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

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

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.

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. FangZ. 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. FangZ. 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]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[3]

Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021063

[4]

Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115

[5]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021031

[6]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056

[7]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001

[8]

Harumi Hattori, Aesha Lagha. Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021071

[9]

Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270

[10]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[11]

Thomas Kappeler, Yannick Widmer. On nomalized differentials on spectral curves associated with the sinh-Gordon equation. Journal of Geometric Mechanics, 2021, 13 (1) : 73-143. doi: 10.3934/jgm.2020023

[12]

Huan Zhang, Jun Zhou. Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021034

[13]

Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019

[14]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060

[15]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006

[16]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237

[17]

Miroslav Bulíček, Victoria Patel, Endre Süli, Yasemin Şengül. Existence of large-data global weak solutions to a model of a strain-limiting viscoelastic body. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021053

[18]

Rong Rong, Yi Peng. KdV-type equation limit for ion dynamics system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021037

[19]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079

[20]

Tadahiro Oh, Yuzhao Wang. On global well-posedness of the modified KdV equation in modulation spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2971-2992. doi: 10.3934/dcds.2020393

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (80)
  • HTML views (85)
  • Cited by (0)

Other articles
by authors

[Back to Top]