2016, 36(9): 5047-5066. doi: 10.3934/dcds.2016019

Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions

1. 

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

2. 

Department of Mathematics, Zhongshan University, Guangzhou, 510275

Received  July 2015 Revised  November 2015 Published  May 2016

In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using Littlewood-Paley theory and transport equations theory, we establish the local well-posedness of the system in the critical Besov space. Moreover, we obtain some weighted $L^p$ estimates of strong solutions to the system. By taking suitable weighted functions, we can get the persistence properties of strong solutions on exponential, algebraic and logarithmic decay rates, respectively.
Citation: Wei Luo, Zhaoyang Yin. Local well-posedness in the critical Besov space and persistence properties for a three-component Camassa-Holm system with N-peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5047-5066. doi: 10.3934/dcds.2016019
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

L. Brandolese, Break down for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not. IMRN, 22 (2012), 5161.

[3]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z.

[4]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5 (2007), 1. doi: 10.1142/S0219530507000857.

[5]

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

[6]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0.

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701.

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.

[10]

A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603.

[11]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.

[13]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[14]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586.

[15]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[16]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12.

[17]

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

[18]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801.

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[20]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050.

[21]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.

[22]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2.

[23]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 19 (2007), 493. doi: 10.3934/dcds.2007.19.493.

[24]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X.

[25]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, J. Differential Equations, 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002.

[26]

C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation,, Contemp. Math., 526 (2010), 199. doi: 10.1090/conm/526/10382.

[27]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa- Holm shallow water system,, J. Funct. Anal., 260 (2011), 1132. doi: 10.1016/j.jfa.2010.11.015.

[28]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 623. doi: 10.1016/j.anihpc.2011.04.003.

[29]

C. Guan, H. He and Z. Yin, Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system,, Nonlinear Anal. Real World Appl., 25 (2015), 219. doi: 10.1016/j.nonrwa.2015.04.001.

[30]

X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions,, Adv. Math., 226 (2011), 827. doi: 10.1016/j.aim.2010.07.009.

[31]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,, J. Funct. Anal., 258 (2010), 4251. doi: 10.1016/j.jfa.2010.02.008.

[32]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104.

[33]

D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified twocomponent Camassa-Holm equation,, Phys. Rev. E (3), 79 (2009). doi: 10.1103/PhysRevE.79.016601.

[34]

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. doi: 10.1007/s00220-006-0172-4.

[35]

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

[36]

W. Luo and Z. Yin, Global existence and local well-posedness for a three-component Camassa-Holm system with N-peakon solutions,, J. Differential Equations, 259 (2015), 201. doi: 10.1016/j.jde.2015.02.005.

[37]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X.

[38]

W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa-Holm shallow water system,, J. Differential Equations, 251 (2011), 3558. doi: 10.1016/j.jde.2011.08.010.

[39]

W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system,, \emph{J. Math. Phys.}, 52 (2011). doi: 10.1063/1.3562928.

[40]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[41]

X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 33 (2013), 3211. doi: 10.3934/dcds.2013.33.3211.

[42]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[43]

K. Yan and Z. Yin, Well-posedness for a modified two-component Camassa-Holm system in critical spaces,, Discrete Contin. Dyn. Syst., 33 (2013), 1699. doi: 10.3934/dcds.2013.33.1699.

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations,, Grundlehren der Mathematischen Wissenschaften, (2011). doi: 10.1007/978-3-642-16830-7.

[2]

L. Brandolese, Break down for the Camassa-Holm equation using decay criteria and persistence in weighted spaces,, Int. Math. Res. Not. IMRN, 22 (2012), 5161.

[3]

A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215. doi: 10.1007/s00205-006-0010-z.

[4]

A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation,, Anal. Appl., 5 (2007), 1. doi: 10.1142/S0219530507000857.

[5]

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

[6]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. doi: 10.1016/S0065-2156(08)70254-0.

[7]

A. Constantin, The Hamiltonian structure of the Camassa-Holm equation,, Exposition. Math., 15 (1997), 53.

[8]

A. Constantin, On the scattering problem for the Camassa-Holm equation,, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 953. doi: 10.1098/rspa.2000.0701.

[9]

A. Constantin, Existence of permanent and breaking waves for a shallow water equation: A geometric approach,, Ann. Inst. Fourier (Grenoble), 50 (2000), 321. doi: 10.5802/aif.1757.

[10]

A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603.

[11]

A. Constantin, The trajectories of particles in Stokes waves,, Invent. Math., 166 (2006), 523. doi: 10.1007/s00222-006-0002-5.

[12]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 26 (1998), 303.

[13]

A. Constantin and J. Escher, Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,, Comm. Pure Appl. Math., 51 (1998), 475. doi: 10.1002/(SICI)1097-0312(199805)51:5<475::AID-CPA2>3.0.CO;2-5.

[14]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations,, Acta Math., 181 (1998), 229. doi: 10.1007/BF02392586.

[15]

A. Constantin and J. Escher, Particle trajectories in solitary water waves,, Bull. Amer. Math. Soc., 44 (2007), 423. doi: 10.1090/S0273-0979-07-01159-7.

[16]

A. Constantin and J. Escher, Analyticity of periodic traveling free surface water waves with vorticity,, Ann. of Math., 173 (2011), 559. doi: 10.4007/annals.2011.173.1.12.

[17]

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

[18]

A. Constantin and L. Molinet, Global weak solutions for a shallow water equation,, Comm. Math. Phys., 211 (2000), 45. doi: 10.1007/s002200050801.

[19]

A. Constantin and W. A. Strauss, Stability of peakons,, Comm. Pure Appl. Math., 53 (2000), 603. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.

[20]

A. Constantin and R. Ivanov, On an integrable two-component Camassa-Holm shallow water system,, Phys. Lett. A, 372 (2008), 7129. doi: 10.1016/j.physleta.2008.10.050.

[21]

R. Danchin, A few remarks on the Camassa-Holm equation,, Differential Integral Equations, 14 (2001), 953.

[22]

R. Danchin, A note on well-posedness for Camassa-Holm equation,, J. Differential Equations, 192 (2003), 429. doi: 10.1016/S0022-0396(03)00096-2.

[23]

J. Escher, O. Lechtenfeld and Z. Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 19 (2007), 493. doi: 10.3934/dcds.2007.19.493.

[24]

B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries,, Phys. D, 4 (): 47. doi: 10.1016/0167-2789(81)90004-X.

[25]

C. Guan and Z. Yin, Global existence and blow-up phenomena for an integrable two-component Camassa-Holm shallow water system,, J. Differential Equations, 248 (2010), 2003. doi: 10.1016/j.jde.2009.08.002.

[26]

C. Guan, K. H. Karlsen and Z. Yin, Well-posedness and blow-up phenomena for a modified two-component Camassa-Holm equation,, Contemp. Math., 526 (2010), 199. doi: 10.1090/conm/526/10382.

[27]

C. Guan and Z. Yin, Global weak solutions for a two-component Camassa- Holm shallow water system,, J. Funct. Anal., 260 (2011), 1132. doi: 10.1016/j.jfa.2010.11.015.

[28]

C. Guan and Z. Yin, Global weak solutions for a modified two-component Camassa-Holm equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 623. doi: 10.1016/j.anihpc.2011.04.003.

[29]

C. Guan, H. He and Z. Yin, Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system,, Nonlinear Anal. Real World Appl., 25 (2015), 219. doi: 10.1016/j.nonrwa.2015.04.001.

[30]

X. G. Geng and B. Xue, A three-component generalization of Camassa-Holm equation with N-peakon solutions,, Adv. Math., 226 (2011), 827. doi: 10.1016/j.aim.2010.07.009.

[31]

G. Gui and Y. Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system,, J. Funct. Anal., 258 (2010), 4251. doi: 10.1016/j.jfa.2010.02.008.

[32]

D. Henry, Persistence properties for a family of nonlinear partial differential equations,, Nonlinear Anal., 70 (2009), 1565. doi: 10.1016/j.na.2008.02.104.

[33]

D. D. Holm, L. Naraigh and C. Tronci, Singular solution of a modified twocomponent Camassa-Holm equation,, Phys. Rev. E (3), 79 (2009). doi: 10.1103/PhysRevE.79.016601.

[34]

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. doi: 10.1007/s00220-006-0172-4.

[35]

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

[36]

W. Luo and Z. Yin, Global existence and local well-posedness for a three-component Camassa-Holm system with N-peakon solutions,, J. Differential Equations, 259 (2015), 201. doi: 10.1016/j.jde.2015.02.005.

[37]

G. Rodríguez-Blanco, On the Cauchy problem for the Camassa-Holm equation,, Nonlinear Anal., 46 (2001), 309. doi: 10.1016/S0362-546X(01)00791-X.

[38]

W. Tan and Z. Yin, Global conservative solutions of a modified two-component Camassa-Holm shallow water system,, J. Differential Equations, 251 (2011), 3558. doi: 10.1016/j.jde.2011.08.010.

[39]

W. Tan and Z. Yin, Global dissipative solutions of a modified two-component Camassa-Holm shallow water system,, \emph{J. Math. Phys.}, 52 (2011). doi: 10.1063/1.3562928.

[40]

J. F. Toland, Stokes waves,, Topol. Methods Nonlinear Anal., 7 (1996), 1.

[41]

X. Wu and B. Guo, Persistence properties and infinite propagation for the modified 2-component Camassa-Holm equation,, Discrete Contin. Dyn. Syst., 33 (2013), 3211. doi: 10.3934/dcds.2013.33.3211.

[42]

Z. Xin and P. Zhang, On the weak solutions to a shallow water equation,, Comm. Pure Appl. Math., 53 (2000), 1411. doi: 10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5.

[43]

K. Yan and Z. Yin, Well-posedness for a modified two-component Camassa-Holm system in critical spaces,, Discrete Contin. Dyn. Syst., 33 (2013), 1699. doi: 10.3934/dcds.2013.33.1699.

[1]

Kai Yan, Zhaoyang Yin. Well-posedness for a modified two-component Camassa-Holm system in critical spaces. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1699-1712. doi: 10.3934/dcds.2013.33.1699

[2]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

[3]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic & Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305

[4]

Lei Zhang, Bin Liu. Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2655-2685. doi: 10.3934/dcds.2018112

[5]

Qiaoyi Hu, Zhijun Qiao. Persistence properties and unique continuation for a dispersionless two-component Camassa-Holm system with peakon and weak kink solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2613-2625. doi: 10.3934/dcds.2016.36.2613

[6]

Joachim Escher, Olaf Lechtenfeld, Zhaoyang Yin. Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 493-513. doi: 10.3934/dcds.2007.19.493

[7]

Jihong Zhao, Ting Zhang, Qiao Liu. Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 555-582. doi: 10.3934/dcds.2015.35.555

[8]

Xi Tu, Zhaoyang Yin. Local well-posedness and blow-up phenomena for a generalized Camassa-Holm equation with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2781-2801. doi: 10.3934/dcds.2016.36.2781

[9]

Ying Fu, Changzheng Qu, Yichen Ma. Well-posedness and blow-up phenomena for the interacting system of the Camassa-Holm and Degasperis-Procesi equations. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1025-1035. doi: 10.3934/dcds.2010.27.1025

[10]

Xinglong Wu. On the Cauchy problem of a three-component Camassa--Holm equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2827-2854. doi: 10.3934/dcds.2016.36.2827

[11]

Shaojie Yang, Tianzhou Xu. Symmetry analysis, persistence properties and unique continuation for the cross-coupled Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 329-341. doi: 10.3934/dcds.2018016

[12]

Zhaoyang Yin. Well-posedness and blow-up phenomena for the periodic generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2004, 3 (3) : 501-508. doi: 10.3934/cpaa.2004.3.501

[13]

Jinlu Li, Zhaoyang Yin. Well-posedness and blow-up phenomena for a generalized Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5493-5508. doi: 10.3934/dcds.2016042

[14]

Zeng Zhang, Zhaoyang Yin. On the Cauchy problem for a four-component Camassa-Holm type system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5153-5169. doi: 10.3934/dcds.2015.35.5153

[15]

Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287

[16]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065

[17]

Xinglong Wu, Boling Guo. Persistence properties and infinite propagation for the modified 2-component Camassa--Holm equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3211-3223. doi: 10.3934/dcds.2013.33.3211

[18]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

[19]

Rossen I. Ivanov. Conformal and Geometric Properties of the Camassa-Holm Hierarchy. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 545-554. doi: 10.3934/dcds.2007.19.545

[20]

Caixia Chen, Shu Wen. Wave breaking phenomena and global solutions for a generalized periodic two-component Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3459-3484. doi: 10.3934/dcds.2012.32.3459

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]