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Traveling waves for a nonlocal dispersal SIR model equipped delay and generalized incidence
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 |
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.
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. |
[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. |
[3] |
A. Degasperis, D. Holm and A. Hone,
A new integrable equation with peakon solution, Theoret. and Math. Phys, 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[4] |
H. R. Dullin, G. 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. |
[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. |
[6] |
Z. Guo, S. 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. |
[7] |
D. Henry,
Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111.
doi: 10.1142/S0219891608001404. |
[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. |
[9] |
A. Himonas, G. Misiolek, G. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
S. Wu, J. 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. |
[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. |
[20] |
R. Xu, M. Zhang, S. Chen, Y. 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. |
[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. |
[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. |
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. |
[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. |
[3] |
A. Degasperis, D. Holm and A. Hone,
A new integrable equation with peakon solution, Theoret. and Math. Phys, 133 (2002), 1463-1474.
doi: 10.1023/A:1021186408422. |
[4] |
H. R. Dullin, G. 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. |
[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. |
[6] |
Z. Guo, S. 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. |
[7] |
D. Henry,
Persistence properties for the Degasperis-Procesi equation, J. Hyperbolic Differ. Equ., 5 (2008), 99-111.
doi: 10.1142/S0219891608001404. |
[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. |
[9] |
A. Himonas, G. Misiolek, G. 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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
S. Wu, J. 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. |
[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. |
[20] |
R. Xu, M. Zhang, S. Chen, Y. 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. |
[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. |
[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. |
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