Article
Kyungpook Mathematical Journal 2021; 61(1): 1-10
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
Blow-up of Solutions for Higher-order Nonlinear Kirchhofftype Equation with Degenerate Damping and Source
Yong Han Kang∗, Jong-Yeoul Park
Francisco College, Daegu Catholic University, Gyeongsan 712-702, Republic of Korea
e-mail : yonghann@cu.ac.kr
Department of Mathematics, Pusan National University, Busan 609-735, Republic of Korea
e-mail : jyepark@pusan.ac.kr
Received: November 1, 2020; Revised: November 29, 2020; Accepted: December 14, 2020
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 appropriate Lyapunov inequality, we prove the finite time blow-up of solution for equation (1.1) as a suitable conditions and the initial data satisfying
Keywords: Kirchoff-type equation, blow up, higher-order nonlinear, degenerate damping and source
1. Introduction
In this paper, we consider the higher-order nonlinear Kirchhoff-type equation with degenerate damping and source:
where
where
where Ω is a bounded domain of
where
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
(1) coercivity:
(2) strict monotonicity:
(3) continuity:
is single-valued and
Here
and
Under the assumptions
Our main results is the Theorem 2.2. Here
Let
In view of (2.6) and
By using (2.2), (2.6) and (2.7), we have
and exploiting (2.1) and (2.3) we obtain
Thus using (2.8) and (2.9), we get
As in [11], we construct a Lyapunov's function
where
and ε being a positive constant to be determined later. By taking a derivative of
On exploiting (2.2) and (2.4), estimate (2.13) takes the form
where
since
From (2.7),(2.15) and Young's inequality it yields
where
Putting
in (2.17), then
Thus, we get
In view of (2.10), we deduce
Since
where
Now, we choose
In the sequel, we may adjust ε again. From (2.21), it follows
where
for
If
In both cases, we have
where
Thus (2.24) is valid for all
Using Höolder's inequality, we obtain the following estimate
which implies
Noting
From (2.12), it easy to see that
Thanks to
where
Then, from (2.26) and (2.27), we infer that
Combing (2.25) and (2.28), we get
Therefore (2.24) follows from (2.23)and (2.29).
From (2.24), we obtain
From (2.30), we deduce that
We obtain from(2.8) and (2.29)
Thus the proof of Theorem 2.2. is complete.
- V. Barbu, I. Lasiecka, and M. A. Rammaha. On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc. 357(7)(2005), 2571-2611.
- 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.
- 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.
- 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.
- 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.
- 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. (2015)
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 pp.
- 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.
- 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), 865-880.