Integral average method for oscillation of second order partial differential equations with delays

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Abstract

Using a generalized Riccati transformation, some new oscillation criteria for second order partial differential equations with delays are found through the method of integral average. These results can be considered as generalizations and improvements of the results due to Kamenev in ordinary differential equations cases.

Introduction

Consider the second order delay partial differential equation of the formtr(t)tu(x,t)+p(t)u(x,t)t=a(t)Δu(x,t)+k=1sak(t)Δu(x,t-ρk(t))-q(t)u(x,t)-j=1mqj(x,t)u(x,t-σj),(x,t)Ω×[0,)G.where Ω is a bounded domain in RN with a piecewise smooth boundary ∂Ω, and Δu(x,t)=j=1N2u(x,t)xj2.

Throughout this paper, we assume that

  • (H1) rC1(R+;(0,)),dsr(s)=,q,pC(R+;R);

  • (H2) qj(G¯;R+),qj(t)=minxΩ¯qj(x,t),jIm=[1,2,,m];

  • (H3) a, ak  (R+; R+), ρk  C(R+; R+), limt→∞(t  ρk(t)) = ∞,

  • σj are nonnegative constants, j  Im, k  Is.

Consider the following boundary condition:u(x,t)γ+g(x,t)u(x,t)=0,(x,t)Ω×R+,where γ is the unit exterior normal vector to ∂Ω and g(x, t) is a nonnegative continuous function on ∂Ω × R+.

Definition 1.1

The solution u(x, t) of problem (1.1), (1.2) is said to be oscillatory in the domain G if for any positive number μ there exists a point (x0, t0)  Ω × [μ, ∞) such that u(x0, t0) = 0 holds.

In the sequel, we introduce linear integral operator Lτρ, which is the main idea of integral average method. For any function h(t, s)  C([t0, ∞) × [t0, t)), τ  t0  0, we defineLτρ(h(t,s))=τtρ(s)(t-s)αh(t,s)ds,where α > 1 is a constant, ρ  C1 [t0, ∞) with ρ > 0. If h(t,s)sC([t0,)×[t0,t)), we getLτρh(t,s)s=-ρ(τ)(t-τ)αh(t,τ)-Lτρ-αt-s+ρ(s)ρ(s)h(t,s).

Recently, the oscillation problem for the functional differential equation has been studied by many authors. We refer the reader to [1], [2], [3] for parabolic equations and to [4], [5], [6], [7], [8], [9], [10], [11] for hyperbolic equations.

In this paper, by using Riccati technique and linear integral operator Lτρ, we obtain several new oscillation criteria. Our results improve and extend the results of Cui [4] and Li [6]. Our methodology is somewhat different from that of previous authors. We believe that our approach is simpler and also provides a more unified account for the study of Kamenev-type oscillation theorems.

Section snippets

Main results

Theorem 2.1

Suppose that there exists f  C1 [t0, ) such thatlimsupt1tαLt0ρχ(s)-Φ(s)r(s)4p(s)r(s)--αt-s+ρ(s)ρ(s)2=,whereΦ(s)=exp-2sf(ξ)dξ,andχ(s)=Φ(s){q(s)+r(s)f2(s)-p(s)f(s)-(r(s)f(s))},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 problem (1.1), (1.2) which has no zero on Ω × [μ0, ∞) for some μ > 0. Without the loss of generality we may assume that u(x, t) > 0, u(x, t  ρk(t)) > 0, and u(x, t  σj) > 0 in Ω × [t0, ∞), t0

Examples

In order to illustrate the effect of theorems, we give the following examples.

Example 1

Consider the partial differential equation2u(x,t)t2-1t2u(x,t)t=1+1t2Δu(x,t)+1t2Δux,t-32π-1t2u(x,t)-2t2u(x,t-π),(x,t)(0,π)×[0,),with the boundary conditionux(0,t)=ux(π,t)=0,t0,here N=1,s=1,m=1,r(t)=1,p(t)=-1t2,a(t)=1+1t2,a1(t)=1t2,q(t)=1t2,q1(x,t)=2t2,ρ1(t)=32π,σ1=π.

Let ρ(t)1,f(t)=-12t then we have Φ(t) = t andχ(t)=Φ(t){q(t)+r(t)f2(t)-p(t)f(t)-(r(t)f(t))}=34·1t-12t2.So, let α = 2limsupt1t21t(t-s)2χ(s)ds=limt1

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This research was partially supported by the NSF of China and NSF of Shandong Province, China (Y2005A06).

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