pISSN 1225-6951 eISSN 0454-8124 c
⃝ Kyungpook Mathematical Journal
Blow-up of Solutions for Higher-order Nonlinear
Kirchhoff-type Equation with Degenerate Damping and Source
Yong Han Kang∗
Francisco College, Daegu Catholic University, Gyeongsan 712-702, Republic of Ko-rea
e-mail : [email protected] Jong-Yeoul Park
Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
e-mail : [email protected]
Abstract. This paper is concerned the finite time blow-up of solution for higher-order nonlinear Kirchhoff-type equation with a degenerate term and a source term. By an appro-priate Lyapunov inequality, we prove the finite time blow-up of solution for equation (1.1) as a suitable conditions and the initial data satisfying ||Dmu0|| > B−(p+2)/(p−2q), E(0) <
E1.
1. Introduction
In this paper, we consider the higher-order nonlinear Kirchhoff-type equation with degenerate damping and source:
utt+
(∫ Ω
|Dmu|2dx )q
(−△)mu +|u|k∂j(ut) =|u|pu, x∈ Ω, t > 0 u(x, t) = 0, ∂
iu
∂νi = 0, i = 1, 2,· · ·, m − 1, x ∈ ∂Ω, t > 0
(1.1)
u(x, 0) = u0(x), ut(x, 0) = u1(x), x∈ Ω,
where m > 1 is an integer constant, p, q, k > 0, Ω is a bounded domain of Rn, with
a smooth boundary ∂Ω and a unit outer normal ν. j(·) : R → R is a given function
* Corresponding Author.
Received November 1, 2020; revised November 29, 2020; accepted December 14, 2020. 2010 Mathematics Subject Classification: 35B40, 35B44, 35L72.
Key words and phrases: Kirchoff-type equation, blow up, higher-order nonlinear, degen-erate damping and source.
The first author was supported by research grants from the Daegu Catholic University in 2018 (Number 20181049).
to be specified later and we use ∂j to denote its sub-differential for example(see [1, 4, 9]). Moreover, in [14], Messaodui and Houari consider the higher-order nonlinear Kirchhoff-type hyperbolic equation
utt+ (∫ Ω |Dmu|2dx )q (−△)mu +|ut|rut=|u|pu, x∈ Ω, t > 0 u(x, t) = 0, ∂ iu ∂νi = 0, i = 1, 2,· · ·, m − 1, x ∈ ∂Ω, t > 0 u(x, 0) = u0(x), ut(x, 0) = u1(x), x∈ Ω,
where m≥ 1, p, q, r ≥ 0, Ω is a bounded domain of Rn, with a smooth boundary ∂Ω and a unit outer normal ν. They established a blow-up result for certain solutions with positive initial energy. In [4], Han and Wang investigated the global existence and blow-up of solutions for nonlinear viscoelastic wave equation with degenerate damping and source term
utt− ∆u +
∫ t 0
g(t− s)∆u(s)ds + |u|k∂j(ut) =|u|p−1u, (x, t)∈ Ω × (0, T ), u(x, t) = 0, (x, t)∈ ∂Ω × (0, T )
u(x, 0) = u0(x), ut(x, 0) = u1(x), x∈ Ω,
where Ω is a bounded domain of Rn, with a smooth boundary ∂Ω in Rn, k, p≥ 0,
and g(·) : R+ → R+, j(·) : R → R are given functions to be specified. Here, we use ∂j to denote its sub-differential(see [1, 4, 9]). Furthermore, in [9], Kim et al. consider a stochastic quasilinear viscoelastic wave equation with degenerate damping and sources
|ut|ρutt(t)− ∆u(t) − ∆utt(t) +
∫ t 0
g(t− s)∆u(s)ds
+|u(t)|k∂j(ut(t)) =|u(t)|p−1u(t) + εσ(x, t)∂tW (x, t), (x, t)∈ D × [0, T ), u(x, t) = 0, (x, t)∈ ∂D × [0, T )
u(x, 0) = u0(x), ut(x, 0) = u1(x), x∈ ¯D,
where ¯D is a bounded domain in Rn(n≥ 1) with a smooth boundary ∂D, k, p ≥ 0 and g(·) : R+ → R+, j(·) : R → R is given functions to be specified. Here, we use ∂j to denote its sub-differential, W (x, t) is an infinite dimensional Winner process, and σ(x, t) is L2(D) valued progressively measurable. Authors proved the blow-up of solution for stochastic/no stochastic quasilinear viscoelastic wave equation with positive probability or explosive in energy sense [2, 3, 5, 6, 7, 8, 10, 11, 13, 15, 16, 17]. Motivated by the previous works, we studied the blow-up of solutions for higher-order nonlinear Kirchhoff-type equation with degenerate damping and source. To the best of our knowledge. there are no results of a higher-order nonlinear Kirchhoff-type equation with degenerate damping and source. The main result was proved in
section 2.
2. Blow-up Result
For j(·) : R → R, similarly to [1, 4, 9], we have the following assumptions :
(H1) Let j(·) : R → R be continuous, convex real valued function, there exist
constants c > 0, c0> 0, c1> 0, c2≥ 0 such that for all s, v ∈ R, j(·) satisfies (1) coercivity: j(s)≥ c|s|r+1,
(2) strict monotonicity: (∂j(s)− ∂j(v))(s − v) ≥ c1|s − v|r+1,
(3) continuity: ∂j(x) is single-valued and|∂j(s)| ≤ c0|s|r+ c 2.
(H2) p≤ max{p2∗,p∗r+1r+k}; p∗=n2n−2.
(H3) k, r≥ 0, p > 0. In addition, k ≤ nn−2, p + 1 <n2n−2, if n≥ 3.
Here||·|| = ||·||2. Let B be the best constant of the embedding inequality||u||p+2≤ B||Dmu||. We set α1= B−(p+2)/(p−2q), E1= ( 1 2(q + 1)− 1 p + 2 ) α2(q+1)1 , (2.1) and E(t) =1 2||ut|| 2+ 1 2(q + 1)||D mu||2(q+1)− 1 p + 2||u|| p+2 p+2. (2.2)
Lemma 2.1.([14]) Let u(t) be solution of Eq.(1.1). Assume that E(0) < E1 and ||Dmu0|| > α1. Then there exists a constant α
2> α1 such that ||Dmu(·, t)|| ≥ α2, ∀t ≥ 0,
(2.3)
||u||p+2≥ Bα2, ∀t ≥ 0.
(2.4)
Under the assumptions (H1)-(H3), we get the two theorems.
Theorem 2.1. Suppose that p≤ k + r − 1 and
0 < p < 2
n− 2m, if n > 2m, and p > 0, if n≤ 2m. Then for any initial data (u0, u1) ∈ H2m(Ω)∩ Hm
0 (Ω)× H0m(Ω), the solution of Eq.(1.1) is global.
Proof. By using Theorem 1.1 of [12] and Theorem 2.3 of [4], we derive to the
solution. 2
Theorem 2.2. Suppose that p > max{k + r − 1, 2q} and
0 < p < 2
n− 2m, if n > 2m, and p > 0, if n≤ 2m. Then for any initial data (u0, u1) ∈ H2m(Ω)∩ Hm
0 (Ω)× H0m(Ω), any solution of Eq.(1.1) with initial data satisfying
||Dmu0|| > B−(p+2)/(p−2q), E(0) < E 1, (2.5)
blows up in finite time T .
Proof. A multiplication of Eq.(1.1) by ut and integration over Ω give
(2.6) E′(t) =− ∫ Ω |u|k∂j(u t)utdx≤ 0. Let G(t) = E1− E(t). In view of (2.6) and (H1), we easily see that (2.7) G′(t) = ∫ Ω |u|k∂j(u t)utdx≥ c1 ∫ Ω |u|k|u t|r+1dx≥ 0.
By using (2.2), (2.6) and (2.7), we have 0 < G(0)≤ G(t) = E1−1 2||ut|| 2− 1 2(q + 1)||D mu||2(q+1)+ 1 p + 2||u|| p+2 p+2 (2.8)
and exploiting (2.1) and (2.3) we obtain
E1− 1 2(q + 1)||D m u||2(q+1)< E1− 1 2(q + 1)α 2(q+1) 1 =− 1 p + 2α 2(q+1) 1 < 0, ∀t ≥ 0. (2.9)
Thus using (2.8) and (2.9), we get
0 < G(0)≤ G(t) ≤ 1 p + 2||u||
p+2
p+2, ∀t ≥ 0.
(2.10)
As in [11], we construct a Lyapunov’s function L(t) = G1−α(t) + ε ∫ Ω uutdx, (2.11) where α = min { p− r − k + 1 r(p + 2) , p 2(p + 2) } , (2.12)
and ε being a positive constant to be determined later. By taking a derivative of L(t) and using equation of (1.1), we have
L′(t) = (1− α)G−α(t)G′(t) + ε||ut||2+ ε||u||p+2p+2+ ε||D mu||2(q+1) (2.13) −ε ∫ Ω |u|k∂j(u t)udx + 2ε(q + 1)G(t)− 2ε(q + 1)E1 +2ε(q + 1) ( 1 2||ut|| 2+ 1 2(q + 1)||D mu||2(q+1)− 1 p + 2||u|| p+2 p+2 ) . On exploiting (2.2) and (2.4), estimate (2.13) takes the form
L′(t)≥ (1 − α)G−α(t)G′(t) + ε||ut||2+ ε||u||p+2p+2+ ε||D mu||2(q+1) −ε ∫ Ω |u|k∂j(u
t)udx + 2ε(q + 1)G(t)− 2ε(q + 1)E1(Bα2)−(p+2)||u|| p+2 p+2 +ε(q + 1)||ut||2+ ε||Dmu||2(q+1)− 2ε(q + 1) p + 2 ||u|| p+2 p+2 = (1− α)H−α(t)G′(t) + ε(q + 2)||ut||2+ εc∗||u|| p+2 p+2+ 2ε||D mu||2(q+1) (2.14) +2ε(q + 1)H(t)− ε ∫ Ω |u|k ∂j(ut)udx, where c∗= 1−2(q + 1) p + 2 − 2(q + 1)E1(Bα2) −(p+2)> 0,
since α2> B−(p+2)/(p−2q). Thanks to p + 1 > k + r, then the continuity of j(·) in
(H1) and using H¨older’s inequality, we infer that | ∫ Ω |u|k∂j(u t)udx| ≤ c0 ∫ Ω |u|k+1− rk r+1|u| rk r+1|ut|rdx (2.15) ≤ c0 (∫ Ω |u|k|u t|r+1dx ) r r+1(∫ Ω |u|r+k+1dx ) 1 r+1 . From (2.7),(2.15) and Young’s inequality it yields
| ∫ Ω |u|k∂j(u t)udx| ≤ KG′(t) r r+1||u|| r+k+1 r+1 p+2 ≤ K(δ||u||r+k+1p+2 + δ−1rG′(t) ) , (2.16) where K = c0c− r r+1 1 |Ω| p−r−k+1
(p+1)(r+1) and δ is a positive constant to be determined later.
From (2.14) and (2.16), we obtain L′(t)≥ [ (1− α)G−α(t)− εKδ−1r ] G′(t) +ε(q + 2)||ut||2+ 2ε||Dmu||2(q+1) +2ε(q + 1)G(t)− εKδ||u||r+k+1p+2 + εc∗||u||p+2p+2. (2.17)
Putting δ = c∗ 2K||u|| p−r−k+1 p+2 > 0 in (2.17), then εKδ||u||r+k+1p+2 = εc∗ 2 ||u|| p+2 p+2. Thus, we get L′(t)≥ [ (1− α)G−α(t)− εKδ−1r ] G′(t) + ε(q + 2)||ut||2+ 2ε||Dmu||2(q+1)+ 2ε(q + 1)G(t) (2.18) + εc∗ 2||u|| p+2 p+2. In view of (2.10), we deduce (1− α)G−α(t)− εKδ−1r = G−α(t) [ 1− α − εKδ−1rGα(t) ] ≥ G−α(t)[1− α − εK1+1 r(c∗ 2 ) −1 r(||u||p−r−k+1 p+2 )− 1 r(p + 2)−α||u||α(p+2) p+2 ] = G−α(t) [ 1− α − εK1+1r(c∗ 2) −1 r(p + 2)−α||u|| r+k−p−1+αr(p+2) r p+2 ] . (2.19) Since||u||p+2 ≥ ((p + 2)G(0)) 1
p+2 > 0 and α was chosen such that r + k− p − 1 +
αr(p + 2)≤ 0, it follows from (2.19) that (1− α)G−α(t)− εKδ−1r ≥ G−α(t)[1− α − εK1+1 r(c∗ 2) −1 r(p + 2) r+k−p−1 r(p+2) G(0) r+k−p−1+αr(p+2) r(p+2) ] = G−α(t) [ 1− α − εK1+1rF ] > 0, (2.20) where F = (c∗ 2) −1 r(p + 2) r+k−p−1 r(p+2) G(0) r+k−p−1+αr(p+2) r(p+2) .
Now, we choose 0 < ε < (1− α)(F K1+1/r)−1 small enough such that 1− α − εK1+1rF > 0.
(2.21)
In the sequel, we may adjust ε again. From (2.21), it follows G(0)≥ cεθ,
(2.22)
where θ = p−r−k+1−αr(p+2)r(p+2) , c := [(c∗/2)1/r(p + 2)(r+k−p−1)/r(p+2)K1+1/r(1 − α)−1]θ. Here and in what follows we use c to denote a generic positive constant
which is independent of ε and initial data. Therefore, we conclude that from (2.18), (2.20) and (2.21) that L′(t)≥ εc [ ||Dmu||2(q+1)+||u t||2+ G(t) +||u||p+2p+2 ] , (2.23)
for t∈ [0, T ). Obviously, (2.23) indicates that L(t) is increasing on [0, T ) and L(t) = G1−α(t) + ε ∫ Ω uutdx ≥ G1−α(0) + ε ∫ Ω u0u1dx = L(0).
If∫Ωu0u1dx≥ 0, then no further restriction on ε is needed. However if∫Ωu0u1dx < 0, we further require ε satisfies
0 < ε < G 1−α(0) −2∫Ωu0u1dx
.
In both cases, we have L(t)≥ L(0) > 0, t ∈ [0, T ). Now we prove that L(t) satisfies the following differential inequality
L′(t)≥ cε1+σL1−α1 (t),
(2.24)
where c is a positive constant and σ = θ(1− 2
1−2α(p+2))≥ 0. To this end, we consider two cases.
Case 1. ∫Ωuutdx≤ 0 for some t ∈ [0, T ). Then we have for such t
L1−α1 (t) = ( G1−α(t) + ε ∫ Ω uutdx ) 1 1−α ≤ G(t). Thus (2.24) is valid for all t∈ [0, T ) for which∫Ωuutdx≤ 0.
Case 2. ∫Ωuutdx > 0 for some t∈ [0, T ). By (2.12), it easily follows 0 < α < 12.
Noting 0 < ε < 1, then by convexity we have L1−α1 (t)≤ 21−α1 −1 [ G(t) +|ε ∫ Ω uutdx| 1 1−α ] . (2.25)
Using H¨older’s inequality, we obtain the following estimate |
∫ Ω
uutdx| ≤ ||u||||ut|| ≤ c||u||p+2||ut||,
which implies | ∫ Ω uutdx| 1 1−α ≤ c||u|| 1 1−α p+2||ut|| 1 1−α.
Noting 1 < 1−α1 < 2, then by Young’s inequality we have | ∫ Ω uutdx| 1 1−α ≤ c ( ||u||1−2α2 p+2 +||ut||2 ) . (2.26)
From (2.12), it easy to see that 2
1−2α ≤ p + 2. Then it follows from (2.10) that ||u||1−2α2 p+2 =||u|| 2 1−2α−(p+2) p+2 ||u|| p+2 p+2≤ ((p + 2)G(0)) 2 (1−2α)(p+2)−1||u||p+2 p+2. Thanks to 2 (1−2α)(p+2) − 1 ≤ 0 and (2.22), it yields ||u||1−2α2 p+2 ≤ cε−σ||u|| p+2 p+2, (2.27) where σ = θ ( 1− 2 (1− 2α)(p + 2) ) = r(p + 2) p− r − k + 1 − αr(p + 2) ( 1− 2 (1− 2α)(p + 2) ) ≥ 0. Then, from (2.26) and (2.27), we infer that
| ∫ Ω uutdx| 1 1−α ≤ cε−σ ( ||ut||2+||u|| p+2 p+2 ) . (2.28)
Combing (2.25) and (2.28), we get L1−α1 (t)≤ cε−σ [ G(t) +||ut||2+||u|| p+2 p+2 ] ≤ cε−σ[||Dmu||2(q+1)+ G(t) +||u t||2+||u||p+2p+2 ] . (2.29)
Therefore (2.24) follows from (2.23)and (2.29). From (2.24), we obtain
L(t)≥ 1
L(0)−α/(1−α)−1αc−αε1+σt. (2.30)
From (2.30), we deduce that limt→T−L(t) = +∞, where
T =1− α αc ε
−1−σL(0)− α
1−α.
We obtain from(2.8) and (2.29) L1/(1−α)(t) ≤ C||u||p+2
p+2 for some constant C > 0
and also ||u||p+2 ≤ B||Dmu||. Hence we have limt→T−||u||p+2 = +∞ and
limt→T−||Dmu|| = +∞ i.e., the solution blow up at finite time T in Lp+2(Ω) and Hm(Ω) norm.
References
[1] V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with
degen-erate damping and source terms, Trans. Amer. Math. Soc., 357(7)(2005), 2571–2611.
[2] Y. Boukhatem and B. Benabderrahmane, Existence and decay of solutions for a
vis-coelastic wave equation with acoustic boundary conditions, Nonlinear Anal., 97(2014),
191–209.
[3] Y. Boukhatem and B. Benabderrahmane, Polynomial decay and blow up of solutions
for variable coefficients viscoelastic wave equation with acoustic boundary conditions,
Acta Math. Sin., 32(2)(2016), 153–174.
[4] X. Han and M. Wang, Global existence and blow-up of solutions for nonlinear
vis-coelastic wave equation with degenerate damping and source, Math. Nachr.,
284(5-6)(2011), 703–716.
[5] J. M. Jeong, J. Y. Park and Y. H. Kang, Energy decay rates for viscoelastic wave
equation with dynamic boundary conditions, J. Comput. Anal. Appl., 19(3)(2015),
500–517.
[6] J. M. Jeong, J. Y. Park and Y. H. Kang, Global nonexistence of solutions for a
quasilinear wave equation with acoustic boundary conditions, Bound. Value Probl.,
(2017), Paper No. 42, 10 pp.
[7] J. M. Jeong, J. Y. Park and Y. H. Kang, Global nonexistence of solutions for a
nonlinear wave equation with time delay and acoustic boundary conditions, Comput.
Math. Appl., 76(2018), 661–671.
[8] Y. H. Kang, J. Y. Park and D. Kim, A global nonexistence of solutions for a quasilinear
viscoelastic wave equation with acoustic boundary conditions, Bound. Value Probl.,
(2018), Paper No. 139, 19 pp.
[9] S. Kim, J. Y. Park and Y. H. Kang, Stochastic quasilinear viscoelastic wave equation
with degenerate damping and source terms, Comput. Math. Appl., 75(2018), 3987–
3994.
[10] S. Kim, J. Y. Park and Y. H. Kang, Stochastic quasilinear viscoelastic wave equation
with nonlinear damping and source terms, Bound. Value Probl., (2018), Paper No.
14, 15 pp.
[11] M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic
wave equation with a delay, Z. Angew. Math. Phys., 62(2011), 1065–1082.
[12] F. Li, Global existence and blow-up of solutions for a higher-order Kirchhoff-type
equation with nonlinear dissipation, Appl. Math. Lett, 17(2004), 1409–1414.
[13] W. Liu and M. Wang, Global nonexistence of solutions with positive initial energy for
a class of wave equations, Math. Methods Appl. Sci., 32(2009), 594–605.
[14] S. A. Messaoudi and B. S. Houari, A blow-up result for a higher-order nonlinear
Kirchhoff-type hyperbolic equation, Appl. Math. Lett., 20(2007), 866–871.
[15] E. Piskin, On the decay and blow up of solutions for a quasilinear hyperbolic equations
with nonlinear damping and source terms, Bound. Value Probl., (2015), 2015:127, 14
[16] F. Q. Sun and M. Wang, Global and blow-up solutions for a system of nonlinear
hyperbolic equations with dissipative terms, Nonlinear Anal., 64(2006), 739–761.
[17] S. T. Wu, Non-existence of global solutions for a class of wave equations with nonlinear
damping and source terms, Proc. Roy. Soc. Edinburgh Sect. A, 141(4)(2011), 865–
