American Institute of Mathematical Sciences

Advanced
February  2021, 20(2): 737-754. doi: 10.3934/cpaa.2020287

Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity

Zheng Han 1,, and  Daoyuan Fang 2,

1. 

Department of Mathematics, Hangzhou Normal University, Hangzhou, 311121, China
2. 

Department of Mathematics, Zhejiang University, Hangzhou, 310027, China

* Corresponding author

Received  May 2020 Revised  October 2020 Published  December 2020

Fund Project: The first author is supported by NSFC grant 11671353, 11401153, Zhejiang Provincial Natural Science Foundation of China under Grant No. LY18A010025. The second author is supported by NSFC grant 11671353

We prove an almost global existence result for the Klein-Gordon equation with the Kirchhoff-type nonlinearity on $ \mathbb{T}^d $ with Cauchy data of small amplitude $ \epsilon $. We show a lower bound $ \epsilon^{-2N-2} $ for the existence time with any natural number $ N $. The proof relies on the method of normal forms and induction. The structure of the nonlinearity is good enough that proceeds normal forms up to any order.

Keywords: Almost global existence, Klein-Gordon equation, Kirchhoff-type nonlinearity, normal forms.
Mathematics Subject Classification: Primary:35L70, 35L15;Secondary:58J45, 70K45.
Citation: Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287
References:
[1]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, T. Am. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2.  Google Scholar

[2]

S. N. Bernstein, Sur une classe d'$\acute{e}$quations fonctionnelles aux d$\acute{e}$riv$\acute{e}$es partielles, Izv. Akad. Nauk SSSR Ser. Mat., 4 (1940), 17-26.   Google Scholar

[3]

P. Baldi and E. Haus, On the existence time for the Kirchhoff equation with periodic boundary conditions, Nonlinearity, 33 (2020), 196-223.  doi: 10.1088/1361-6544/ab4c7b.  Google Scholar

[4]

R. W. Dickey, Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc., 23 (1969), 459-468.  doi: 10.1090/S0002-9939-1969-0247189-8.  Google Scholar

[5]

J. M. Delort, On long time existence for small solutions of semi-linear Klein-Gordon equaitons on the torus, J. Anal. Math., 107 (2009), 161-194.  doi: 10.1007/s11854-009-0007-2.  Google Scholar

[6]

J. M. Delort and J. Szeftel, Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Am. J. Math., 128 (2006), 1187-1218.  doi: 10.1353/ajm.2006.0038.  Google Scholar

[7]

D. Y. Fang, Z. Han and Q. D. Zhang, Almost global existence for the semi-linear Klein-Gordon equation on the circle, J. Differ. Equ., 262 (2017), 4610-4634.  doi: 10.1016/j.jde.2016.12.013.  Google Scholar

[8]

G. Kirchhoff, Vorlesungen $\ddot{u}$ber mathematische Physik: Mechanik, ch. 29, Teubner, Leipzig, 1876. Google Scholar

[9]

L. A. Medeiros and M. M. Miranda, Solutions for the equation of nonlinear vibrations in Sobolev spaces of fractionary order, Mat. Apl. Comput., 6 (1987), 257-276.   Google Scholar

[10]

T. Matsuyama and M. Ruzhansky, Global well-posedness of Kirchhoff system, J. Math. Pures Appl., 100 (2013), 220-240.  doi: 10.1016/j.matpur.2012.12.002.  Google Scholar

[11]

S. Spagnolo, The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano, 62 (1992), 17-51.  doi: 10.1007/bf02925435.  Google Scholar

[12]

T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three, Math. methods Appl. Sci., 27 (2004), 1893-1916.  doi: 10.1002/mma.530.  Google Scholar

show all references

References:
[1]

A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, T. Am. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2.  Google Scholar

[2]

S. N. Bernstein, Sur une classe d'$\acute{e}$quations fonctionnelles aux d$\acute{e}$riv$\acute{e}$es partielles, Izv. Akad. Nauk SSSR Ser. Mat., 4 (1940), 17-26.   Google Scholar

[3]

P. Baldi and E. Haus, On the existence time for the Kirchhoff equation with periodic boundary conditions, Nonlinearity, 33 (2020), 196-223.  doi: 10.1088/1361-6544/ab4c7b.  Google Scholar

[4]

R. W. Dickey, Infinite systems of nonlinear oscillation equations related to the string, Proc. Amer. Math. Soc., 23 (1969), 459-468.  doi: 10.1090/S0002-9939-1969-0247189-8.  Google Scholar

[5]

J. M. Delort, On long time existence for small solutions of semi-linear Klein-Gordon equaitons on the torus, J. Anal. Math., 107 (2009), 161-194.  doi: 10.1007/s11854-009-0007-2.  Google Scholar

[6]

J. M. Delort and J. Szeftel, Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Am. J. Math., 128 (2006), 1187-1218.  doi: 10.1353/ajm.2006.0038.  Google Scholar

[7]

D. Y. Fang, Z. Han and Q. D. Zhang, Almost global existence for the semi-linear Klein-Gordon equation on the circle, J. Differ. Equ., 262 (2017), 4610-4634.  doi: 10.1016/j.jde.2016.12.013.  Google Scholar

[8]

G. Kirchhoff, Vorlesungen $\ddot{u}$ber mathematische Physik: Mechanik, ch. 29, Teubner, Leipzig, 1876. Google Scholar

[9]

L. A. Medeiros and M. M. Miranda, Solutions for the equation of nonlinear vibrations in Sobolev spaces of fractionary order, Mat. Apl. Comput., 6 (1987), 257-276.   Google Scholar

[10]

T. Matsuyama and M. Ruzhansky, Global well-posedness of Kirchhoff system, J. Math. Pures Appl., 100 (2013), 220-240.  doi: 10.1016/j.matpur.2012.12.002.  Google Scholar

[11]

S. Spagnolo, The Cauchy problem for Kirchhoff equations, Rend. Sem. Mat. Fis. Milano, 62 (1992), 17-51.  doi: 10.1007/bf02925435.  Google Scholar

[12]

T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three, Math. methods Appl. Sci., 27 (2004), 1893-1916.  doi: 10.1002/mma.530.  Google Scholar

[1]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[2]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[3]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912
[4]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[5]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136
[6]

Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1
[7]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[8]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094
[9]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[10]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973
[11]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005
[12]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[13]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[14]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[15]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[16]

Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617
[17]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206
[18]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[19]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213
[20]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (49)
  • HTML views (67)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences