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

[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

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

[1]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011

[2]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032

[3]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[4]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194

[5]

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

[6]

Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, , () : -. doi: 10.3934/era.2021024

[7]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, 2021, 14 (2) : 199-209. doi: 10.3934/krm.2021002

[8]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[9]

Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018

[10]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376

[11]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008

[12]

Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3579. doi: 10.3934/dcdsb.2020246

[13]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[14]

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

[15]

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

[16]

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

[17]

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

[18]

Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012

[19]

Rafael López, Óscar Perdomo. Constant-speed ramps for a central force field. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3447-3464. doi: 10.3934/dcds.2021003

[20]

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

 Impact Factor: 0.263

Article outline

[Back to Top]