16th Conference on Applied Mathematics, Univ. of Central Oklahoma, Electronic Journal of Differential Equations, Conf. 07, 2001, pp. ??–??. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp)

Almost periodic solutions of nonlinear hyperbolic equations with time delay ^{∗}

Hushang Poorkarimi & Joseph Wiener

Abstract The almost periodicity of bounded solutions is established for a nonlinear hyperbolic equation with piecewise continuous time delay. The equation represents a mathematical model for the dynamics of gas absorption.

1 Introduction

In this paper we are interested in determining almost periodicity for a unique bounded solution of nonlinear hyperbolic equations with time delay. The initial value problem under investigation is the following:

u_{xt}(x, t) + a(x, t)u_{x}(x, t) = C(x, t, u(x, [t]))  (1)
u(0, t) = u_{0}(t),  (2)

where a and C are defined in the domain D : (0, l) × ℝ → ℝ, and [t] denotes the greatest integer function: [t] = n when n ≤ t < n + 1, for an integer n. In this case the delay function is piecewise constant. The existence of a unique bounded solution of problem (1)–(2) has been discussed earlier [1].

Equation (1) with condition (2) under assumption

a(x, t) ≥ m > 0 in D

has a unique bounded solution via Volterra integral equation

u(x, t) = u_{0}(t) + ∫_{0}^{x} ∫_{−∞}^{t} e^{− ∫_{τ}^{t} a(ξ,θ) dθ}C(ξ, τ, u(ξ, n)) dτ dξ  (3)

Let us notice that, in case of periodicity, the period has to be the same for all the functions involved [2]. This result is based on the equivalence of (1)–(2) with integral equation (3), and it can be stated as the following assertion.

Theorem 1 If u_{0}(t), a(x, t), and C(x, t, u(x, [t]) are periodic in t with period T, then the unique bounded solution of (3) is also periodic in t, with the same period T.

∗Mathematics Subject Classifications: 35B10, 35B15, 35J60, 35L70.

Key words: Nonlinear hyperbolic equation, time delay, almost periodic solution.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published July 20, 2001.

Proof From (3), one obtains

u(x, t + T) = u_{0}(t + T) + ∫_{0}^{x} ∫_{−∞}^{t+T} e^{− ∫_{τ}^{t+T} a(ξ,θ) dθ}C(ξ, τ, u(ξ, n)) dτ dξ.

Making the substitution τ = η + T and taking into account

∫_{η+T}^{t+T} a(ξ, θ) dθ = ∫_{η+T}^{η} a(ξ, θ) dθ + ∫_{η}^{t} a(ξ, θ) dθ + ∫_{t}^{t+T} a(ξ, θ) dθ

we have

u(x, t + T) = u_{0}(t) + ∫_{0}^{x} ∫_{−∞}^{t} e^{− ∫_{η}^{t} a(ξ,θ) dθ}C(ξ, η, u(ξ, n)) dηdξ = u(x, t)

which proves the periodicity of u in t with period T.

Definition 2 (Bohr’s Definition of ϵ–almost periodicity) For any ϵ > 0, there exists a number l(ϵ) > 0 with property that any interval of length l(ϵ) of the real line contains at least one point with abscissa δ, such that

|u(x, t + δ) − u(x, t)| < ϵ, (x, t) ∈ D ,

the number δ is called translation number of u(x, t) corresponding to ϵ, or an ϵ-almost period of u(x, t).

The following lemma will be used to prove that the unique bounded solution (in D) of equation (3) is almost periodic in t.

Lemma 3 Assume the following conditions hold true in regard to the equation

V_{t}(x, t) + a(x, t)V (x, t) = f(x, t), in D : (0, l) × ℝ → ℝ  (4)

1. a(x, t), f(x, t) are almost periodic in t, uniformly with respect to x;

2. a(x, t) ≥ m > 0 in D.

Then the unique bounded solution of (4), given by

V (x, t) = ∫_{−∞}^{t} e^{− ∫_{τ}^{t} a(x,θ) dθ}f(x, τ) dτ,  (5)

is almost periodic in t, uniformly with respect to x, and

|V (x, t)| ≤ \frac{1}{m} sup |f(x, t)| , (x, t) ∈ D.  (6)

Proof We obtain from (4), changing t to t + δ:

V_{t}(x, t + δ) + a(x, t + δ)V (x, t + δ) = f(x, t + δ),

and subtracting (4) from it,

[V (x, t + δ) − V (x, t)]_{t} + a(x, t + δ) [V (x, t + δ) − V (x, t)]
= f(x, t + δ) − f(x, t) − [a(x, t + δ) − a(x, t)] V (x, t).

Taking into account the almost periodicity of a(x, t), f(x, t) and boundedness of V (x, t) in D, one obtains in D, according to (6):

sup |V (x, t + δ) − V (x, t)| ≤ \frac{1}{m} sup |f(x, t + δ) − f(x, t)|

+\frac{M}{m} sup |a(x, t + δ) − a(x, t)| ,

where M = sup |V (x, t)|, (x, t) ∈ D. We choose δ such that,

|f(x, t + δ) − f(x, t)| < \frac{mϵ}{2}, and |a(x, t + δ) − a(x, t)| < \frac{mϵ}{2M}

for sufficiently large t, i.e., f(x, t) must be an (mϵ)/2-almost periodic and a(x, t) is (mϵ)/2M-almost periodic. Then

sup |V (x, t + δ) − V (x, t)| ≤ \frac{ϵ}{2} + \frac{ϵ}{2} = ϵ for all such δ ∈ ℝ.  (7)

In other words, for any ϵ > 0, there exists a number l(ϵ) > 0 with the property that any interval (a, a + l) ∈ ℝ contains an ϵ-almost period of V (x, t). This means that V (x, t) is an almost periodic function in t, uniformly with respect to x ∈ [0, l] by Bohr’s definition of almost periodicity. Let us conclude now with the result on almost periodicity of the unique bounded solution of (3) in D.

Theorem 4 Consider equation (1) in D, and assume u_{0}(t), a(x, t), and C(x, t, u(x, [t])) are almost periodic in t, uniformly with respect to x ∈ [0, l], and a(x, t) ≥ m > 0. Also assume that C(x, t, u(x, [t])) is continuous on D × ℝ, with C(x, t, 0) bounded on D, and satisfies the Lipschitz condition

|C(x, t, u(x, [t])) − C(x, t, V (x, [t]))| ≤ L |u(x, [t]) − V (x, [t])|

where L is a positive constant. Then the unique bounded solution of (1)–(2) in D is almost periodic in t, uniformly with respect to x ∈ [0, l].

Proof Let the first approximation be u_{0}(x, t) ≡ 0. Next approximation is then

u_{1}(x, t) = u_{0}(t) + ∫_{0}^{x} ∫_{−∞}^{t} e^{− ∫_{τ}^{t} a(ξ,θ) dθ}C(ξ, τ, 0) dτ dξ .

Since V (x, t) = \frac{∂}{∂x}u_{1}(x, t), then from the equation

V_{t}(x, t) + a(x, t)V (x, t) = C(x, t, 0)

by Lemma 2 we obtain the almost periodicity of V (x, t). But

u_{1}(x, t) = u_{0}(t) + ∫_{0}^{x} V (ξ, t) dξ.

This shows that u_{1}(x, t) is almost periodic in t, uniformly with respect to x ∈ [0, l], and

u_{2}(x, t) = u_{0}(t) + ∫_{0}^{x} ∫_{−∞}^{t} e^{− ∫_{τ}^{t} a(ξ,θ) dθ}C(ξ, τ, u_{1}(ξ, τ)) dτ dξ.

The relation \overline{V} (x, t) = \frac{∂}{∂x}u_{2}(x, t) and equation

\overline{V_{t}}(x, t) + a(x, t)\overline{V} (x, t) = C(x, t, u_{1}(x, t))

implies almost periodicity of

u_{2}(x, t) = u_{0}(t) + ∫_{0}^{x} \overline{V} (ξ, t) dξ,

by Lemma 2. Then u_{3}(x, t) is almost periodic by a similar argument. Consequently, all successive approximations u_{n}(x, t), n = 1, 2, … are almost periodic functions in t, uniformly with respect to x ∈ [0, l]. Hence the solution

u(x, t) = lim_{n→∞} u_{n}(x, t)

is also almost periodic in t, uniformly with respect to x ∈ [0, l].

References

[1] Poorkarimi, H., and Wiener, J., (1986), “Bounded Solutions of Non-linear Hyperbolic Equations with Delay”, Proceedings of the VII International Conference on Non-Linear Analysis, V. Lakshmikantham, Ed., 471–478

[2] Poorkarimi, H., “Asymptotically Periodic Solutions for Some Hyperbolic Equations”, Libertas Mathematica, Vol.8, 1998, 117–122.

[3] Tikhonov, A. N., and Samarskii, A. A., Equations of Mathematical Physics, Pergamon Press, New York, 1963.

[4] Corduneanu, C., Almost Periodic Functions, Wiley, New York, 1968.

Hushang Poorkarimi & Joseph Wiener University of Texas-Pan American Department of Mathematics Edinburg, TX 78539, USA

poorkar@panam.edu & jwiener@panam.edu