# American Institute of Mathematical Sciences

• Previous Article
Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space
• DCDS-S Home
• This Issue
• Next Article
Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system
December  2016, 9(6): 2149-2165. doi: 10.3934/dcdss.2016089

## Wave breaking and persistent decay of solution to a shallow water wave equation

 1 Business School of Central South University, Changsha 410012, China 2 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China

Received  June 2015 Revised  September 2016 Published  November 2016

As we all know, wave breaking of the water wave is important and interesting to physicist and mathematician. In the article, we devote to the study of blow-up phenomena, the decay of solution and traveling wave solution to a shallow water wave equation. First, based on the blow-up scenario, some new blow-up phenomena is derived. By virtue of a weighted function, the persistent decay of solution is established. Finally, we explore the analytic solutions and traveling wave solutions.
Citation: Xue Yang, Xinglong Wu. Wave breaking and persistent decay of solution to a shallow water wave equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2149-2165. doi: 10.3934/dcdss.2016089
##### References:
 [1] L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Notices, 22 (2012), 5161-5181. [2] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [3] R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [4] G. M. Coclite, K. H. Karlsen and N. H. Risebro, Numberical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal., 28 (2008), 80-105. doi: 10.1093/imanum/drm003. [5] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. [6] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equation, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [7] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328. [8] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [9] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [10] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. [11] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [12] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [13] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solution, Theoret. and Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [14] A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, (1999), 23-37. [15] J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022. [16] J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117. doi: 10.1512/iumj.2007.56.3040. [17] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [18] A. Geyer, Solitary traveling waves of moderate amplitude, J. Nonlinear Math. Phys., 19 (2012), 1240010, 12pp. doi: 10.1142/S1402925112400104. [19] A. A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584. doi: 10.1007/s00208-003-0466-1. [20] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380. doi: 10.1137/S1111111102410943. [21] T. Kato and K. Masuda, Nonlinear evolution equations and analyticity, I. Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467. [22] T. Kato and G. Ponce, Commutator estimation and the Euler and Navier-Stokes Equation, Comm. Pure Appl. Math., 41 (1998), 891-907. doi: 10.1002/cpa.3160410704. [23] M. C. Lombardo, M. Sammartino and V. Sciacca, A Note on the analytic solutions of the Camassa-Holm equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 659-664. doi: 10.1016/j.crma.2005.10.006. [24] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3. [25] N. D. Mutlubas and A. Geyer, Orbital stability of solitary waves of moderate amplitude, J. Differential Equations, 255 (2013), 254-263. doi: 10.1016/j.jde.2013.04.010. [26] G. Rodrigues Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [27] M. V. Safonov, An abstract Cauchy-Kovalevskaya thm in a weighted Banach space, Comm. Pure Appl. Math., 48 (1995), 629-637. doi: 10.1002/cpa.3160480604. [28] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math., 30 (1977), 321-337. doi: 10.1002/cpa.3160300305. [29] X. Wu, On some wave breaking for the nonlinear integrable shallow water wave equations, Nonlinear Analysis, TMA., 127 (2015), 352-361. doi: 10.1016/j.na.2015.07.015. [30] X. Wu, Global Analytic Solutions and Traveling wave Solutions of the Cauchy problem for the Novikov Equation, Pro. AMS. [31] X. Wu and B. Guo, The exponential decay of solutions and traveling wave solutions for a modified Camassa-Holm equation with cubic nonlinearity, J. Math. Phys., 55 (2014), 081504, 17pp. doi: 10.1063/1.4891989. [32] X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Annali Sc. Norm. Sup. Pisa, 11 (2012), 707-727. [33] X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735. [34] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutins, Illinois J. Math., 47 (2003), 649-666. [35] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010.

show all references

##### References:
 [1] L. Brandolese, Breakdown for the Camassa-Holm equation using decay criteria and persistence in weighted spaces, Int. Math. Res. Notices, 22 (2012), 5161-5181. [2] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Letters, 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661. [3] R. Camassa, D. Holm and J. Hyman, An integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0. [4] G. M. Coclite, K. H. Karlsen and N. H. Risebro, Numberical schemes for computing discontinuous solutions of the Degasperis-Procesi equation, IMA J. Numer. Anal., 28 (2008), 80-105. doi: 10.1093/imanum/drm003. [5] A. Constantin, On the scattering problem for the Camassa-Holm equation, Proc. R. Soc. London A, 457 (2001), 953-970. doi: 10.1098/rspa.2000.0701. [6] A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equation, Acta Math., 181 (1998), 229-243. doi: 10.1007/BF02392586. [7] A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Annali Sc. Norm. Sup. Pisa, 26 (1998), 303-328. [8] A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity, Ann. Math., 173 (2011), 559-568. doi: 10.4007/annals.2011.173.1.12. [9] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2. [10] A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Comm. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801. [11] A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L. [12] R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988. [13] A. Degasperis, D. D. Holm and A. N. W. Hone, A new integrable equation with peakon solution, Theoret. and Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422. [14] A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, (1999), 23-37. [15] J. Escher, Y. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485. doi: 10.1016/j.jfa.2006.03.022. [16] J. Escher, Y. Liu and Z. Yin, Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation, Indiana Univ. Math. J., 56 (2007), 87-117. doi: 10.1512/iumj.2007.56.3040. [17] A. Fokas and B. Fuchssteiner, Symplectic structures, their Bäcklund transformation and hereditary symmetries, Physica D, 4 (1981), 47-66. doi: 10.1016/0167-2789(81)90004-X. [18] A. Geyer, Solitary traveling waves of moderate amplitude, J. Nonlinear Math. Phys., 19 (2012), 1240010, 12pp. doi: 10.1142/S1402925112400104. [19] A. A. Himonas and G. Misiolek, Analyticity of the Cauchy problem for an integrable evolution equation, Math. Ann., 327 (2003), 575-584. doi: 10.1007/s00208-003-0466-1. [20] D. D. Holm and M. F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst., 2 (2003), 323-380. doi: 10.1137/S1111111102410943. [21] T. Kato and K. Masuda, Nonlinear evolution equations and analyticity, I. Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 455-467. [22] T. Kato and G. Ponce, Commutator estimation and the Euler and Navier-Stokes Equation, Comm. Pure Appl. Math., 41 (1998), 891-907. doi: 10.1002/cpa.3160410704. [23] M. C. Lombardo, M. Sammartino and V. Sciacca, A Note on the analytic solutions of the Camassa-Holm equation, C. R. Math. Acad. Sci. Paris, 341 (2005), 659-664. doi: 10.1016/j.crma.2005.10.006. [24] H. Lundmark, Formation and dynamics of shock waves in the Degasperis-Procesi equation, J. Nonlinear Sci., 17 (2007), 169-198. doi: 10.1007/s00332-006-0803-3. [25] N. D. Mutlubas and A. Geyer, Orbital stability of solitary waves of moderate amplitude, J. Differential Equations, 255 (2013), 254-263. doi: 10.1016/j.jde.2013.04.010. [26] G. Rodrigues Blanco, On the Cauchy problem for the Camassa-Holm equation, Nonlinear Anal., 46 (2001), 309-327. doi: 10.1016/S0362-546X(01)00791-X. [27] M. V. Safonov, An abstract Cauchy-Kovalevskaya thm in a weighted Banach space, Comm. Pure Appl. Math., 48 (1995), 629-637. doi: 10.1002/cpa.3160480604. [28] E. Trubowitz, The inverse problem for periodic potentials, Comm. Pure Appl. Math., 30 (1977), 321-337. doi: 10.1002/cpa.3160300305. [29] X. Wu, On some wave breaking for the nonlinear integrable shallow water wave equations, Nonlinear Analysis, TMA., 127 (2015), 352-361. doi: 10.1016/j.na.2015.07.015. [30] X. Wu, Global Analytic Solutions and Traveling wave Solutions of the Cauchy problem for the Novikov Equation, Pro. AMS. [31] X. Wu and B. Guo, The exponential decay of solutions and traveling wave solutions for a modified Camassa-Holm equation with cubic nonlinearity, J. Math. Phys., 55 (2014), 081504, 17pp. doi: 10.1063/1.4891989. [32] X. Wu and Z. Yin, Well-posedness and global existence for the Novikov equation, Annali Sc. Norm. Sup. Pisa, 11 (2012), 707-727. [33] X. Wu and Z. Yin, A note on the Cauchy problem of the Novikov equation, Appl. Anal., 92 (2013), 1116-1137. doi: 10.1080/00036811.2011.649735. [34] Z. Yin, On the Cauchy problem for an integrable equation with peakon solutins, Illinois J. Math., 47 (2003), 649-666. [35] Z. Yin, Global weak solutions to a new periodic integrable equation with peakon solutions, J. Funct. Anal., 212 (2004), 182-194. doi: 10.1016/j.jfa.2003.07.010.
 [1] Donghao Li, Hongwei Zhang, Shuo Liu, Qingiyng Hu. Blow-up of solutions to a viscoelastic wave equation with nonlocal damping. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022009 [2] Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171 [3] Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 [4] Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280 [5] Hayato Miyazaki. Strong blow-up instability for standing wave solutions to the system of the quadratic nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2411-2445. doi: 10.3934/dcds.2020370 [6] Kyouhei Wakasa. Blow-up of solutions to semilinear wave equations with non-zero initial data. Conference Publications, 2015, 2015 (special) : 1105-1114. doi: 10.3934/proc.2015.1105 [7] Mingqi Xiang, Die Hu. Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4609-4629. doi: 10.3934/dcdss.2021125 [8] Ahmad Z. Fino, Mohamed Ali Hamza. Blow-up of solutions to semilinear wave equations with a time-dependent strong damping. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022006 [9] Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583 [10] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [11] Mengxian Lv, Jianghao Hao. General decay and blow-up for coupled Kirchhoff wave equations with dynamic boundary conditions. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021058 [12] Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106 [13] Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117 [14] Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations and Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019 [15] Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121 [16] C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88 [17] Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155 [18] Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267 [19] Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 [20] Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations and Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

2021 Impact Factor: 1.865