Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 59, 1-14; http://www.math.u-szeged.hu/ejqtde/

OSCILLATION OF SOLUTIONS TO A HIGHER-ORDER NEUTRAL PDE WITH DISTRIBUTED DEVIATING ARGUMENTS

JULIO G. DIX

Abstract. This article presents conditions for the oscillation of solutions to neutral partial differential equations. The order of these equations can be even or odd, and the deviating arguments can be distributed over an interval. We also extend our results to a nonlinear equation and to a system of equations.

1. Introduction

We study the oscillation of solutions to the neutral differential equation

\begin{matrix}\frac{∂^{n}}{∂t^{n}} (u(x, t) + ∫_{a}^{b} p(t, ξ)u(x, r(t, ξ)) dμ(ξ)) \\ = ∑_{j=1}^{m_{2}} a_{j}(t)Δu(x, h_{j}(t)) − ∫_{c}^{d} q(x, t, ξ)u(x, g(t, ξ)) dμ(ξ), \end{matrix}  (1.1)

where x is in a bounded domain Ω of ℝ^{d}, with smooth boundary ∂Ω, t ≥ 0, and the delayed arguments satisfy r(t, ξ) ≤ t, h_{j}(t) ≤ t, g(t, ξ) ≤ t. To this equation we attach one of the following two boundary conditions:

u(x, t) = 0 x ∈ ∂Ω, t ≥ 0;  (1.2)
\frac{∂u}{∂ν} + γ(x, t)u(x, t) = 0 x ∈ ∂Ω, t ≥ 0,  (1.3)

where ν is the unit exterior normal vector to ∂Ω, and γ(x, t) is a positive function in C(Ω × [0, ∞), ℝ). Here n, m_{2} are positive integers with n ≥ 2; a_{j}(t), h_{j}(t) are in C([0, ∞), ℝ); g(t, ξ) is in C([0, ∞) × [c, d], ℝ); p(t, ξ) is in C([0, ∞) × [a, b], ℝ); q(x, t, ξ) is in C(Ω × [0, ∞) × [a, b], ℝ); r(t, ξ) is in C([0, ∞) × [a, b], ℝ); p, r have n continuous derivatives with respect to time; Δ is the Laplacian operator, Δ = ∂_{x_{1}^{2}}^{2} + ⋯ + ∂_{x_{d}^{2}}^{2}; and the integrals are in the Stieltjes sense with μ non-decreasing. Note that these integrals can represent summations of the form ∑_{j} p_{j}(t)u(x, r_{j}(t)) and ∑_{j} q_{j}(x, t)u(x, g_{j}(t)), which we call the summation case. The study of solutions to neutral differential equations has practical importance, because they appear in population models, chemical reactions, control systems, etc. There are many publications related to the oscillation of solutions to neutral ordinary differential equations; see for example [2, 6, 11, 12, 13] and the books [1, 3, 4, 7]. There are also some publications for neutral partial differential equations, see for example [8, 9, 10, 14]. Li [8] stated that solutions to a system of type (1.1) are oscillatory for n odd. However their article has many mistakes: On page 527 “There exist

EJQTDE, 2010 No. 59, p. 1 M > 0 such that v(t) ≥ M” is not true when y decreases to zero; Lemma 2 needs the assumption that W is eventually positive; etc. Lin [9] studied a system of neutral PDEs, with n even; we will compare their hypotheses and ours in Section 3. Wang [14] stated that for a particular case of (1.1) all solutions are oscillatory. This is not true for n odd; it is easy to build an example with solution sin(x)e^{−t}, which is non-oscillatory for 0 ≤ x ≤ π. On page 570, it says “By choosing i = 1, we have z′(t) > 0”, which is used later. However, by Lemma 2.1 with n odd, i can be zero and their proof fails. Luo [10] studied a system of PDEs. Their proof follows the steps in [14], including mistakes, so it fails for n odd. The main objective of this article is to present verifiable hypotheses for the oscillation of solutions to (1.1) for even and odd order, with various ranges for the coefficient p(t). In Section 3, we extend our results to a nonlinear neutral equation and to an equation of the type type studied in [9]. In Section 4, we apply our results to a system of neutral partial differential equations.

2000 Mathematics Subject Classification. 35B05, 35R10.

Key words and phrases. Oscillatory solution; neutral PDE; distributed deviating argument.

2. Oscillation for the neutral PDE

By a solution, u(x, t), we mean a function in C(Ω × [t , ∞), ℝ) that is twice continuously differentiable for x ∈ Ω, and n times continuously differentiable for t ≥ 0, and that satisfies (1.1) with a boundary condition (1.2) or (1.3). The value t is the minimum of value of functions r, h_{j}, g when t ≥ 0. A solution u(x, t) is called eventually positive if there exists t_{0} such that u(x, t) > 0 for t ≥ t_{0} and all x in the interior of Ω. Eventually negative solutions are defined similarly. Solutions that are not eventually positive and not eventually negative are called oscillatory; i.e., for every t_{0} ≥ 0, there exist t_{1} ≥ t_{0} and x_{1} in the interior of Ω, such that u(x_{1}, t_{1}) = 0. The following hypotheses will be used in this article.

(H1) 0 ≤ a_{j}(t) with 0 < ∑_{j=1}^{m_{2}} a_{j}(t) for t ≥ 0. There exists a continuous function ĝ(t) such that ĝ(t) ≤ g(t, ξ) and for all t, ξ (in the summation case: ĝ(t) ≤ g_{j}(t) for all j). h_{j}(t), ĝ(t), r(t, ξ) approach +∞ as t → ∞, for all ξ ∈ [a, b], j ∈ {1, …, m_{2}}; Also,

0 ≤ min_{x∈Ω} q(x, t, ξ) := Q(t, ξ), 0 ≤ min_{x∈Ω} q_{j}(x, t) := Q_{j}(t) .

We use the well known “averaging technique” to transform the partial differential equation into a delay differential inequality. The existence (and non-existence) of eventually positive solutions to this inequality, provides oscillation results for neutral differential equations; see for example [3, Theorem 5.1.1]. Let λ_{1} be the smallest eigenvalue of the elliptic problem

Δϕ + λϕ = 0 in Ω
ϕ(x) = 0 on ∂Ω .

It is well know that λ_{1} > 0 and that the corresponding eigenfunction ϕ_{1} does not have zeros in the interior of Ω; we select ϕ_{1}(x) > 0. See for example [5, Theorem 8.5.4]. Assuming that u(x, t) is a solution to (1.1)-(1.2) with u(x, t) > 0 for t ≥ t_{0},

EJQTDE, 2010 No. 59, p. 2 we define the “average function”

v(t) = ∫_{Ω} u(x, t)ϕ_{1}(x) dx  (2.1)

which is positive because both u and ϕ_{1} are positive. Note that v is the projection of u on the first eigenspace of the Laplacian. By Green’s formula,

\begin{aligned}∫_{Ω} Δu(x, t)ϕ_{1}(x) dx & = ∫_{∂Ω} ϕ_{1} \frac{∂u}{∂ν} − u\frac{∂ϕ_{1}}{∂ν} dS + ∫_{∂Ω} uΔϕ_{1} dS \\ & = −λ_{1} ∫_{Ω} u(x, t)ϕ_{1}(x) dx < 0 . \end{aligned}  (2.2)

We multiply each term in (1.1) by the eigenfunction ϕ_{1}, and integrate over Ω. Using (2.1), (2.2), (H1), and the notation

z(t) = v(t) + ∫_{a}^{b} p(t, ξ)v(r(t, ξ)) dμ(ξ) ,  (2.3)

the PDE (1.1) is transformed into the delay differential inequality

z^{(n)}(t) < − ∫_{a}^{b} Q(t, ξ)v(g(t, ξ)) dμ(ξ) ,
z^{(n)}(t) < − ∑_{j=1}^{m_{1}} Q_{j}(t)v(g_{j}(t)) (for the summation case) .

(2.4)

Now for the boundary condition (1.3), assuming that u(x, t) is a positive solution to (1.1)-(1.3), we define the “average function”

v(t) = ∫_{Ω} u(x, t) dx  (2.5)

which is positive. By Green’s formula,

∫_{Ω} Δu(x, t) dx = ∫_{∂Ω} \frac{∂u}{∂ν} = − ∫_{Ω} γ(x, t)u(x, t) dx < 0 .  (2.6)

Using this inequality, (2.5) and (H1), we obtain (2.4) again.

Lemma 2.1 ([7, Lemma 5.2.1]). Let z(t) be an n times differentiable function of constant sign, z^{(n)}(t) be of constant sign and not identically zero in any interval [t_{0}, ∞) and z^{(n)}(t)z(t) ≤ 0. Then

(i) There exists a time t_{1} such that z^{(0)}, z^{(1)}, …, z^{(n−1)} are of constant sign on [t_{1}, ∞).

(ii) There exists an integer k in {1, 3, 5, …, n − 1} when n is even, and k in {0, 2, 4, …, n − 1} when n is odd, such that

\begin{matrix}z^{(i)}(t)z(t) > 0 for i = 0, 1, …, k, \\ (−1)^{n+i−1}z^{(i)}(t)z(t) > 0 for i = k + 1, …, n . \end{matrix}  (2.7)

Remark: In our settings z^{(n)}(t) < 0 for t ≥ t_{1}, which satisfies the “not identically zero” condition in the above lemma. However, this part was not shown in Wang [14] and Luo [10]. There it was also wrongly assumed that z(t) is always increasing.

EJQTDE, 2010 No. 59, p. 3 Lemma 2.2 ([7, Lemma 5.2.2]). Assume that z^{(0)}, z^{(1)}, …, z^{(n−1)} are absolutely continuous and of constant sign on the interval (t_{0}, ∞). Moreover, z^{(n)}(t)z(t) ≥ 0. Then either

z^{(i)}(t)z(t) ≥ 0 for i = 0, 1, …, n,

or there exists an integer k in {0, 2, …, n−2} when n is even, and k in {1, 3, …, n− 2} when n is odd, such that

\begin{matrix}z^{(i)}(t)z(t) ≥ 0 for i = 0, 1, …, k, \\ (−1)^{n+i}z^{(i)}(t) ≥ 0 for i = k + 1, …, n, \end{matrix}  (2.8)

Lemma 2.3 ([7, Lemma 5.2.3]). Let z(t) be an n times differentiable function of constant sign, z^{(n)}(t) be of constant sign and not identically zero in any interval [t_{0}, ∞), and z^{(n)}(t)z^{(n−1)}(t) ≤ 0 for every t ≥ t_{0}. Then for each 0 < λ < 1,

|z(λt)| ≥ M_{1}t^{n−1}|z^{(n−1)}(t)|, M_{1} = \frac{λ^{k}(1 − λ)^{n−1−k}}{2^{k}k!(n − 1 − k)!} ,

where K is defined in Lemma 2.1 (ii).

Our first result concerns the equation

\begin{matrix}\frac{∂^{n}}{∂t^{n}} (u(x, t) + ∫_{a}^{b} p(t, ξ)u(x, r(t, ξ)) dμ(ξ)) \\ = ∑_{j=1}^{m_{2}} a_{j}(t)Δu(x, h_{j}(t)) − ∑_{j=1}^{m_{1}} q_{j}(x, t)u_{j}(x, g(t)) \end{matrix}  (2.9)

with boundary conditions (1.2) or (1.3). This equation is a particular case of (1.1), when μ is constant on [c, d], except at m_{1} values of ξ, where it has jumps of discontinuity.

Theorem 2.4. Assume (H1), 0 ≤ p(t, ξ) and

∫_{0}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t) dt = ∞ .  (2.10)

Then every solution of (2.9) is oscillatory or its “average” v(t) converges to zero, as t → ∞.

Proof. Assuming that u(x, t) is an eventually positive solution of (2.9), we show that the “average” function approaches zero. By (H1) there exists a time t_{0} such that u(x, t), u(x, r(t, ξ)), u(x, h_{j}(t)), and u(x, g_{i}(t)) are positive for all t ≥ t_{0} and all j, ξ. Then we define z by (2.3), so that z(t) > 0, and (2.4) and (2.7) hold. For the value k defined in (2.7), the function z^{(k)}(t) is positive and decreasing. Therefore, L := lim_{t→∞} z^{(k)}(t) exists as a finite number. Note that

0 ≤ z^{(k)}(t) − L = \frac{(−1)^{n−k}}{(n − 1 − k)!} ∫_{t}^{∞}(s − t)^{n−1−k}z^{(n)}(s) ds .  (2.11)

Note that the left-hand side is a finite number for each t; therefore, the integral on the right-hand side is convergent. From (2.4), it follows that for every j ∈

EJQTDE, 2010 No. 59, p. 4 {1, …, m_{1}},

0 ≤ ∫_{0}^{∞} t^{n−1−k}Q_{j}(t)v(g_{j}(t)) dt < ∞ .

Using (2.10) and the limit comparison test,

lim sup_{t→∞} \frac{t^{n−1−k}Q_{j}(t)v(g_{j}(t))}{Q_{j}(t)} = 0

for at least one index j. Since 0 ≤ k ≤ n − 1, for this index, lim_{t→∞} v(g_{j}(t)) = 0. Since g_{j} is continuous and approaches ∞ as t → ∞, we have lim_{t→∞} v(t) = 0. For an eventually negative solution u, we note that −u is also a solution and it is eventually positive. This completes the proof. □

In the next theorem, we relax the conditions on Q_{j}, but restrict the values of p(t, ξ).

Theorem 2.5. Assume: (H1) holds; there exist a positive constant p̂ such that 0 ≤ ∫_{a}^{b} p(t, ξ) dμ(ξ) ≤ p̂ < 1 for t > 0; ĝ(t) is differentiable and strictly increasing; and there exist positive constants α, γ such that γt^{α} ≤ ĝ(t) for t sufficiently large, and β with 0 ≤ β < α(n − 1) such that

∫_{0}^{∞} t^{β} ∑_{j=1}^{m_{1}} Q_{j}(t) dt = ∞ .  (2.12)

Then every solution of (2.9) is oscillatory or its “average” v(t) converges to zero as t → ∞.

Proof. Assuming that u(x, t) is an eventually positive solution, we show that the “average” function approaches zero. Define z by (2.3), so that z(t) is positive, and (2.4) and (2.7) hold.

Case 1: z(t) is decreasing. In this case k = 0 in (2.7); thus L := lim_{t→∞} z(t) exists as a finite number. The same process as in the proof of Theorem 2.4 shows that lim_{t→∞} v(t) = 0.

Case 2: z(t) is increasing. This happens when n is even, because k ≥ 1 in (2.7), and sometimes when n is odd. Note that r(t, ξ) ≤ t and z(r) ≤ z(t). Also note that v(r) ≤ z(r), so that by (2.3),

\begin{aligned}(1 − p̂)z(t) & ≤ (1 − ∫_{a}^{b} p(t, ξ) dμ(ξ))z(t) \\ & ≤ z(t) − ∫_{a}^{b} p(t, ξ)z(r) dμ(ξ) \\ & ≤ z(t) − ∫_{a}^{b} p(t, ξ)v(r) dμ(ξ) = v(t) . \end{aligned}  (2.13)

From (2.4), using that ĝ(t) ≤ g(t, ξ), we have

z^{(n)}(t) < −z(ĝ(t))(1 − p̂) ∑_{j=1}^{m_{1}} Q_{j}(t) ,  (2.14)

Then for β ≥ 0, we define

w(t) = \frac{z^{(n−1)}(t)}{z(\frac{1}{2} ĝ(t))}t^{β}

and differentiate with respect to t,

w′(t) = \frac{z^{(n)}(t)}{z(\frac{1}{2} ĝ)}t^{β} − \frac{z^{(n−1)}(t)z′(\frac{1}{2} ĝ)\frac{1}{2}ĝ′(t)}{(z(\frac{1}{2} ĝ))^{2}}t^{β} + \frac{z^{(n)}(t)}{z(\frac{1}{2} ĝ)}βt^{β−1} .

To estimate the first term in the right-hand side, we use (2.14) and the fact that z(ĝ)/z(\frac{1}{2} ĝ) ≥ 1 because z is increasing. To estimate the second term, we use Lemma 2.3. Since 0 ≤ k ≤ n − 1, we can make M_{1} independent of k, hence independent of the function z. By setting λ = 1/2 and using z′ instead of z and ĝ(t) instead of t, we have constants M and t_{2} such that

z′(\frac{1}{2}ĝ(t)) ≥ M(ĝ(t))^{n−2}z^{(n−1)}(ĝ(t)), for t ≥ t_{2} .  (2.15)

To estimate the third term, we multiply and divide by t^{β}. Then

w′(t) ≤ −t^{β}(1 − p̂) ∑_{j=1}^{m_{1}} Q_{j}(t) − [\frac{Mĝ^{n−2}ĝ′}{2t^{β}} w^{2} − \frac{β}{t} w] .

By completing the square in the brackets,

w′(t) ≤ −t^{β}(1 − p̂) ∑_{j=1}^{m_{1}} Q_{j}(t) + \frac{β^{2}t^{β−2}}{2Mĝ^{n−2}(t)ĝ′(t)} .

Integrating from t_{1} to s,

w(s) ≤ w(t_{1}) − ∫_{t_{1}}^{s} t^{β}(1 − p̂) ∑_{j=1}^{m_{1}} Q_{j}(t) + ∫_{t_{1}}^{s} \frac{β^{2}t^{β−2}}{2Mĝ^{n−2}(t)ĝ′(t)} dt .

Note that the left-hand side remains positive while the right-hand side approaches −∞ as x → ∞. By (2.12) the first integral approaches ∞ while the second integral converges as explained below. This contradiction indicates that there are no eventually positive solutions under assumption (2.12). To study the convergence of the second integral, we use the limit comparison test and L’Hôpital’s Rule, so that ∫^{∞} t^{β−2} \frac{1}{ĝ^{n−2}ĝ′} and ∫^{∞} t^{β−2} \frac{t}{ĝ^{n−1}} both converge or both diverge. Now, we use the comparison test,

0 < \frac{t^{β−1}}{ĝ^{n−1}} ≤ \frac{t^{β−1}}{γ^{n−1}t^{α(n−1)}} = \frac{1}{γ^{n−1}} t^{β−1−α(n−1)} .

By the p-test, the integral converges if β−1−α(n−1) < −1; i.e., β < α(n−1) which is assumed in this theorem. For an eventually negative solution u, we note that −u is also a solution and it is eventually positive. This completes the proof. □

Remark. Instead of t^{β}, Wang [14] and Luo [10] used a positive nondecreasing function. They also used a function H(t, s)ρ(s). However, their hypotheses are not easy to verify, and do not seem to cover a much wider range of coefficients for (1.1). An increasing function ϕ(t) played the role of t^{β} in [9], for n even. In the next

EJQTDE, 2010 No. 59, p. 6 theorem, we restrict n to be even, so we can study the case when ∑ Q is replaced by ∫ Q. Also we obtain results stronger than in Theorem 2.5.

Theorem 2.6. Assume: (H1) holds; n is even; there exist a positive constant p̂ such that 0 ≤ ∫_{a}^{b} p(t, ξ) dμ(ξ) ≤ p̂ < 1 for t > 0; ĝ(t) is differentiable and strictly increasing; there exist positive constants α, γ such that γt^{α} ≤ ĝ(t) for t sufficiently large; and there exists β with 0 ≤ β < α(n − 1) such that

∫_{0}^{∞} t^{β} ∫_{c}^{d} Q(t, ξ) dμ(ξ) = ∞ .  (2.16)

Then every solution of (1.1) is oscillatory.

Proof. Assuming that u(x, t) is an eventually positive solution, we find a contradiction. Define z by (2.3), so that z(t) is positive, and (2.4) and (2.7) hold. Because n is even, k ≥ 1 in (2.7); therefore z(t) is positive and increasing. The rest of the proof is as in the proof of case 2 in Theorem 2.5, except for using ∫_{c}^{d} Qt, ξ) instead of ∑_{j=1}^{m_{1}} Q_{j}(t). □

Note that α in Theorems 2.5 and 2.6 can not exceed 1, because ĝ(t) ≤ t. Also note that when α = 1, the exponent β can be close to n − 1, which seems to be the optimal exponent, even for special cases of (1.1); see [3, Theorem 5.2.6] Next, we allow the coefficient p_{1} to be negative in the equation

\begin{matrix}\frac{∂^{n}}{∂t^{n}} (u(x, t) + p_{1}(t)u(x, r_{1}(t))) \\ = ∑_{j=1}^{m_{2}} a_{j}(t)Δu(x, h_{j}(t)) − ∑_{j=1}^{m_{1}} q_{j}(x, t)u(x, g_{j}(t)) , \end{matrix}  (2.17)

with boundary conditions (1.2) or (1.3).

Theorem 2.7. Assume (H1) holds; there exists a constant p̂ such that p̂ < p_{1}(t) ≤ 0; and

∫_{0}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t) dt = ∞ .

Then every solution of (2.17) is oscillatory, or its “average” v(t) converges to zero, or v(t) approaches infinity at least at the rate of t^{n−2} (as t → ∞).

Proof. Assuming that u(x, t) is an eventually positive solution, we show that the “average” function approaches zero, or the “average” is bounded below a constant times t^{n−2} for t large. Define z by (2.3). Then (2.4) holds and z^{(n)}(t) < 0, for t ≥ t_{0}. As in Lemma 2.1(i), there exists a t_{1} ≥ t_{0} such that z^{(0)}(t), …, z^{(n−1)}(t) are of constant sign on [t_{1}, ∞).

Case 1: z^{(i)}(t) > 0 for some i ∈ {0, …, n − 1}. Then the conditions of Lemma 2.1 are satisfied with z^{(i)}(t) instead of z(t). We proceed as in the proof of Theorem 2.4 to show that lim_{t→∞} v(t) = 0.

Case 2: z^{(i)}(t) < 0 for all i ∈ {0, …, n − 1}. By a repeated integration of z^{(n)}, we obtain a negative constant M, such that z(t) < Mt^{n−2} for t sufficiently large.

EJQTDE, 2010 No. 59, p. 7 From (2.3) and p̂ < p_{1}(t), we obtain z(t) = v(t) + p_{1}(t)v(r_{1}(t)) > p̂v(r_{1}(t)). Since r_{1}(t) ≤ t, we have

\frac{v(r_{1}(t))}{(r_{1}(t))^{n−2}} ≥ \frac{v(r_{1}(t))}{t^{n−2}} > \frac{M}{p̂} > 0 .

Recall that lim_{t→∞} r_{1}(t) = ∞. Therefore, v(t) ≥ Mt^{n−2}/p̂ for all t sufficiently large. This completes the proof. □

In the next theorem, we impose restrictions on n, r_{1} and p_{1}, so that we obtain results stronger than those in Theorem 2.7.

Theorem 2.8. Assume (H1) holds, n is odd, −1 ≤ p_{1}(t) ≤ 0, r_{1}(t) is strictly increasing, and

∫_{0}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t) dt = ∞ .

Then every solution of (2.17) is oscillatory, or its “average” v(t) converges to zero.

Proof. Assuming that u(x, t) is an eventually positive solution, we show that the “average” function approaches zero. Define z by (2.3). Then (2.4) holds and z^{n}(t) < 0, for t ≥ t_{0}. As in Lemma 2.1(i), there exists a t_{1} ≥ t_{0} such that z^{(0)}(t), …, z^{(n−1)}(t) are of constant sign on [t_{1}, ∞). Claim: z(t) > 0 for t ≥ t_{1}. The proof of this claim is a generalization of the proof in [3, Lemma 5.14]. On the contrary assume that z(t) < 0 for t ≥ t_{1}. Since n is odd and z^{(n)}(t) < 0, by Lemma 2.2, z^{(1)}(t) < 0. Thus z(t) is negative and decreasing. For t > t_{1}, we have

0 > z(t_{1}) > z(t) = v(t) + p_{1}(t)v(r_{1}(t)) ≥ v(t) − v(r_{1}(t)) .

For t, r_{1}^{−1}(t), r_{1}^{−1}(r_{1}^{−1}(t)), …, the above inequality yields z(t_{1}) > v(t) − v(r_{1}(t)), z(t_{1}) > v(r_{1}^{−1}(t)) − v(t), z(t_{1}) > v(r_{1}^{−1}(r_{1}^{−1}(t))) − v(r_{1}^{−1}(t)), …. Adding k of these inequalities, we have

kz(t_{1}) > v(r_{1}^{−1}(r_{1}^{−1} … (t))) − v(r_{1}(t)) ≥ −v(r_{1}(t))

For a fixed value of t, the left-hand side approaches −∞ as k → ∞, while the righthand side is a finite number. This contradiction proves the claim. Once we know that z(t) is positive, we proceed as in Theorem 2.4 to show that lim_{t→∞} v(t) = 0. □

3. Oscillation for related neutral PDEs

In this section, we study a nonlinear PDE, and then an equation more general than (1.1). We consider the neutral differential equation

\begin{matrix}\frac{∂^{n}}{∂t^{n}} (u(x, t) + ∫_{a}^{b} p(t, ξ)u(x, r(t, ξ)) dμ(ξ)) \\ = ∑_{j=1}^{m_{2}} a_{j}(t)Δu(x, h_{j}(t)) − q(x, t)F (u(x, g_{1}(t)), …, u(x, g_{m_{1}}(t))) , \end{matrix}  (3.1)

where F ∈ C(ℝ^{m_{1}} → ℝ) satisfies the following two properties: If u_{j} > 0 for all j ∈ {1, …, m_{1}}, then F (u_{1}, …, u_{m_{1}}) ≥ u_{j} for all j; if u_{j} < 0 for all j ∈ {1, …, m_{1}},

EJQTDE, 2010 No. 59, p. 8 then F (u_{1}, …, u_{m_{1}}) ≤ u_{j} for all j. The conditions in (H1) are also assumed with the part corresponding to q replaced by

0 ≤ min_{x∈Ω} q(x, t) := Q(t) .

For defining the “average” v(t), we consider the eventually positive and eventually negative solutions separately. When u(x, t) > 0, define v(t) by (2.1), and z(t) by (2.3). Then instead of 2.4 we obtain

z^{(n)}(t) < −Q(t)v(g_{j}(t)) for all j ∈ {1, …, m_{1}} .  (3.2)

When u(x, t) < 0, for the boundary condition (1.2), define v(t) = − ∫_{Ω} ϕ(x)u(x, t) dx so that v(t) > 0. Then multiply each term in (3.1) by −ϕ(x) and integrate over Ω. Using the assumptions on F, we obtain (3.2) again. For the boundary condition (1.3), define v(t) = − ∫_{Ω} u(x, t) dx so that v(t) > 0. Then multiply each term in (3.1) by −1 and integrate over Ω. Using the assumptions on F, we obtain (3.2) again. Once the differential inequality (3.2) is established for non-oscillatory solutions, the results in Theorems 2.4, 2.5, 2.6, 2.7, 2.8 follow with a minor change in notation: Use ∫_{0}^{∞} Q(t) = ∞ instead of ∫_{0}^{∞} ∑ Q_{j}(t) = ∞. The second equation to be considered in this section is the neutral equation

\begin{matrix}\frac{∂}{∂t}(b(t)\frac{∂^{n−1}}{∂t^{n−1}} (u(x, t) + ∫_{a}^{b} p(t, ξ)u(x, r(t, ξ)) dμ(ξ))) \\ = ∑_{j=1}^{m} a_{j}(t)Δu(x, h_{ij}(t)) − ∑_{j=1}^{m_{1}} q_{j}(x, t)u(x, g_{j}(t)), \end{matrix}  (3.3)

where n ≥ 3, and b(t) is a positive function in C^{1}([0, ∞), ℝ) such that

∫_{0}^{∞} \frac{1}{b(t)} dt = ∞.  (3.4)

A system of this form was studied in [9] when n is odd, with ∫ p replaced by ∑ c_{j}, where 0 ≤ ∑ c_{j} < 1. Assuming that u(x, t) > 0 for t ≥ t_{0}, we define z(t) by (2.3). From (3.3) and each one of the two boundary conditions, (1.2) and (1.3), we have

(b(t)z^{(n−1)}(t))′ < − ∑_{j=1}^{m_{1}} Q_{j}(t)v(g_{j}(t)) for t ≥ t_{0} .  (3.5)

Therefore, b(t)z^{(n−1)}(t) is a decreasing function; hence, eventually positive or eventually negative. Claim: b(t)z^{(n−1)}(t) and z^{(n−1)}(t) are eventually positive, when z(t) > 0. We proceed as in [9]: If there is a time for which b(t)z^{(n−1)}(t) ≤ 0, then because b(t)z^{(n−1)}(t) is decreasing, there is t_{1} such that for all t ≥ t_{1},

z^{(n−1)}(t) ≤ \frac{1}{b(t)}b(t_{1})z^{(n−1)}(t_{1}) < 0 .

Integrating the above inequality, we obtain

z^{(n−2)}(s) ≤ z^{(n−2)}(t_{1}) + b(t_{1})z^{(n−1)}(t_{1}) ∫_{t_{1}}^{s} \frac{1}{b(t)} dt .

By (3.4), the right-hand side approaches −∞ as s → ∞; thus lim_{t→∞} z^{(n−2)}(t) = −∞. Integrating z^{(n−2)}(t), we show that lim_{t→∞} z^{(n−3)}(t) = −∞. Repeating the integration, we show that z^{(n−4)}(t), …, z^{(0)}(t) approach −∞ as t → ∞. This contradicts z(t) > 0, and proves that b(t)z^{(n−1)}(t) is eventually positive, and so is z^{(n−1)}(t). The main result for (3.3) reads as follows.

Theorem 3.1. Assume: (H1) holds; there exist a positive constant p̂ such that 0 ≤ ∫_{a}^{b} p(t, ξ) dμ(ξ) ≤ p̂ < 1 for t > 0; b is bounded; and

∫_{0}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t) dt = ∞ .

Then every solution of (3.3) is oscillatory, or its “average” v(t) converges to zero, or v(t) approaches infinity at least at the rate of t^{n−3} (as t → ∞).

Proof. Assuming that u(x, t) is an eventually positive solution of (3.3), we show that the “average” function approaches zero, or the “average” is bounded below by a constant times t^{n−3}. By (H1) there exists a time t_{0} such that u(x, t), u(x, r(t, ξ)), u(x, h_{j}(t)), and u(x, g_{i}(t)) are positive for all t ≥ t_{0} and all j, ξ. Then we define z by (2.3), so that z(t) > 0. Then by the claim after (3.5), z^{(n−1)}(t) > 0. As in Lemma 2.1(i), there exists a t_{1} ≥ t_{0} such that z^{(0)}(t), …, z^{(n−1)}(t) are of constant sign on [t_{1}, ∞). By Lemma 2.3 with n − 1 instead of n, we have only two possible cases.

Case 1: z^{(i)}(t) < 0 for some i ∈ {1, …, n − 2}. By Lemma 2.1, z^{(n−2)}(t) is negative and increasing; so L := lim_{t→∞} z^{(n−2)}(t) exists as finite number. Then L − z^{(n−2)}(t) = ∫_{t}^{∞} z^{(n−1)}(s) ds. Therefore, lim_{t→∞} z^{(n−1)}(t) = 0 and because b(t) is bounded, lim_{t→∞} b(t)z^{(n−1)}(t) = 0. Then using (3.2)

0 − b(t)z^{(n−1)}(t) = ∫_{t}^{∞} (b(s)z^{(n−1)}(s))′ < − ∫_{t}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t)v(g(_{j}(t)) .

Therefore, 0 ≤ ∫_{0}^{∞} Q_{j}(t)v(g_{j}(t)) < +∞ for each j ∈ {0, …, m_{1}}. We proceed as in the proof of Theorem 2.4 to show that lim_{t→∞} v(t) = 0.

Case 2: z^{(i)}(t) > 0 for all i ∈ {0, …, n − 1}. Note that z(t) is positive and increasing. By repeated integration of z^{(n−1)}, we obtain a positive constant M such that z(t) ≥ Mt^{n−3} for t sufficiently large. Since z(t) is increasing and v(r(t)) ≤ z(r(t)) ≤ z(t), by (2.13), (1 − p̂)z(t) ≤ v(t). Therefore, (1 − p̂)Mt^{n−3} < v(t), which completes the proof. □

Now, we allow the coefficient p_{1} to be negative in the equation

\begin{matrix}\frac{∂}{∂t}(b(t)\frac{∂^{n−1}}{∂t^{n−1}} (u(x, t) + p_{1}(t)u(x, r_{1}(t)))) \\ = ∑_{j=1}^{m} a_{j}(t)Δu(x, h_{ij}(t)) − ∑_{j=1}^{m_{1}} q_{j}(x, t)u(x, g_{j}(t)), \end{matrix}  (3.6)

Theorem 3.2. Assume (H1) holds; b(t) is bounded; there is constant p̄ such that p̄ < p_{1}(t) ≤ 0; and

∫_{0}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t) dt = ∞ .

Then every solution of (3.6) is oscillatory, or its “average” v(t) converges to zero, or v(t) approaches infinity at least at the rate of t^{n−3} (as t → ∞).

Proof. Assuming that u(x, t) is an eventually positive solution of (3.6), we show that the “average” function approaches zero, or the “average” is bounded below by a constant times t^{n−3}. By (H1) there exists a time t_{0} such that u(x, t), u(x, r(t, ξ)), u(x, h_{j}(t)), and u(x, g_{i}(t)) are positive for all t ≥ t_{0} and all j, ξ. Then we define z by (2.3). By (3.5) b(t)z^{(n−1)}(t) is decreasing; thus b(t)z^{(n−1)}(t) is eventually positive or eventually negative. Using that b(t) > 0, we consider four possible cases.

Case 1.1: z^{(n−1)}(t) > 0 and z^{(i)}(t) < 0 for some i ∈ {0, …, n − 2}. By Lemma 2.1, z^{(n−2)}(t) negative and increasing. So L := lim_{t→∞} z^{(n−2)}(t) exists as finite number. Then by (3.5), L − z^{(n−2)}(t) = ∫_{t}^{∞} z^{(n−1)}(s) ds. Therefore, lim_{t→∞} z^{(n−1)}(t) = 0 and because b(t) is bounded, lim_{t→∞} b(t)z^{(n−1)}(t) = 0. Then

0 − b(t)z^{(n−1)}(t) = ∫_{t}^{∞} (b(s)z^{(n−1)}(s))′ < − ∫_{t}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t)v(g(_{j}(t)) .

Therefore, 0 ≤ ∫_{0}^{∞} Q_{j}(t)v(g_{j}(t)) < +∞ for each j ∈ {0, …, m_{1}}. We proceed as in the proof of Theorem 2.4 to show that lim_{t→∞} v(t) = 0.

Case 1.2: z^{(n−1)}(t) > 0 and z^{(i)}(t) > 0 for all i ∈ {0, …, n − 2}. By repeated integration of z^{(n−1)}, we obtain a positive constant M such that z(t) ≥ Mt^{n−3} for t sufficiently large. Since p_{1}(t) ≤ 0, z(t) < v(t). Therefore, Mt^{n−3} < v(t).

Case 2.1: z^{(n−1)}(t) < 0 and z^{(i)}(t) > 0 for some i ∈ {0, …, n − 2}. By Lemma 2.1, z^{(n−2)}(t) is positive and decreasing, As in case 1.1.

0 − b(t)z^{(n−1)}(t) = ∫_{t}^{∞} (b(s)z^{(n−1)}(s))′ < − ∫_{t}^{∞} ∑_{j=1}^{m_{1}} Q_{j}(t)v(g(_{j}(t)) .

Note that the left-hand side is positive, while the right-hand side is negative. This contradiction indicates that this case does not happen.

Case 2.2: z^{(n−1)}(t) < 0 and z^{(i)}(t) < 0 for all i ∈ {0, …, n − 2}. By repeated integration of z^{(n−1)}, we obtain a negative constant M such that z(t) ≤ Mt^{n−3} for t sufficiently large. Since p̄ ≤ p_{1}(t) ≤ 0, z(t) > p_{1}(t)v(r(t)) ≥ p̄v(r(t)). Using that r(t) ≤ t, we have M(r(t))^{n−3} ≥ Mt^{n−3} > p̄v(r(t)). Recall that p̄ and M are negative, and that lim_{t→∞} r(t) = ∞. Then v(t) ≥ M_{1}t^{n−3} for t large, where M_{1} is a positive constant. This completes the proof. □

EJQTDE, 2010 No. 59, p. 11

4. Oscillation for a system of neutral PDEs

In this section, we study the system

\begin{matrix}\frac{∂^{n}}{∂t^{n}} (u_{i}(x, t) + ∫_{a}^{b} p_{i}(t)u_{i}(x, r_{i}(t))) \\ = ∑_{j=1}^{m_{2}} a_{ij}(t)Δu_{i}(x, h_{ij}(t)) − ∑_{j=1}^{m} ∫_{c}^{d} q_{ij}(x, t, ξ)u_{j}(x, g_{j}(t, ξ)) dμ(ξ), \end{matrix}  (4.1)

for x ∈ Ω, t ≥ 0, i = 1, …, m. To the above system we attach one the following two boundary conditions

u_{i}(x, t) = 0 x ∈ ∂Ω, t ≥ 0, i = 1, …, m  (4.2)
\frac{∂u_{i}}{∂ν} + γ_{i}(x, t)u_{i}(x, t) = 0 x ∈ ∂Ω, t ≥ 0, i = 1, …, m .  (4.3)

A system of this type was studied by [9], when n is even and the delays are constants. Their results correspond to Theorem 2.6 with β = 0, but their hypotheses need additional assumptions on p, or q, or f to guarantee that [λ(t)V^{(n−1)}(t)]′ < 0 on page 110. A solution (u_{1}, u_{2}, …, u_{m}) of (4.1) is said to be oscillatory, if at least one component is oscillatory. For eventually positive or eventually negative components, we define δ_{i} = sign(u_{i}); thus, |u_{i}| = δ_{i}u_{i}. Our next step is to transform the coupled system of PDEs (4.1) into an uncoupled system of differential inequalities. When the boundary condition (4.2) is satisfied, we multiply each equation in (4.1) by the first eigenvalue of the Laplacian and by δ_{i}. Then integrate over Ω and add over i = 1, …, m. Let

v_{i}(t) = ∫_{Ω} ϕ_{1}(x)δ_{i}u_{i}(x, t) dx ,  (4.4)

which is eventually positive when u_{i} is non-oscillatory. Let

z_{i}(t) = v_{i}(t) + ∫_{a}^{b} p_{i}(t)v_{i}(r_{i}(t)) .  (4.5)

By Green’s formula and (4.2), the first summation in the right-hand side of (4.1) leads to a negative quantity. Therefore,

∑_{i=1}^{m} z_{i}^{(n)}(t) < − ∑_{i,j}^{m} ∫_{Ω} ∫_{a}^{b} q_{ij}(x, t, ξ)δ_{i}u_{j}(x, g_{j}(t, ξ)) dμ(ξ) .

Let q̄_{ij}(t, ξ) = min_{x∈Ω} q_{ij}(x, t, ξ) and q̂_{ij}(t, ξ) = max_{x∈Ω} q_{ij}(x, t, ξ). Note that

∫_{Ω} q_{jj}(x, t, ξ)δ_{j}u_{j}(x, g_{j}) ≥ q̄_{jj}(t, ξ) ∫_{Ω} δ_{j}u_{j}(x, g_{j}) = q̄_{jj}(t, ξ)v_{j}(g_{j}(t, ξ)) ,

where v_{j} is defined by (4.4). Also note that δ_{i}u_{j} ≤ δ_{j}u_{j} and

∫_{Ω} q_{ij}(x, t, ξ)δ_{i}u_{j}(x, g_{j}) ≤ q̂_{ij}(t, ξ) ∫_{Ω} δ_{j}u_{j}(x, g_{j}) = q̂_{ij}(t, ξ)v_{j}(g_{j}(t, ξ)) .

The three inequalities above imply

∑_{i=1}^{m} z_{i}^{(n)}(t) < − ∑_{j=1}^{m} ∫_{c}^{d} [q̄_{jj}(t, ξ) − ∑_{i≠j}^{m} q̂_{ij}(t, ξ)]v_{j}(g_{j}(t, ξ)) dμ(ξ) .  (4.6)

With the notation in (H2) below, provided that z_{i}(t) ≥ 0, we obtain the uncoupled system of inequalities

z_{i}^{(n)}(t) < − ∫_{c}^{d} Q_{i}(t, ξ)v_{i}(g_{i}(t, ξ)) dμ(ξ),
z_{i}^{(n)}(t) < − ∑_{k=1}^{m_{1}} Q_{ik}(t)v_{i}(g_{ik}(t)) (for the summation case)

(4.7)

where i = 1, …, m. The hypotheses in (H1) are modified as follows

(H2) 0 ≤ a_{ij}(t) with 0 < ∑_{j=1}^{m_{2}} a_{ij}(t) for t ≥ 0, 1 ≤ i ≤ m. There exist continuous functions ĝ_{i}(t) such that ĝ_{i}(t) ≤ g_{i}(t, ξ) and for all t, ξ, i (in the summation case: ĝ_{i}(t) ≤ g_{ik}(t) for all k). h_{ij}(t), ĝ_{i}(t), r_{i}(t, ξ) approach +∞ as t → ∞, for all ξ ∈ [a, b], j ∈ {1, … m_{2}}, i ∈ {1, … m}; let q̄_{ij}(t, ξ) = min_{x∈Ω} q_{ij}(x, t, ξ) and q̂_{ij}(t, ξ) = max_{x∈Ω} q_{ij}(x, t, ξ); assume that

0 ≤ q̄_{jj}(t, ξ) − ∑_{i≠j}^{m} q̂_{ij}(t, ξ) := Q_{j}(t, ξ) .  (4.8)

When p is non-negative, each component z_{i} is positive, then the inequality (4.7) holds. So that the proofs of Theorems 2.4, 2.5 and 2.6 apply to each component. However, when p ≤ 0, the inequality (4.7) may not hold. So we cannot state analogs for Theorems 2.7 and 2.8. We state only the analog to Theorem 2.4. The other theorems require similar changes in notation. Consider the system

\begin{matrix}\frac{∂^{n}}{∂t^{n}} (u_{i}(x, t) + ∫_{a}^{b} p_{i}(t)u_{i}(x, r_{i}(t))) \\ = ∑_{j=1}^{m_{2}} a_{ij}(t)Δu_{i}(x, h_{ij}(t)) − ∑_{j=1}^{m} ∑_{k=1}^{m_{1}} q_{ijk}(x, t)u_{j}(x, g_{jk}(t)) . \end{matrix}  (4.9)

Theorem 4.1. Assume (H2) holds; 0 ≤ p_{i}(t, ξ) for all i, t, ξ; and

∫_{0}^{∞} ∑_{k=1}^{m_{1}} Q_{ik}(t) dt = ∞ .

Then each component of each solution of (4.9) is oscillatory, or its “average” v_{i}(t) converges to zero, as t → ∞.

Concluding Remarks. We studied oscillation only for a few range intervals of the coefficient p(t), but there are many intervals to be considered. The case when p changes sign is also an open question. Another open question is oscillation for nonlinearities more general than those in Section 3.

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EJQTDE, 2010 No. 59, p. 13

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(Received July 15, 2010)

Department of Mathematics, Texas State University, 601 University Dr., San Marcos, TX 78666, USA

E-mail address: jd01@txstate.edu