, on (x, t) R, in which α is constnt. It is well known from 11] thtthe smllest eigenvlue λ 1 of problem (2.1) is positive nd tht the corresponding eigenfunction ψ(x) 0forx. Let u(x, t) be solution of problem (1.1) (B 1 );wedefine throughout this section u(x, t)ψ(x) dx U(t) = ψ(x) dx. (2.3) (2.1) (2.2) Let u(x, t) be solution of problem (1.1) (B i ), i = 2, 3; we lwys define u(x, t) dx V (t) = dx. (2.4) To obtin our results, we first introduce lemm s follows: Lemm 2.1. Suppose tht (H 1 ) (H 4 ) hold; then (z(t) p(t)z(t τ)) (n) h 1 (t)ϕ 1 (z) Q(t,ξ)f (z(g 1 (t,ξ)),...,z(g l (t,ξ)))dσ(ξ) 0, t R, (2.5) hs n eventully positive solution if nd only if (z(t) p(t)z(t τ)) (n) h 1 (t)ϕ 1 (z) Q(t,ξ)f (z(g 1 (t,ξ)),...,z(g l (t,ξ))) dσ(ξ) = 0, t R (2.6) hs n eventully positive solution. Proof. The sufficiency is obvious. We only need to prove the necessity. Assume tht (2.5) hs n eventully positive solution z(t); thismens tht there is number T > 0suchthtz(t) > 0, z(t τ) > 0, z(g i (t,ξ)) > 0for t > T, i = 1,...,l. Itfollows tht there exists nonnegtive integer n n 1(ifn is odd, then n is even; if n is even, then n is odd), such tht z (i) (t) >0, i = 0, 1,...,n ( 1) i z (i) (t) >0, i = n (2.7),...,n 1, By using (2.7), (H 1 ) (H 3 ) nd integrting (2.5) from t to, n n times, we obtin (z(t) p(t)z(t τ)) (n ) (s t) n n 1 t (n n h 1 (s)ϕ 1 (z(s)) ] Q(s,ξ)f (z(g 1 (s,ξ)),..., z(g l (s,ξ))) dσ(ξ) ds. (2.8) Let T > 0islrge enough tht (2.8) holds nd z(t τ) > 0, z(g i (t,ξ)) > 0, i = 1,...,l. Integrting (2.8) from T to t nd using (2.8),wehve t (t s) n 1 z(t) p(t)z(t τ) T (n s (n n h 1 (r)ϕ 1 (z(r)) ] Q(r,ξ)f (z(g 1 (r,ξ)),...,z(g l (r,ξ))) dσ(ξ) dr ds. (2.9)

4 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) We define setoffunctions s follows: K = C(T τ, ), 0, 1]),nddefine n opertor S on K : (Sy)(T { ), T τ t < T, 1 t (t s) n 1 p(t)y(t τ)z(t τ) z(t) T (n s (n n h (Sy)(t) = 1 (r)ϕ 1 (y(r)z(r)) Q(r,ξ)f (y(g 1 (r,ξ))z(g 1 (r,ξ)),..., y(g l (r,ξ))z(g l (r,ξ))) ] } dσ(ξ) dr ds, t T. It is esy to see by (2.9) tht S mps K into itself, nd for ny y K,wehve(Sy)(t) >0, for T τ t < T.Now, we definethesequences y k (t) in K : nd y 0 (t) 1, t T τ, y k1 (t) = (Sy k )(t), for t T τ,k = 0, 1,..., lim k(t) = y(t), k t T τ. This mens, from Lebesgue s dominted convergence theorem, tht there exists function y(t) tht stisfies { y(t) = 1 t (t s) n 1 p(t)y(t τ)z(t τ) z(t) T (n s (n n h 1 (r)ϕ 1 (y(r)z(r)) ] } Q(r,ξ)f (y(g 1 (r,ξ))z(g 1 (r,ξ)),..., y(g l (r,ξ))z(g l (r,ξ))) dσ(ξ) dr ds, t T. nd y(t) = (Sy)(T ), T τ t < T. This being the cse, define w(t) = y(t)z(t). It is obvious tht w(t) >0forT τ t < T nd t (t s) n 1 w(t) = p(t)w(t τ) (n T Q(r,ξ)f (w(g 1 (r,ξ)),..., w(g l (r,ξ))) dσ(ξ) s (n n h 1 (r)ϕ 1 (w(r)) ] dr ds t T. Thus, w(t) is nonnegtive solution of Eq. (2.6) for t T.Finlly, it remins to show tht w(t) >0fort > T µ. Assume tht there exists t > T µ such tht w(t) >0forT τ<t < t nd w(t ) = 0. Then t > T,nd t 0 = w(t ) = p(t )w(t (t s) n 1 τ) T (n s (n n h 1 (r)ϕ 1 (w(r)) ] Q(r,ξ)f (w(g 1 (r,ξ)),...,w(g l (r,ξ))) dσ(ξ) dr ds, t T. which contrdicts the ssumption. Thus, w(t) is n eventully positive solution of Eq. (2.6); thisiscontrdiction to the supposition too, nd the proof is completed. If we let y(t) = z(t), thenlemm 2.1 chnges into the following:

5 762 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) Corollry 2.1. Suppose tht (H 1 ) (H 4 ) hold; then (z(t) p(t)z(t τ)) (n) h 1 (t)ϕ 1 (z) Q(t,ξ)f (z(g 1 (t,ξ)),...,z(g l (t,ξ))) dσ(ξ) 0, t R, (2.10) hs n eventully negtive solution if nd only if (2.6) hs n eventully negtive solution. Theorem 2.1. Suppose tht (H 1 ) (H 4 ) hold; if every solution of the differentil eqution (z(t) p(t)z(t τ)) (n) λ 1 (t)z(t) h 1 (t)ϕ 1 (z) Q(t,ξ)f (z(g 1 (t,ξ)),...,z(g l (t,ξ))) dσ(ξ) = 0, t R, (2.11) oscilltes, then every solution of (1.1) (B 1 ) oscilltes. Proof. Assume tht there is non-oscilltory solution u(x, t) of the problem (1.1) (B 1 ).Without loss of generlly, let u(x, t) be n eventully positive solution of problem (1.1) (B 1 );thenthere exists number T > 0suchtht u(x, t) >0, u(x, t τ) > 0, u(x, g k (t,ξ)) >0fort > T, k = 1,...,l. Multiplying both sides of Eq. (1.1) by ψ(x), ndintegrting both sides over the domin with respect to x, we hve d n dt n (u(x, t) p(t)u(x, t τ))ψ(x) dx (t) uψ(x) dx c(x, t, u)ψ(x) dx ψ(x) q(x, t,ξ)f (u(x, g 1 (t,ξ)),...,u(x, g l (t,ξ))) dσ(ξ)dx = 0, t T. (2.12) Using (2.1) nd (2.2), ndgreen s formul, we hve uψ(x) dx = λ 1 uψ(x) dx, t T. (2.13) Combining (2.12), (2.13), (H 3 ), (H 4 ),using Jensen s inequlity nd chnging the order of integrtion, we hve d n dt n (u(x, t) p(t)u(x, t τ))ψ(x) dx λ 1 (t) uψ(x) dx h 1 (t) ϕ 1 (u)ψ(x) dx ( Q(t,ξ)f ψ(x)u(x, g 1(t,ξ))dx ψ(x) dx,..., ψ(x)u(x, g ) l(t,ξ))dx ψ(x) dx t T. Using (2.3),thismens d n dt n (U(t) p(t)u(t τ)) λ 1(t)U(t) h 1 (t)ϕ 1 (U) dσ(ξ) ψ(x) dx 0, Q(t,ξ)f (U(g 1 (t,ξ)),...,u(g l (t,ξ))) dσ(ξ) 0, t T. (2.14) Becuse u(x, t) is positive, from (2.3) gin, we hve tht U(t) is eventully positive. It follows tht U(t) is n eventully positive solution of (2.5); ccording to Lemm 2.1, Eq.(2.6) hs n eventully positive solution, nd this contrdicts the supposition. If u(x, t) is n eventully negtive solution of problem (1.1) (B 1 ),thenthere exists number T > 0suchtht u(x, t) <0, u(x, t τ) < 0, u(x, g k (t,ξ)) <0fort > T, k = 1,...,l; then(2.12) chnges into d n dt n (( u(x, t)) p(t) (u(x, t τ)))ψ(x) dx (t) ( u)ψ(x) dx c(x, t, u)ψ(x) dx ψ(x) q(x, t,ξ)f ( u(x, g 1 (t,ξ)),..., u(x, g l (t,ξ))) dσ(ξ)dx = 0, t T.

6 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) Let v(x, t) = u(x, t), Using method similr to the bove, we cn derive contrdiction too. The proof is completed. If p(t) = 0, c(x, t, u) = 0, then (1.1) hs the following specil form: n u(x, t) t n (t) u q(x, t,ξ) f (u(x, g 1 (t,ξ)),...,u(x, g l (t,ξ))) dσ(ξ) = 0, (x, t) G. (2.15) We consider the following differentil eqution: z (n) (t) α 1 (t)z(t) Using Theorem 2.1, wehve: Q(t,ξ)f (z(g 1 (t,ξ)),..., z(g l (t,ξ))) dσ(ξ) = 0, t R. (2.16) Corollry 2.2. Suppose tht (H 1 ) (H 4 )hold, nd every solution of differentil eqution (2.16) oscilltes; then every solution of problem (2.15) (B 1 ) oscilltes in G. Now, we investigte the oscilltion of problem (1.1) with boundry vlue condition (B 2 ). Theorem 2.2. Suppose tht (H 1 ) (H 4 ) hold; if every solution of the differentil eqution (z(t) p(t)z(t τ)) (n) h 1 (t)ϕ 1 (z) Q(t,ξ)f (z(g 1 (t,ξ)),...,z(g l (t,ξ))) dσ(ξ) = 0, t R (2.17) oscilltes, then every solution of (1.1) (B 2 ) oscilltes. Proof. Assume tht there is non-oscilltory solution u(x, t) of the problem (1.1) (B 2 ).Without loss of generlly, we ssume u(x, t) is n eventully positive solution of (1.1) (B 2 )(if it is n eventully negtive solution, the proof is similr). This mens tht there exists number T 0suchtht u(x, t) > 0, u(x, g k (t,ξ)) > 0,for t > T, k = 1,...,l. Integrting (1.1) on both sides overthe domin with respect to x,itfollows tht d n dt n (u(x, t) p(t)u(x, t τ))dx (t) u dx c(x, t, u) dx q(x, t,ξ)f (u(x, g 1 (t,ξ)),...,u(x, g l (t,ξ))) dσ(ξ)dx = 0, t T. (2.18) Using Green s formul nd (B 2 ),weobtin u(x, t) u dx = ds = 0, t T. (2.19) N Combining (2.18), (2.19), (H 2 ), (H 3 ) nd using Jensen s inequlity, we obtin d n dt n (u(x, t) p(t)u(x, t τ))dx h 1 (t) ϕ 1 (u) dx ( Q(t,ξ)f u(g 1(t,ξ))dx dx,..., u(g ) l(t,ξ))dx dx dσ(ξ) dx 0, t T. Accordingto (2.4), itfollows tht d n dt n (V (t) p(t)v (t τ)) h 1(t)ϕ 1 (V (t)) Q(t,ξ)f (V (g 1 (t,ξ)),...,v (g l (t,ξ))) dσ(ξ) 0, t T. (2.20)

7 764 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) Becuse u(x, t) is positive, from (2.4) gin, we hve tht V (t) is positive eventully. According to Lemm 2.1, V (t) is n eventully positive solution of Eq. (2.17); this contrdicts the supposition. The proof is completed. From Theorems 2.1 nd 2.2, Lemm 2.1, itisesy toprove: Corollry 2.3. Suppose tht (H 1 ) (H 4 )hold; then every solution of the boundry vlue problem (2.15) (B 2 ) oscilltes if nd only if every solution of z (n) (t) Q(t,ξ)f (z(g 1 (t,ξ)),...,z(g l (t,ξ))) dσ(ξ) = 0, t R (2.21) is oscilltory. Using Theorem 2.2 nd Lemm 2.1, itisesy toshow. Theorem 2.3. Suppose tht (H 1 ) (H 4 ) hold; if every solution of the differentil eqution (z(t) p(t)z(t τ)) (n) h 1 (t)ϕ 1 (z) f (z(g 1 (t,ξ)),...,z(g l (t,ξ))) dσ(ξ) = 0, t R, (2.22) oscilltes, then every solution of (1.1) (B 3 ) oscilltes. 3. Asymptotic behvior of non-oscilltory solutions In this section, we will estblish the symptotic behvior of the non-oscilltory solutions (1.1) (B 1 ), (B 2 ), (B 3 ). For convenience, we let nd y(t) = z(t) p(t)z(t τ), v(x, t) = u(x, t) p(t)u(x, t τ). (3.1) (3.2) Theorem 3.1. Suppose tht (H 1 ) (H 4 ) hold; if z(t) is n eventully positive solution of (z(t) p(t)z(t τ)) (n) h 2 (t)ϕ 2 (z) Q(t,ξ)f (z(g 1 (t,ξ)),...,z(g l (t,ξ))) dσ(ξ) = 0, t R,(3.3) then there exists non-oscilltory solution of the boundry vlue problem (1.1) (B 1 ), (B 2 ), (B 3 ) such tht: (1) If lim t z(t) = 0,thenlim t u(x, t) = 0, lim t v(x, t) = 0, i v(x,t) is monotonic nd t i lim i v(x,t) t = 0, i v(x,t) i1 v(x,t) < 0,fori = 1,...,n 1. t i t i t (2) If z(t) 0 s t,thenu(x, i1 t) >0,v(x, t) >0, oru(x, t) <0,v(x, t) <0,wherey(t), v(x, t) re defined by (3.1) nd (3.2) respectively. Proof. (1) We only prove tht there exists solution of (1.1) (B 1 ) which stisfies (1), (2) in Theorem 3.1; therest is similr. Without loss of generlity, ssume (3.3) hs n eventully positive solution z(t) (if it hs n eventully negtivesolution, the proof is similr) nd let u(x, t) = z(t); thenitisneventully positive solution of t n (t) u h 2 (t)ϕ 2 (u) q(x, t,ξ)f (u(x, g 1 (t,ξ)),..., u(x, g l (t,ξ))) dσ(ξ) 0, (x, t) G, From (H 4 ),itfollows tht u(x, t) stisfies t n (t) u c(x, t, u) q(x, t,ξ)f (u(x, g 1 (t,ξ)),..., u(x, g l (t,ξ))) dσ(ξ) 0, (x, t) G, (3.4)

8 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) by Lemm 2.1, u(x, t) is n eventully positive solution of (1.1) (B 1 ),nd u = 0, tht is t n = c(x, t, u) This mens tht n v(x,t) t n it follows tht lim n 1 v(x,t) t q(x, t,ξ)f (u(x, g 1 (t,ξ)),..., u(x, g l (t,ξ))) dσ(ξ) 0, (x, t) G. (3.5) 0, nd v(x, t) 0; it follows tht i v(x,t) t i t n 1 nd number T > t 0,suchtht is monotone, nd n 1 v(x,t) t n 1 is decresing. Thus exists. Assume tht lim t n 1 v(x,t) t n 1 = L < 0; then there exists number L 1 < 0, n 1 v(x, t) lim t t n 1 L 1, t > T. This contrdicts lim t v(x, t) = 0. On the other hnd, if L > 0, then n 1 v(x,t) L > 0, which contrdicts lim t n 1 t v(x, t) = 0lso.Thus, we hve lim n 1 v(x,t) t = 0. Moreover, n 1 v(x,t) > 0, t > T ;thismens tht n 2 v(x,t) is incresing for n > 2. Let t n 1 t n 1 t n 2 lim n 2 v(x,t) t = L t n 2 2.Itisobvious tht L 2 = 0; by the similr proof, we cn esily prove tht i v(x, t) lim t t i = 0, i = 1,...,n 1, nd i v(x, t) i1 v(x, t) t i t i1 < 0, i = 1,...,n 1. (2) If lim t z(t) 0, then lim sup t z(t) > 0, becuse u(x, t) = z(t) is eventully positive nd lim sup t u(x, t) = U(x) >0. Assume tht v(x, t) is not eventully positive; then it iseventully negtive. If z(t) is unbounded, then there exists sequence t k,ndx 0 such tht v(x 0, t k ) = mx t tk v(x 0, t), nd lim sup tk z(t k) = lim sup tk u(x, t k) =.By(3.2),ndtking x 0,wehve v(x 0, t k ) = u(x 0, t k ) p(t k )u(x 0, t k τ) u(x 0, t k )1 p(t k )] u(x 0, t k )1 p]. (3.6) Tking the limits of both sides of (3.6) s t k,wecn get lim sup v(x 0, t) = ;thiscontrdicts v(x, t) <0. If v(x, t) is bounded, then there exists sequence t k such tht lim v(x, t k) = lim sup v(x, t) = V (x) >0. t k t Tking x 0, wehvev (x 0 )>0. Combining (3.2) nd (3.6) nd tking the limit s t k,wehve 0 lim v(x 0, t k ) t k = lim u(x 0, t k ) lim (p(t k)u(x 0, t k τ)) t k t k lim u(x 0, t k )1 p(t k )] t k lim u(x 0, t k )1 p] > 0. t k This is lso contrdiction. The proof is completed. Exmple 3.1. Consider the following boundry vlue problem: n (u(x, t) 1 u(x, t 4)) 3e 4 t n u xx 1 u(x, t 4) 3e4 2 π e (t e t) ξe u(x,t 4) dξ = 0, (x, t) (0,π) R G, (3.8) 3π 2 e e4 0 (3.7)

9 766 Q. Wng et l. / Applied Mthemtics Letters 19 (2006) with the following boundry vlue condition: u(0, t) x = u(π, t) x = 0, t 0, (3.9) 2 π where n is n odd number, f = 3π 2 e e4 0 e (t e t) ξe u(x,t 4) dξ = 1 e (t e t ) e u(x,t 4), p = 1, c = 3e e4 3e 4 1 u(x, t 4); its corresponding dely differentil eqution is s follows: 3e 4 ( z(t) 1 z(t 4) 3e4 ) (n) 1 1 z(t 4) 3e4 3e e4 e (t e t ) e z(t 4) = 0, t t 0, ), t 0 R, (3.10) nd it is obvious tht z = e t is n eventully positive solution of (3.10), ndlim t z(t) = 0. It follows tht u(x, t) = z(t) = e t is n eventully positive solution of problem (3.8) nd (3.9),ndlim t u(x, t) = 0,v(x, t) = 2 3 e t, lim t v(x, t) = 0, k v(x,t) = ( 1) k 2 t k 3 e t ;itisesy to show tht k v(x,t) k1 v(x,t) < 0, lim k v(x,t) t k t k1 t = t 0, k = 1,...,n 1. k Acknowledgements The uthor thnks the referees nd Professor Yunqing Hung who red the mnuscript crefully nd put forwrd some helpful opinions. References 1] D.P. Mishev, Oscilltion properties of the hyperbolic differentil equtions with mximum, Hiroshim Mth. J. 16 (1986) ] D.P. Mishev, D.D. Binov, Oscilltion properties of the solutions of clss of hyperbolic equtions of neutrl type, Funkcil Ekvc. 29 (2) (1986) ] N. Yoshid, On the zeros of solutions of hyperbolic equtions of neutrl types, Differ. Integrl Equ. 3 (1990) ] Y.K. Li, Oscilltion of systems of hyperbolic differentil equtions with deviting rgument, Act Mth. Sinic. 40 (1) (1997) ] X.L. Fu, Oscilltion of clss of prbolic differentil equtions, Act Mth. Sci. 13 (3) (1993) ] B.T. Cui, B.S. Llli, Y.H. Yu, Oscilltion of hyperbolic differentil equtions with deviting rguments, Act Mth. Appl. Sinic. 11 (4) (1995) ] B.T. Cui, Oscilltion properties of the solutions of hyperbolic equtions with deviting rguments, Demonstrtio Mth. 29 (1996) ] Y.H. Yu, X.L. Fu, oscilltions of second order nonliner neutrl equtions with continuous distributed deviting rguments, Rd Mt. 7 (1) (1991) ] Z.G.Ouyng, Necessry nd sufficient conditions for oscilltion of odd order neutrl dely prbolic differentil equtions, Appl. Mth. Lett. 16 (2003) ] P. Wng, Oscilltory criteri of nonliner hyperbolic equtions with continuous deviting rguments, Appl. Mth. Comput. 106 (1999) ] V.S. Vldimirov, Equtions of Mthemtics Physics, Nuk, Moscow, 1981.