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Oscillation of a kind of second order quasilinear equation with mixed arguments
Journal Pre-proof Oscillation of a kind of second order quasilinear equation with mixed arguments
Ying Sui, Huimin Yu
PII: DOI: Reference:
S0893-9659(19)30520-8 https://doi.org/10.1016/j.aml.2019.106193 AML 106193
To appear in:
Applied Mathematics Letters
Received date : 31 October 2019 Accepted date : 14 December 2019 Please cite this article as: Y. Sui and H. Yu, Oscillation of a kind of second order quasilinear equation with mixed arguments, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106193. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
Journal Pre-proof
Oscillation of a kind of second order quasilinear equation with mixed arguments∗ Ying Sui† Huimin Yu‡ School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China.
Abstract: In this paper, we investigate the oscillation of a kind of quasilinear equation with mixed arguments and Dirichlet/Robin boundary conditions. By using inequality technique, generalized Riccati transformation and comparison method, we obtain three sufficient conditions. At last, the main results are illustrated by examples. Keywords: Oscillation, quasilinear equation, delay, mixed arguments.
Introduction
na lP repr oo f
1
In this paper, we consider the oscillatory of equation in the form r(t)uα−1 utt + p(x, t)uα−2 u2t + f (u, x, m(t)) = a(t)4u(x, t) +
s X
ak (t)4u(x, η(t)),
k=1
(x, t) ∈ G,
(1.1) where α ≥ 2 is the ratio of positive odd integers, G = Ω × (t0 , ∞), t0 > 0, Ω is a bounded domain ∂ 2 u(x,t) in RN with a piecewise smooth boundary ∂Ω, and 4u(x, t) = ΣN j=1 ∂x2 . j
Throughout this paper, we assume that (H1) r(t), a(t), ak (t) ∈ C((t0 , ∞), R+ ), k ∈ Is = {1, 2, · · ·, s}, p(x, t) ∈ C(G, R+ ), and (α − 1)r(t) ≤ p(x, t) for t > t0 ; (H2) m(t) ∈ C((t0 , ∞), R+ ), η(t) ∈ ((t0 , ∞), R+ ) are increasing w.r.t time t, m(t) ≥ t, and lim m(t) = lim η(t) = ∞ ; t→∞
t→∞
(H3) f ∈ C(R × Ω × (t0 , ∞), R), q(x, t) ∈ C(G, R+ ), and f (u, x, t)/uα (x, t) ≥ q(x, t) ≥ q(t) = minx∈Ω¯ q(x, t) > 0 for any u 6= 0. Problem (1.1) is governed by the following boundary conditions: (1.2)
u(x, t) = 0, (x, t) ∈ ∂Ω × R+ ,
(1.3)
ur
or
∂u(x, t) + ψ(x, t)u(x, t) = 0, (x, t) ∈ ∂Ω × R+ , ∂γ
Jo
where γ is the unit exterior normal vector of Ω and ψ(x, t) is a nonnegative continuous function on ∂Ω × R+ . Oscillation theory is an important part of equation qualitative theory, and it has important applications in the fields of control engineering, mechanical vibration and mechanics. In recent years, there are many results about the oscillation of functional differential equations, such as [1–6] and the references therein. 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 [7] through the method of integral average and generalized Riccati transformation. In 2017, the authors [8] obtained some sufficient conditions for oscillation ∗ This
work is supported in part by the National Natural Science Foundation of China (Grant No. 11671237) [email protected] ‡ Corresponding author, Email: [email protected] † Email:
1
Journal Pre-proof
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criteria to a kind of linear impulsive partial fractional differential equations by differential inequality method. Both of the above articles [7, 8] studied linear partial differential equations. Also, the oscillation of semilinear wave equations are studied, we refer to [9,10]. At the same time, there are many results on delay functional differential equations, for example see [11, 12] and the references therein. Moreover, the evolution rate of advanced functional differential equations can be used in many application problems. These kind of equations depend not only on the present but also on the future. The typical fields of advanced phenomenon are population dynamics, economic problems or mechanical control engineering. Therefore, the advanced functional differential equations highlight the influence of potential future actions, which are very useful in the decision-making process [13]. However, to the best of our knowledge, there are very few results dealing with the oscillation of the solutions for quasilinear partial differential equations with advanced arguments. In this article, we study the quasilinear partial differential equation (1.1) with mixed arguments. In equation (1.1), m(t) ≥ t, so m(t) is the advanced arguments. And the relationship between η(t) and t is not limited, so η(t) can be either advanced arguments or delayed arguments. In order to overcome the difficulties brought by the quasilinear terms, we construct two new integral mean functions of space variable, which transform quasilinear partial differential equation (1.1) into a functional differential equation with advanced arguments. Evidently, the advanced arguments could influent the oscillation of our problem. Fortunately, we find the monotonicity of some important functions and then deal with the advanced arguments skillfully. By using inequality method, generalized Riccati transformation and comparison theorem, we obtain some new oscillation criteria for nonlinear problems (1.1)(1.2) and (1.1)(1.3). We believe that our oscillation results are simpler and completely new. For later use, we give the following definition and lemma.
Definition 1.1. The solution u(x, t) of problem (1.1), (1.2) or (1.3) 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. Lemma 1.1. The smallest eigenvalue η0 of the Dirichlet problem ( ∆u(x) + ηu(x) = 0, in Ω, u(x) = 0, on ∂Ω
(1.4)
is positive and the corresponding eigenfunction ϕ(x) is positive in Ω.
2
Main results
Now, we present the main results of this paper.
Theorem 2.1. Assume that (H1)-(H3) hold, Q(t) = αq(m(t)) r(t) . If Z ∞ tQ(t)dt = ∞,
(2.1)
ur
t0
and for sufficiently large t ≥ T ≥ t0 , Z t Z −1 m (t) (s − T )Q(s)m(s)ds + (t − T )
(2.2)
Q(s)ds > 1.
t
Jo
T
∞
Then every smooth solution u(x, t) of (1.1), (1.2) or (1.3) is oscillatory in G. Proof. Suppose to the contrary that there is a nonoscillatory smooth solution u(x, t) of problem (1.1), (1.2) or (1.3), and there exists a t1 ≥ t0 such that u(x, t) has no zero on Ω × (t1 , ∞). Without ¯ × (t1 , ∞). the loss of generality, we may assume that u(x, t) > 0, u(x, m(t)) > 0, u(x, η(t)) > 0 in Ω By (H3), we have f (u, x, m(t)) ≥ q(m(t))uα (x, m(t)), then from (1.1) and (H1), r(t)(uα−1 utt + (α − 1)uα−2 u2t ) + q(m(t))uα (x, m(t)) ≤ a(t)4u(x, t) +
2
s X
ak (t)4u(x, η(t)),
k=1
(2.3)
Journal Pre-proof
Case (1): We consider (1.1) and Robin boundary condition (1.2). Integrating (2.3) with respect to x over the domain Ω, we have Z Z r(t) (uα−1 utt + (α − 1)uα−2 u2t )dx + q(m(t)) uα (x, m(t))dx Ω
≤a(t)
Z
Ω
s X
4u(x, t)dx +
ak (t)
Ω
Z
Ω
k=1
4u(x, η(t))dx,
(2.4)
(x, t) ∈ G.
From Green’s formula and boundary condition (1.2), it follows that Z Z Z ∂u(x, t) ds = − ψ(x, t)u(x, t)ds < 0, 4u(x, t)dx = ∂γ ∂Ω Ω ∂Ω and
Z
Ω
4u(x, η(t))dx =
Z
∂Ω
∂u(x, η(t)) ds = − ∂γ
where ds is surface element on ∂Ω. Set
Z
1 α
then v(m(t)) =
Z
uα (x, t)dx,
(2.7)
Ω
v 00 (t) + Q(t)v(m(t)) < 0,
where Q(t) =
(2.6)
uα (x, m(t))dx and v(t) > 0 for big enough t > 0. In view of (2.5)-(2.7),
Ω
(2.4)yields
1 α
ψ(x, η(t))u(x, η(t))ds < 0,
∂Ω
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v(t) =
Z
(2.5)
αq(m(t)) r(t)
(2.8)
> 0.
Case (2): We consider (1.1) and Dirichlet boundary condition (1.3). Lemma 1.1 is needed here. Now, multiplying (2.3) by the eigenfunction ϕ(x) > 0 and integrating with respect to x over the domain Ω, we have Z Z α−1 α−2 2 r(t) ϕ(x)(u utt + (α − 1)u ut )dx + q(m(t)) ϕ(x)uα (x, m(t))dx Ω
≤a(t)
Z
Ω
4u(x, t)ϕ(x)dx +
s X
ak (t)
Z
Ω
k=1
Ω
4u(x, η(t))ϕ(x)dx,
(2.9)
(x, t) ∈ G.
From Green’s formula and boundary condition (1.3), it follows that Z Z Z 4u(x, t)ϕ(x)dx = u(x, t)4ϕ(x)dx = −η0 u(x, t)ϕ(x)dx < 0, Ω
and
Ω
4u(x, η(t))ϕ(x)dx =
ur
Z
Ω
Ω
u(x, η(t))4ϕ(x)dx = −η0 1 v(t) = α
Jo
Set
Z
(2.10)
Ω
Z
Z
u(x, η(t))ϕ(x)dx < 0.
(2.11)
Ω
ϕ(x)uα (x, t)dx,
(2.12)
Ω
Z 1 ϕ(x)uα (x, m(t))dx. In view of (2.10)-(2.12), (2.9) yields (2.8) holds. In α Ω conclusion, for (1.1) and two boundary conditions (1.2) and (1.3), the inequality (2.8) holds. then v(m(t)) =
By (2.8), we get v 00 (t) < −Q(t)v(m(t)) < 0, thus v 0 (t) is decreasing for all t ≥ t1 , which implies that v 0 (t) does not change sign eventually, then there exists a T ≥ t1 such that either v 0 (t) < 0 or v 0 (t) > 0 for any t ≥ T . If we assume that v 0 (t) < 0, then v(t) = v(T ) +
Z
t
T
v 0 (s)ds ≤ v(T ) + v 0 (T ) 3
Z
t
T
ds, for any t > T,
Journal Pre-proof
imply v(t) → −∞ as t → ∞, which is a contradiction to the positivity of v(t). Therefore, v 0 (t) > 0, v 00 (t) < 0, for any t ≥ T , and limt→∞ v 0 (t) exists.
0 If v(t) → ∞ as t → ∞, it follows from L’Hospital rule that limt→∞ v(t) t = limt→∞ v (t). We 0 0 claim that (2.1) implies limt→∞ v (t) = 0. If we admit that limt→∞ v (t) = l > 0, then integrate (2.8) from T ≥ t1 ≥ t0 to ∞, by m(t) ≥ t, yields Z Z ∞ Z ∞ l ∞ v(s) 0 sQ(s)ds. ds ≥ v (T ) > Q(s)v(m(s))ds ≥ sQ(s) s 2 T T T
This contradicts to (2.1). If v(t) < ∞ as t → ∞, we also claim that limt→∞ v 0 (t) = 0. If we admit that limt→∞ v 0 (t) = Rt Rt Rt L > 0, then v(t) ≥ T v 0 (s)ds ≥ v 0 (t) T ds ≥ L2 T ds, which is a contradiction to v(t) < ∞ as t → ∞. Then Z t v 0 (s)ds ≥ v(T ) − T v 0 (t) + tv 0 (t) ≥ tv 0 (t), for sufficiently large t. v(t) = v(T ) + T
And
v(t) >
Z tZ
∞
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v(t) 0 tv 0 (t) − v(t) ≤ 0, for sufficiently large t. = t t2 Next, integrating (2.8) twice, for t ≥ h ≥ T , we have Z tZ
t
Q(s)v(m(s))dsdh + T h T h Z t Z ∞ = (s − T )Q(s)v(m(s))ds + (t − T ) Q(s)v(m(s))ds. T
Q(s)v(m(s))dsdh =
Z tZ T
∞
(2.13)
Q(s)v(m(s))dsdh
t
(2.14)
t
From (2.14), (2.13) and m(t) ≥ t, we obtain v(m(t)) v(m(t)) > m(t)
Z
t
T
(s − T )Q(s)m(s)ds + (t − T )v(m(t))
Z
∞
Q(s)ds,
t
Rt R∞ then 1 > m−1 (t) T (s − T )Q(s)m(s)ds + (t − T ) t Q(s)ds. This contradicts (2.2) and we complete the proof of Theorem 2.1. Theorem 2.2. Let (H1)-(H3) hold. Assume that there exists a funtion b(t) ∈ C 1 ((t0 , ∞), R+ ) such that Z t (b0 (s))2 (2.15) lim sup b(s)Q(s) − ds = ∞. 4b(s) t→∞ T Then every smooth solution u(x, t) of (1.1), (1.2) or (1.3) is oscillatory in G.
ur
Proof. Proceeding as in the proof of Theorem 2.1, we have v 0 (t) > 0 holds, for all t ≥ T . And the corresponding function v(t) satisfies (2.8). Define the following generalized Riccati 0 (t) transformation: w(t) = b(t) vv(t) , then w(t) > 0. By (2.8), v 0 (t) > 0 and m(t) ≥ t, we obtain b0 (t) v 00 (t) (v 0 (t))2 b0 (t) −Q(t)v(m(t)) (v 0 (t))2 w(t) + b(t) − b(t) 2 < w(t) + b(t) − b(t) 2 b(t) v(t) v (t) b(t) v(t) v (t) 2 0 0 0 b (t) 1 2 1 b (t) (b (t))2 ≤ w(t) − b(t)Q(t) − w (t) = −b(t)Q(t) − w(t) − + b(t) b(t) b(t) 2 4b(t) 0 2 (b (t)) ≤ −b(t)Q(t) + . 4b(t) (2.16) Rt (b0 (s))2 Integration (2.16) from T ≥ t1 ≥ t0 to t, yields w(T ) > T b(s)Q(s) − 4b(s) ds, which contra-
Jo
w0 (t) =
dicts (2.15). This completes the proof.
4
Journal Pre-proof
Theorem 2.3. Let (H1)-(H3) hold. Assume that there exist a constant β and a function τ (t) ∈ C((t0 , ∞), R+ ), such that τ (t) ≤ t, τ 0 (t) ≥ 0, τ 00 (t) ≤ 0 and the first order inequation v 0 (t) − Q1 (t)v(m(τ (t))) > 0
(2.17)
R t+β has no positive solution, where Q1 (t) = t Q0 (s)ds, Q0 (t) = min{Q(t), Q(τ (t))(τ 0 (t))2 }, and Q(t) as defined in Theorem 2.1. Then every smooth solution u(x, t) of (1.1), (1.2) or (1.3) is oscillatory in G. Proof. Proceeding as in the proof of Theorem 2.1, we have v 0 (t) > 0 holds, for all t ≥ T . And the corresponding function v(t) satisfies (2.8). Further, it follows from (2.8) that vτ τ (τ (t)) + Q(τ (t))v(m(τ (t))) < 0. Then we have v 00 (τ (t)) = vτ τ (τ (t))(τ 0 (t))2 + vτ (τ (t))τ 00 (t) < −Q(τ (t))v(m(τ (t)))(τ 0 (t))2 + vτ (τ (t))τ 00 (t), therefore
v 00 (τ (t)) + Q(τ (t))v(m(τ (t)))(τ 0 (t))2 < vτ (τ (t))τ 00 (t) ≤ 0.
(2.18)
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Combining (2.8) and (2.18), we have v 00 (t)+Q(t)v(m(t))+v 00 (τ (t))+Q(τ (t))(τ 0 (t))2 v(m(τ (t))) < 0. On the other hand, Q0 (t) = min{Q(t), Q(τ (t))(τ 0 (t))2 }, τ (t) ≤ t, m(t) is increasing and v 0 (t) > 0 provides v 00 (t) + v 00 (τ (t)) < −2Q0 (t)v(m(τ (t))). (2.19) R t+β Integrating (2.19) from t to t+β, β is a constant, we get v 0 (t)+v 0 (τ (t)) > 2 t Q0 (s)v(m(τ (s)))ds. On the other hand, since v 0 (t) is decreasing and τ (t) ≤ t, then v 0 (τ (t)) >
Z
t+β
(2.20)
Q0 (s)v(m(τ (s)))ds.
t
Integrating (2.20) from T to t, and using that v(t) and m(t) is increasing, yields v(τ (t)) >
Z tZ T
t+β
h
Q0 (s)v(m(τ (s)))dsdh ≥
Z
t
v(m(τ (h)))
T
Z
t+β
Q0 (s)dsdh.
(2.21)
h
Rt Rt By (2.21) we have v(τ (t)) > T Q1 (h)v(m(τ (h)))dh. Let V (t) := T Q1 (h)v(m(τ (h)))dh. Then V (t) > 0. Noting that v(t) ≥ v(τ (t)) > V (t), one can see that V 0 (t) = Q1 (t)v(m(τ (t))) > Q1 (t)V (m(τ (t))). Thus V (t) is a positive solution of (2.17). This contradicts our assumptions.
3
Examples
Example 3.1 As an illustrative example, we consider the following equation
ur
tu6 utt + 6(x2 + 1)tu5 u2t + t2 u7 (x, 2t) = 4u(x, t) + with the boundary conditions: 1 2,
∂u(x,t) ∂γ
s X
(2 + sin kt)4u(x, t − 3),
(3.1)
k=1
+ tu(x, t) = 0, (x, t) ∈ ∂Ω × R+ .
Jo
Here α = 7, t0 = r(t) = t, q(t) = t, p(x, t) = 6(x2 + 1)t, m(t) = 2t, T = 1, a(t) = 1, ak (t) = 2 + sin kt, η(t) = t − 3, f (u, x, t) = t2 u7 (x, t), ψ(x, t) = t. And it is easy to verify Rt R∞ R∞ R∞ 1 28s(s − 1)ds + (t − 1) t 14ds > 1. It is easy to Q(t) = 14, t0 tQ(t)dt = 1 14tdt = ∞, and 2t 1 2 check that all hypotheses of Theorem 2.1 are satisfied, so we get the equation (3.1) is oscillatory. Example 3.2. Consider the equation u2 utt + (t + 3)uu2t + u3 (x, t + 1) = t2 4u(x, t) + with the boundary conditions: u(x, t) = 0, (x, t) ∈ ∂Ω × R+ . 5
s X
k=1
kt4u(x, t3 ),
(3.2)
Journal Pre-proof
Here α = 3, t0 = 1, r(t) = 1, q(t) = 1, p(x, t) = t + 3 ≥ 2, m(t) = t + 1, T = 2, a(t) = t2 , Rt ak (t) = kt, η(t) = t3 , f (u, x, t) = u3 (x, t). By taking b(t) = 1, we get lim supt→∞ T b(s)Q(s) − Rt (b0 (s))2 4b(s) ds = lim supt→∞ 2 3ds = ∞. It is easy to check that all hypotheses of Theorem 2.2 are satisfied, so we get the equation (3.2) is oscillatory. Example 3.3. Consider the following equation 4
1
7
2tu 3 utt + (x2 + 6t)u 3 u2t + 2tu 3 (x, t2 ) = 4u(x, t) +
s X k
k=1
t
1 4u(x, ), t
(x, t) ∈ G,
(3.3)
References
na lP repr oo f
with the boundary conditions: u(x, t) = 0, (x, t) ∈ ∂Ω × R+ . Here α = 37 , r(t) = 2t, q(t) = t, p(x, t) = x2 + 6t, m(t) = t2 , a(t) = 1, ak (t) = kt , 7 7 η(t) = 1t , f (u, x, t) = 2tu 3 (x, t), τ (t) = 12 t. Let β = 1, we obtain Q(t) = 67 t, Q0 (t) = 48 t, R t+1 7 7 7 and Q1 (t) = t 48 sds = 48 t + 96 . When t → ∞, proceeding as in the proof of Theorem 2.1, 2 7 7 we have limt→∞ v 0 (t) = 0, so we obtain equation v 0 (t) − ( 48 t + 96 )v( t4 ) > 0 has no a positive solution as t → ∞. It is easy to check that all hypotheses of Theorem 2.3 are satisfied, so we get the equation (3.3) is oscillatory.
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[9] P. Prakash, S. Harikrishnan, M. Benchohra, Oscillation of certain nonlinear fractional partial differential equation with damping term, Applied Mathematics Letters, 43 (2015) 72–79.
Jo
[10] Q. Yang, On the oscillation of certain nonlinear neutral partial differential equations, Applied Mathematics Letters, 20 (2007) 900–907. [11] X. Li, T. Caraballo, R.Rakkiyappan, X.Han, On the stability of impulsive functional differential equations with infinite delays, Mathematical Methods in the Applied Sciences, 38 (2015) 3130-3140. [12] H. Li, Y. Zheng, F.Alsaadi, Algebraic formulation and topological structure of Boolean networks with state-dependent delay, Journal of computational and applied mathematics, 350 (2019) 87-97. [13] L. Elsgolts, S. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Elsevier, 1973.
6
Ying Sui, Huimin Yu
PII: DOI: Reference:
S0893-9659(19)30520-8 https://doi.org/10.1016/j.aml.2019.106193 AML 106193
To appear in:
Applied Mathematics Letters
Received date : 31 October 2019 Accepted date : 14 December 2019 Please cite this article as: Y. Sui and H. Yu, Oscillation of a kind of second order quasilinear equation with mixed arguments, Applied Mathematics Letters (2019), doi: https://doi.org/10.1016/j.aml.2019.106193. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Elsevier Ltd. All rights reserved.
Journal Pre-proof
Oscillation of a kind of second order quasilinear equation with mixed arguments∗ Ying Sui† Huimin Yu‡ School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China.
Abstract: In this paper, we investigate the oscillation of a kind of quasilinear equation with mixed arguments and Dirichlet/Robin boundary conditions. By using inequality technique, generalized Riccati transformation and comparison method, we obtain three sufficient conditions. At last, the main results are illustrated by examples. Keywords: Oscillation, quasilinear equation, delay, mixed arguments.
Introduction
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1
In this paper, we consider the oscillatory of equation in the form r(t)uα−1 utt + p(x, t)uα−2 u2t + f (u, x, m(t)) = a(t)4u(x, t) +
s X
ak (t)4u(x, η(t)),
k=1
(x, t) ∈ G,
(1.1) where α ≥ 2 is the ratio of positive odd integers, G = Ω × (t0 , ∞), t0 > 0, Ω is a bounded domain ∂ 2 u(x,t) in RN with a piecewise smooth boundary ∂Ω, and 4u(x, t) = ΣN j=1 ∂x2 . j
Throughout this paper, we assume that (H1) r(t), a(t), ak (t) ∈ C((t0 , ∞), R+ ), k ∈ Is = {1, 2, · · ·, s}, p(x, t) ∈ C(G, R+ ), and (α − 1)r(t) ≤ p(x, t) for t > t0 ; (H2) m(t) ∈ C((t0 , ∞), R+ ), η(t) ∈ ((t0 , ∞), R+ ) are increasing w.r.t time t, m(t) ≥ t, and lim m(t) = lim η(t) = ∞ ; t→∞
t→∞
(H3) f ∈ C(R × Ω × (t0 , ∞), R), q(x, t) ∈ C(G, R+ ), and f (u, x, t)/uα (x, t) ≥ q(x, t) ≥ q(t) = minx∈Ω¯ q(x, t) > 0 for any u 6= 0. Problem (1.1) is governed by the following boundary conditions: (1.2)
u(x, t) = 0, (x, t) ∈ ∂Ω × R+ ,
(1.3)
ur
or
∂u(x, t) + ψ(x, t)u(x, t) = 0, (x, t) ∈ ∂Ω × R+ , ∂γ
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where γ is the unit exterior normal vector of Ω and ψ(x, t) is a nonnegative continuous function on ∂Ω × R+ . Oscillation theory is an important part of equation qualitative theory, and it has important applications in the fields of control engineering, mechanical vibration and mechanics. In recent years, there are many results about the oscillation of functional differential equations, such as [1–6] and the references therein. 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 [7] through the method of integral average and generalized Riccati transformation. In 2017, the authors [8] obtained some sufficient conditions for oscillation ∗ This
work is supported in part by the National Natural Science Foundation of China (Grant No. 11671237) [email protected] ‡ Corresponding author, Email: [email protected] † Email:
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Journal Pre-proof
na lP repr oo f
criteria to a kind of linear impulsive partial fractional differential equations by differential inequality method. Both of the above articles [7, 8] studied linear partial differential equations. Also, the oscillation of semilinear wave equations are studied, we refer to [9,10]. At the same time, there are many results on delay functional differential equations, for example see [11, 12] and the references therein. Moreover, the evolution rate of advanced functional differential equations can be used in many application problems. These kind of equations depend not only on the present but also on the future. The typical fields of advanced phenomenon are population dynamics, economic problems or mechanical control engineering. Therefore, the advanced functional differential equations highlight the influence of potential future actions, which are very useful in the decision-making process [13]. However, to the best of our knowledge, there are very few results dealing with the oscillation of the solutions for quasilinear partial differential equations with advanced arguments. In this article, we study the quasilinear partial differential equation (1.1) with mixed arguments. In equation (1.1), m(t) ≥ t, so m(t) is the advanced arguments. And the relationship between η(t) and t is not limited, so η(t) can be either advanced arguments or delayed arguments. In order to overcome the difficulties brought by the quasilinear terms, we construct two new integral mean functions of space variable, which transform quasilinear partial differential equation (1.1) into a functional differential equation with advanced arguments. Evidently, the advanced arguments could influent the oscillation of our problem. Fortunately, we find the monotonicity of some important functions and then deal with the advanced arguments skillfully. By using inequality method, generalized Riccati transformation and comparison theorem, we obtain some new oscillation criteria for nonlinear problems (1.1)(1.2) and (1.1)(1.3). We believe that our oscillation results are simpler and completely new. For later use, we give the following definition and lemma.
Definition 1.1. The solution u(x, t) of problem (1.1), (1.2) or (1.3) 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. Lemma 1.1. The smallest eigenvalue η0 of the Dirichlet problem ( ∆u(x) + ηu(x) = 0, in Ω, u(x) = 0, on ∂Ω
(1.4)
is positive and the corresponding eigenfunction ϕ(x) is positive in Ω.
2
Main results
Now, we present the main results of this paper.
Theorem 2.1. Assume that (H1)-(H3) hold, Q(t) = αq(m(t)) r(t) . If Z ∞ tQ(t)dt = ∞,
(2.1)
ur
t0
and for sufficiently large t ≥ T ≥ t0 , Z t Z −1 m (t) (s − T )Q(s)m(s)ds + (t − T )
(2.2)
Q(s)ds > 1.
t
Jo
T
∞
Then every smooth solution u(x, t) of (1.1), (1.2) or (1.3) is oscillatory in G. Proof. Suppose to the contrary that there is a nonoscillatory smooth solution u(x, t) of problem (1.1), (1.2) or (1.3), and there exists a t1 ≥ t0 such that u(x, t) has no zero on Ω × (t1 , ∞). Without ¯ × (t1 , ∞). the loss of generality, we may assume that u(x, t) > 0, u(x, m(t)) > 0, u(x, η(t)) > 0 in Ω By (H3), we have f (u, x, m(t)) ≥ q(m(t))uα (x, m(t)), then from (1.1) and (H1), r(t)(uα−1 utt + (α − 1)uα−2 u2t ) + q(m(t))uα (x, m(t)) ≤ a(t)4u(x, t) +
2
s X
ak (t)4u(x, η(t)),
k=1
(2.3)
Journal Pre-proof
Case (1): We consider (1.1) and Robin boundary condition (1.2). Integrating (2.3) with respect to x over the domain Ω, we have Z Z r(t) (uα−1 utt + (α − 1)uα−2 u2t )dx + q(m(t)) uα (x, m(t))dx Ω
≤a(t)
Z
Ω
s X
4u(x, t)dx +
ak (t)
Ω
Z
Ω
k=1
4u(x, η(t))dx,
(2.4)
(x, t) ∈ G.
From Green’s formula and boundary condition (1.2), it follows that Z Z Z ∂u(x, t) ds = − ψ(x, t)u(x, t)ds < 0, 4u(x, t)dx = ∂γ ∂Ω Ω ∂Ω and
Z
Ω
4u(x, η(t))dx =
Z
∂Ω
∂u(x, η(t)) ds = − ∂γ
where ds is surface element on ∂Ω. Set
Z
1 α
then v(m(t)) =
Z
uα (x, t)dx,
(2.7)
Ω
v 00 (t) + Q(t)v(m(t)) < 0,
where Q(t) =
(2.6)
uα (x, m(t))dx and v(t) > 0 for big enough t > 0. In view of (2.5)-(2.7),
Ω
(2.4)yields
1 α
ψ(x, η(t))u(x, η(t))ds < 0,
∂Ω
na lP repr oo f
v(t) =
Z
(2.5)
αq(m(t)) r(t)
(2.8)
> 0.
Case (2): We consider (1.1) and Dirichlet boundary condition (1.3). Lemma 1.1 is needed here. Now, multiplying (2.3) by the eigenfunction ϕ(x) > 0 and integrating with respect to x over the domain Ω, we have Z Z α−1 α−2 2 r(t) ϕ(x)(u utt + (α − 1)u ut )dx + q(m(t)) ϕ(x)uα (x, m(t))dx Ω
≤a(t)
Z
Ω
4u(x, t)ϕ(x)dx +
s X
ak (t)
Z
Ω
k=1
Ω
4u(x, η(t))ϕ(x)dx,
(2.9)
(x, t) ∈ G.
From Green’s formula and boundary condition (1.3), it follows that Z Z Z 4u(x, t)ϕ(x)dx = u(x, t)4ϕ(x)dx = −η0 u(x, t)ϕ(x)dx < 0, Ω
and
Ω
4u(x, η(t))ϕ(x)dx =
ur
Z
Ω
Ω
u(x, η(t))4ϕ(x)dx = −η0 1 v(t) = α
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Set
Z
(2.10)
Ω
Z
Z
u(x, η(t))ϕ(x)dx < 0.
(2.11)
Ω
ϕ(x)uα (x, t)dx,
(2.12)
Ω
Z 1 ϕ(x)uα (x, m(t))dx. In view of (2.10)-(2.12), (2.9) yields (2.8) holds. In α Ω conclusion, for (1.1) and two boundary conditions (1.2) and (1.3), the inequality (2.8) holds. then v(m(t)) =
By (2.8), we get v 00 (t) < −Q(t)v(m(t)) < 0, thus v 0 (t) is decreasing for all t ≥ t1 , which implies that v 0 (t) does not change sign eventually, then there exists a T ≥ t1 such that either v 0 (t) < 0 or v 0 (t) > 0 for any t ≥ T . If we assume that v 0 (t) < 0, then v(t) = v(T ) +
Z
t
T
v 0 (s)ds ≤ v(T ) + v 0 (T ) 3
Z
t
T
ds, for any t > T,
Journal Pre-proof
imply v(t) → −∞ as t → ∞, which is a contradiction to the positivity of v(t). Therefore, v 0 (t) > 0, v 00 (t) < 0, for any t ≥ T , and limt→∞ v 0 (t) exists.
0 If v(t) → ∞ as t → ∞, it follows from L’Hospital rule that limt→∞ v(t) t = limt→∞ v (t). We 0 0 claim that (2.1) implies limt→∞ v (t) = 0. If we admit that limt→∞ v (t) = l > 0, then integrate (2.8) from T ≥ t1 ≥ t0 to ∞, by m(t) ≥ t, yields Z Z ∞ Z ∞ l ∞ v(s) 0 sQ(s)ds. ds ≥ v (T ) > Q(s)v(m(s))ds ≥ sQ(s) s 2 T T T
This contradicts to (2.1). If v(t) < ∞ as t → ∞, we also claim that limt→∞ v 0 (t) = 0. If we admit that limt→∞ v 0 (t) = Rt Rt Rt L > 0, then v(t) ≥ T v 0 (s)ds ≥ v 0 (t) T ds ≥ L2 T ds, which is a contradiction to v(t) < ∞ as t → ∞. Then Z t v 0 (s)ds ≥ v(T ) − T v 0 (t) + tv 0 (t) ≥ tv 0 (t), for sufficiently large t. v(t) = v(T ) + T
And
v(t) >
Z tZ
∞
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v(t) 0 tv 0 (t) − v(t) ≤ 0, for sufficiently large t. = t t2 Next, integrating (2.8) twice, for t ≥ h ≥ T , we have Z tZ
t
Q(s)v(m(s))dsdh + T h T h Z t Z ∞ = (s − T )Q(s)v(m(s))ds + (t − T ) Q(s)v(m(s))ds. T
Q(s)v(m(s))dsdh =
Z tZ T
∞
(2.13)
Q(s)v(m(s))dsdh
t
(2.14)
t
From (2.14), (2.13) and m(t) ≥ t, we obtain v(m(t)) v(m(t)) > m(t)
Z
t
T
(s − T )Q(s)m(s)ds + (t − T )v(m(t))
Z
∞
Q(s)ds,
t
Rt R∞ then 1 > m−1 (t) T (s − T )Q(s)m(s)ds + (t − T ) t Q(s)ds. This contradicts (2.2) and we complete the proof of Theorem 2.1. Theorem 2.2. Let (H1)-(H3) hold. Assume that there exists a funtion b(t) ∈ C 1 ((t0 , ∞), R+ ) such that Z t (b0 (s))2 (2.15) lim sup b(s)Q(s) − ds = ∞. 4b(s) t→∞ T Then every smooth solution u(x, t) of (1.1), (1.2) or (1.3) is oscillatory in G.
ur
Proof. Proceeding as in the proof of Theorem 2.1, we have v 0 (t) > 0 holds, for all t ≥ T . And the corresponding function v(t) satisfies (2.8). Define the following generalized Riccati 0 (t) transformation: w(t) = b(t) vv(t) , then w(t) > 0. By (2.8), v 0 (t) > 0 and m(t) ≥ t, we obtain b0 (t) v 00 (t) (v 0 (t))2 b0 (t) −Q(t)v(m(t)) (v 0 (t))2 w(t) + b(t) − b(t) 2 < w(t) + b(t) − b(t) 2 b(t) v(t) v (t) b(t) v(t) v (t) 2 0 0 0 b (t) 1 2 1 b (t) (b (t))2 ≤ w(t) − b(t)Q(t) − w (t) = −b(t)Q(t) − w(t) − + b(t) b(t) b(t) 2 4b(t) 0 2 (b (t)) ≤ −b(t)Q(t) + . 4b(t) (2.16) Rt (b0 (s))2 Integration (2.16) from T ≥ t1 ≥ t0 to t, yields w(T ) > T b(s)Q(s) − 4b(s) ds, which contra-
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w0 (t) =
dicts (2.15). This completes the proof.
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Journal Pre-proof
Theorem 2.3. Let (H1)-(H3) hold. Assume that there exist a constant β and a function τ (t) ∈ C((t0 , ∞), R+ ), such that τ (t) ≤ t, τ 0 (t) ≥ 0, τ 00 (t) ≤ 0 and the first order inequation v 0 (t) − Q1 (t)v(m(τ (t))) > 0
(2.17)
R t+β has no positive solution, where Q1 (t) = t Q0 (s)ds, Q0 (t) = min{Q(t), Q(τ (t))(τ 0 (t))2 }, and Q(t) as defined in Theorem 2.1. Then every smooth solution u(x, t) of (1.1), (1.2) or (1.3) is oscillatory in G. Proof. Proceeding as in the proof of Theorem 2.1, we have v 0 (t) > 0 holds, for all t ≥ T . And the corresponding function v(t) satisfies (2.8). Further, it follows from (2.8) that vτ τ (τ (t)) + Q(τ (t))v(m(τ (t))) < 0. Then we have v 00 (τ (t)) = vτ τ (τ (t))(τ 0 (t))2 + vτ (τ (t))τ 00 (t) < −Q(τ (t))v(m(τ (t)))(τ 0 (t))2 + vτ (τ (t))τ 00 (t), therefore
v 00 (τ (t)) + Q(τ (t))v(m(τ (t)))(τ 0 (t))2 < vτ (τ (t))τ 00 (t) ≤ 0.
(2.18)
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Combining (2.8) and (2.18), we have v 00 (t)+Q(t)v(m(t))+v 00 (τ (t))+Q(τ (t))(τ 0 (t))2 v(m(τ (t))) < 0. On the other hand, Q0 (t) = min{Q(t), Q(τ (t))(τ 0 (t))2 }, τ (t) ≤ t, m(t) is increasing and v 0 (t) > 0 provides v 00 (t) + v 00 (τ (t)) < −2Q0 (t)v(m(τ (t))). (2.19) R t+β Integrating (2.19) from t to t+β, β is a constant, we get v 0 (t)+v 0 (τ (t)) > 2 t Q0 (s)v(m(τ (s)))ds. On the other hand, since v 0 (t) is decreasing and τ (t) ≤ t, then v 0 (τ (t)) >
Z
t+β
(2.20)
Q0 (s)v(m(τ (s)))ds.
t
Integrating (2.20) from T to t, and using that v(t) and m(t) is increasing, yields v(τ (t)) >
Z tZ T
t+β
h
Q0 (s)v(m(τ (s)))dsdh ≥
Z
t
v(m(τ (h)))
T
Z
t+β
Q0 (s)dsdh.
(2.21)
h
Rt Rt By (2.21) we have v(τ (t)) > T Q1 (h)v(m(τ (h)))dh. Let V (t) := T Q1 (h)v(m(τ (h)))dh. Then V (t) > 0. Noting that v(t) ≥ v(τ (t)) > V (t), one can see that V 0 (t) = Q1 (t)v(m(τ (t))) > Q1 (t)V (m(τ (t))). Thus V (t) is a positive solution of (2.17). This contradicts our assumptions.
3
Examples
Example 3.1 As an illustrative example, we consider the following equation
ur
tu6 utt + 6(x2 + 1)tu5 u2t + t2 u7 (x, 2t) = 4u(x, t) + with the boundary conditions: 1 2,
∂u(x,t) ∂γ
s X
(2 + sin kt)4u(x, t − 3),
(3.1)
k=1
+ tu(x, t) = 0, (x, t) ∈ ∂Ω × R+ .
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Here α = 7, t0 = r(t) = t, q(t) = t, p(x, t) = 6(x2 + 1)t, m(t) = 2t, T = 1, a(t) = 1, ak (t) = 2 + sin kt, η(t) = t − 3, f (u, x, t) = t2 u7 (x, t), ψ(x, t) = t. And it is easy to verify Rt R∞ R∞ R∞ 1 28s(s − 1)ds + (t − 1) t 14ds > 1. It is easy to Q(t) = 14, t0 tQ(t)dt = 1 14tdt = ∞, and 2t 1 2 check that all hypotheses of Theorem 2.1 are satisfied, so we get the equation (3.1) is oscillatory. Example 3.2. Consider the equation u2 utt + (t + 3)uu2t + u3 (x, t + 1) = t2 4u(x, t) + with the boundary conditions: u(x, t) = 0, (x, t) ∈ ∂Ω × R+ . 5
s X
k=1
kt4u(x, t3 ),
(3.2)
Journal Pre-proof
Here α = 3, t0 = 1, r(t) = 1, q(t) = 1, p(x, t) = t + 3 ≥ 2, m(t) = t + 1, T = 2, a(t) = t2 , Rt ak (t) = kt, η(t) = t3 , f (u, x, t) = u3 (x, t). By taking b(t) = 1, we get lim supt→∞ T b(s)Q(s) − Rt (b0 (s))2 4b(s) ds = lim supt→∞ 2 3ds = ∞. It is easy to check that all hypotheses of Theorem 2.2 are satisfied, so we get the equation (3.2) is oscillatory. Example 3.3. Consider the following equation 4
1
7
2tu 3 utt + (x2 + 6t)u 3 u2t + 2tu 3 (x, t2 ) = 4u(x, t) +
s X k
k=1
t
1 4u(x, ), t
(x, t) ∈ G,
(3.3)
References
na lP repr oo f
with the boundary conditions: u(x, t) = 0, (x, t) ∈ ∂Ω × R+ . Here α = 37 , r(t) = 2t, q(t) = t, p(x, t) = x2 + 6t, m(t) = t2 , a(t) = 1, ak (t) = kt , 7 7 η(t) = 1t , f (u, x, t) = 2tu 3 (x, t), τ (t) = 12 t. Let β = 1, we obtain Q(t) = 67 t, Q0 (t) = 48 t, R t+1 7 7 7 and Q1 (t) = t 48 sds = 48 t + 96 . When t → ∞, proceeding as in the proof of Theorem 2.1, 2 7 7 we have limt→∞ v 0 (t) = 0, so we obtain equation v 0 (t) − ( 48 t + 96 )v( t4 ) > 0 has no a positive solution as t → ∞. It is easy to check that all hypotheses of Theorem 2.3 are satisfied, so we get the equation (3.3) is oscillatory.
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[9] P. Prakash, S. Harikrishnan, M. Benchohra, Oscillation of certain nonlinear fractional partial differential equation with damping term, Applied Mathematics Letters, 43 (2015) 72–79.
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[10] Q. Yang, On the oscillation of certain nonlinear neutral partial differential equations, Applied Mathematics Letters, 20 (2007) 900–907. [11] X. Li, T. Caraballo, R.Rakkiyappan, X.Han, On the stability of impulsive functional differential equations with infinite delays, Mathematical Methods in the Applied Sciences, 38 (2015) 3130-3140. [12] H. Li, Y. Zheng, F.Alsaadi, Algebraic formulation and topological structure of Boolean networks with state-dependent delay, Journal of computational and applied mathematics, 350 (2019) 87-97. [13] L. Elsgolts, S. Norkin, Introduction to the theory and application of differential equations with deviating arguments, Elsevier, 1973.
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