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

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.

Here n, m 2 are positive integers with n ≥ 2; a j (t), h j (t) are in C([0, ∞), R); b], R); p, r have n continuous derivatives with respect to time; ∆ is the Laplacian operator, ∆ = ∂ 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 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.

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.
Let λ 1 be the smallest eigenvalue of the elliptic problem 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 , we define the "average function" 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, 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 the PDE (1.1) is transformed into the delay differential inequality Q j (t)v(g j (t)) (for the summation case) . ( 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" which is positive.By Green's formula, Using this inequality, (2.5) and (H1), we obtain (2.4) again.
Our first result concerns the equation q j (x, t)u j (x, g(t)) (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.
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 (2.11) EJQTDE, 2010 No. 59, p. 4 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 ∈ {1, . . ., m 1 }, Using (2.10) and the limit comparison test, lim sup 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).
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), (2.13) From (2.4), using that ĝ(t) ≤ g(t, ξ), we have EJQTDE, 2010 No. 59, p. 5 Then for β ≥ 0, we define and differentiate with respect to t, To estimate the first term in the right-hand side, we use (2.14) and the fact that z(ĝ)/z( 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 (2.15) To estimate the third term, we multiply and divide by t β .Then By completing the square in the brackets, .
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 1 ĝn−2 ĝ′ and ∞ t β−2 t ĝn−1 both converge or both diverge.Now, we use the comparison test, 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.
EJQTDE, 2010 No. 59, p. 6 In the next 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.
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 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 a j (t)∆u(x, h j (t)) − m1 j=1 q j (x, t)u(x, g j (t)) , (2.17) with boundary conditions (1.2) or (1.3).
Theorem 2.7.Assume (H1) holds; there exists a constant p such that p < p 1 (t) ≤ 0; and 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 → ∞).
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.
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 For t, r −1 1 (t), r −1 1 (r −1 1 (t)), . . ., the above inequality yields z(t For a fixed value of t, the left-hand side approaches −∞ as k → ∞, while the right-hand 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.
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 The second equation to be considered in this section is the neutral equation q j (x, t)u(x, g j (t)), (3.3)where n ≥ 3, and b(t) is a positive function in 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 Therefore, b(t)z (n−1) (t) is a decreasing function; hence, eventually positive or eventually negative.
Theorem 3.1.Assume: (H1) holds; there exist a positive constant p such that 0 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 → ∞).
EJQTDE, 2010 No. 59, p. 10 Now, we allow the coefficient p 1 to be negative in the equation Theorem 3.2.Assume (H1) holds; b(t) is bounded; there is constant p such that p < p 1 (t) ≤ 0; and 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 → ∞).
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) ≤ M t n−3 for t sufficiently large.Since p ≤ p 1 (t) ≤ 0, z(t) > p 1 (t)v(r(t)) ≥ pv(r(t)).Using that r(t) ≤ t, we have M (r(t)) n−3 ≥ M t n−3 > pv(r(t)).Recall that p 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 for x ∈ Ω, t ≥ 0, i = 1, . . ., m.To the above system we attach one the following two boundary conditions 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, 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 which is eventually positive when u i is non-oscillatory.Let By Green's formula and (4.2), the first summation in the right-hand side of (4.1) leads to a negative quantity.Therefore, Let qij (t, ξ) = min x∈Ω q ij (x, t, ξ) and qij (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 ) ≥ qjj (t, ξ) Ω δ j u j (x, g j ) = qjj (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 ) ≤ qij (t, ξ) Ω δ j u j (x, g j ) = qij (t, ξ)v j (g j (t, ξ)) .With the notation in (H2) below, provided that z i (t) ≥ 0, we obtain the uncoupled system of inequalities Q ik (t)v i (g ik (t)) (for the summation case) (4.7) where i = 1, . . ., m.
We state only the analog to Theorem 2.4.The other theorems require similar changes in notation.Consider the system ∂ n ∂t n u i (x, t) + q ijk (x, t)u j (x, g jk (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.