March  2020, 28(1): 15-25. doi: 10.3934/era.2020002

Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion

1. 

Department of Mathematics, Daegu University, Gyeongsan, Gyeongbuk 38453, Republic of Korea

2. 

Department of Mathematics, Incheon National University, Incheon 22012, Republic of Korea

* Corresponding author: Byungsoo Moon

Received  September 2019 Revised  October 2019 Published  March 2020

Fund Project: The first author is supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03031180). The second author is supported by Incheon National University Research Grant in 2017.

In this paper, we consider the dissipative Degasperis-Procesi equation with arbitrary dispersion coefficient and compactly supported initial data. We establish the simple condition on the initial data which lead to guarantee that the solution exists globally. We also investigate the propagation speed for the equation under the initial data is compactly supported.

Citation: Guenbo Hwang, Byungsoo Moon. Global existence and propagation speed for a Degasperis-Procesi equation with both dissipation and dispersion. Electronic Research Archive, 2020, 28 (1) : 15-25. doi: 10.3934/era.2020002
References:
[1]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[2]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[3]

A. DegasperisD. Holm and A. Hone, A new integrable equation with peakon solution, Theoret. and Math. Phys, 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422.  Google Scholar

[4]

H. R. DullinG. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg -de Veris-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res., 33 (2003), 73-95.  doi: 10.1016/S0169-5983(03)00046-7.  Google Scholar

[5]

Z. Guo, Some properties of solutions to the weakly dissipative Degasperis-Procesi equation, J. Differential Equations, 246 (2009), 4332-4344.  doi: 10.1016/j.jde.2009.01.032.  Google Scholar

[6]

Z. GuoS. Lai and Y. Wang, Global weak solutions to the weakly dissipative Degasperis-Procesi equation, Nonlinear Anal., 74 (2011), 4961-4973.  doi: 10.1016/j.na.2011.04.051.  Google Scholar

[7]

D. Henry, Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111.  doi: 10.1142/S0219891608001404.  Google Scholar

[8]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606.  doi: 10.3934/dcdsb.2009.12.597.  Google Scholar

[9]

A. HimonasG. MisiolekG. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa–Holm equation, Comm. Math. Phys., 271 (2007), 511-522.  doi: 10.1007/s00220-006-0172-4.  Google Scholar

[10]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.  Google Scholar

[11]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448(1975), 25–70.  Google Scholar

[12]

J. Lenells and M. Wunsch, On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations, J. Differential Equations, 255 (2013), 441-448.  doi: 10.1016/j.jde.2013.04.015.  Google Scholar

[13]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

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Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[15]

Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171.  Google Scholar

[16]

O. G. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14.  doi: 10.2991/jnmp.2005.12.1.2.  Google Scholar

[17]

E. Novruzova and A. Hagverdiyevb, On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient, J. Differential Equations, 257 (2014), 4525-4541.  doi: 10.1016/j.jde.2014.08.016.  Google Scholar

[18]

S. WuJ. Escher and Z. Yin, Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 633-645.  doi: 10.3934/dcdsb.2009.12.633.  Google Scholar

[19]

S. Wu and Z. Yin, Blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi equation, SIAM J. Math. Anal., 40 (2008), 475-490.  doi: 10.1137/07070855X.  Google Scholar

[20]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[21]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.  doi: 10.1215/ijm/1258138186.  Google Scholar

[22]

R. Zheng and Z. Yin, Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system, Discrete Contin. Dyn. Syst., 38 (2018), 329-341.  doi: 10.3934/dcds.2018016.  Google Scholar

show all references

References:
[1]

R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar

[2]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rational Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.  Google Scholar

[3]

A. DegasperisD. Holm and A. Hone, A new integrable equation with peakon solution, Theoret. and Math. Phys, 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422.  Google Scholar

[4]

H. R. DullinG. A. Gottwald and D. D. Holm, Camassa-Holm, Korteweg -de Veris-5 and other asymptotically equivalent equations for shallow water waves, Fluid Dynam. Res., 33 (2003), 73-95.  doi: 10.1016/S0169-5983(03)00046-7.  Google Scholar

[5]

Z. Guo, Some properties of solutions to the weakly dissipative Degasperis-Procesi equation, J. Differential Equations, 246 (2009), 4332-4344.  doi: 10.1016/j.jde.2009.01.032.  Google Scholar

[6]

Z. GuoS. Lai and Y. Wang, Global weak solutions to the weakly dissipative Degasperis-Procesi equation, Nonlinear Anal., 74 (2011), 4961-4973.  doi: 10.1016/j.na.2011.04.051.  Google Scholar

[7]

D. Henry, Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111.  doi: 10.1142/S0219891608001404.  Google Scholar

[8]

D. Henry, Infinite propagation speed for a two component Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 597-606.  doi: 10.3934/dcdsb.2009.12.597.  Google Scholar

[9]

A. HimonasG. MisiolekG. Ponce and Y. Zhou, Persistence properties and unique continuation of solutions of the Camassa–Holm equation, Comm. Math. Phys., 271 (2007), 511-522.  doi: 10.1007/s00220-006-0172-4.  Google Scholar

[10]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  doi: 10.1017/S0022112001007224.  Google Scholar

[11]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Lecture Notes in Math., Springer Verlag, Berlin, 448(1975), 25–70.  Google Scholar

[12]

J. Lenells and M. Wunsch, On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations, J. Differential Equations, 255 (2013), 441-448.  doi: 10.1016/j.jde.2013.04.015.  Google Scholar

[13]

W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[14]

Y. Liu and Z. Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys., 267 (2006), 801-820.  doi: 10.1007/s00220-006-0082-5.  Google Scholar

[15]

Y. Liu and R. Xu, Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.  doi: 10.3934/dcdsb.2007.7.171.  Google Scholar

[16]

O. G. Mustafa, A note on the Degasperis-Procesi equation, J. Nonlinear Math. Phys., 12 (2005), 10-14.  doi: 10.2991/jnmp.2005.12.1.2.  Google Scholar

[17]

E. Novruzova and A. Hagverdiyevb, On the behavior of the solution of the dissipative Camassa-Holm equation with the arbitrary dispersion coefficient, J. Differential Equations, 257 (2014), 4525-4541.  doi: 10.1016/j.jde.2014.08.016.  Google Scholar

[18]

S. WuJ. Escher and Z. Yin, Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 633-645.  doi: 10.3934/dcdsb.2009.12.633.  Google Scholar

[19]

S. Wu and Z. Yin, Blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi equation, SIAM J. Math. Anal., 40 (2008), 475-490.  doi: 10.1137/07070855X.  Google Scholar

[20]

R. XuM. ZhangS. ChenY. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

[21]

Z. Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math., 47 (2003), 649-666.  doi: 10.1215/ijm/1258138186.  Google Scholar

[22]

R. Zheng and Z. Yin, Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system, Discrete Contin. Dyn. Syst., 38 (2018), 329-341.  doi: 10.3934/dcds.2018016.  Google Scholar

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