Integral average method for oscillation of second order partial differential equations with delays☆
Introduction
Consider the second order delay partial differential equation of the formwhere Ω is a bounded domain in RN with a piecewise smooth boundary ∂Ω, and .
Throughout this paper, we assume that
(H1) ;
(H2) ;
(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: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 , which is the main idea of integral average method. For any function h(t, s) ∈ C([t0, ∞) × [t0, t)), τ ⩾ t0 ⩾ 0, we definewhere α > 1 is a constant, ρ ∈ C1 [t0, ∞) with ρ > 0. If , we get
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 , 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 thatwhereandthen 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 equationwith the boundary conditionhere Let then we have Φ(t) = t andSo, let α = 2
References (11)
- et al.
Oscillation of solutions of parabolic differential equations of neutral type
Appl. Math. Comput.
(1988) - et al.
Oscillation of neutral delay parabolic equations
J. Math. Anal. Appl.
(1995) - et al.
Oscillation of hyperbolic equtions with functional arguments
Appl. Math. Comput.
(1993) - et al.
Oscillation for solutions of systems of neutral type partial functional differential equations
Comput. Math. Appl.
(2002) - et al.
Forced oscillation for certain systems hyperbolic differential equations
Appl. Math. Comput.
(2003)
Cited by (16)
Oscillation of a kind of second order quasilinear equation with mixed arguments
2020, Applied Mathematics LettersCitation Excerpt :And there are also some results about the oscillation of partial differential equations [7–10]. In 2007, some oscillation criteria for a kind of linear second order partial differential equations with delays are found in [8] through the method of integral average and generalized Riccati transformation. In 2017, the authors [7] obtained some sufficient conditions for oscillation criteria to a kind of linear impulsive partial fractional differential equations by differential inequality method.
Oscillation criteria of neutral type impulsive hyperbolic equations
2014, Acta Mathematica ScientiaExact solutions and numerical approximations of mixed problems for the wave equation with delay
2012, Applied Mathematics and ComputationCitation Excerpt :Wave equations with delay terms are basic modeling tools in the analysis of oscillatory phenomena including aftereffects, time lags or hereditary characteristics [1–4], as the deformation of viscoelastic materials [5] or the retarded control of the dynamics of flexible structures [6–8].
Oscillations of nonlinear hyperbolic equations with functional arguments via Riccati method
2010, Applied Mathematics and ComputationApproximate solutions for fuzzy stochastic differential equations with Markovian switching
2024, Applied Mathematics in Science and EngineeringMULTI-VALUED RANDOM DYNAMICS OF STOCHASTIC WAVE EQUATIONS WITH INFINITE DELAYS
2022, Discrete and Continuous Dynamical Systems - Series B
- ☆
This research was partially supported by the NSF of China and NSF of Shandong Province, China (Y2005A06).