Guo, Zhengguang; Ni, Lidiao Persistence properties and unique continuation of solutions to a two-component Camassa-Holm equation. (English) Zbl 1245.35108 Math. Phys. Anal. Geom. 14, No. 2, 101-114 (2011). Summary: We consider a two-component Camassa-Holm system which arises in shallow water theory. The present work is mainly concerned with persistence properties and unique continuation to this new kind of system, in view of the classical Camassa-Holm equation. Firstly, it is shown that there are three results about these properties of the strong solutions. Then we also investigate the infinite propagation speed in the sense that the corresponding solution does not have compact spatial support for \(t > 0\) although the initial data belongs to \(C_{0}^{\infty}(\mathbb{R})\). Cited in 13 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 35D35 Strong solutions to PDEs Keywords:two-component Camassa-Holm equation; persistence properties; propagation speed PDFBibTeX XMLCite \textit{Z. Guo} and \textit{L. Ni}, Math. Phys. Anal. Geom. 14, No. 2, 101--114 (2011; Zbl 1245.35108) Full Text: DOI References: [1] Beals, R., Sattinger, D., Szmigielski, J.: Multi-peakons and a theorem of Stieltjes. Inverse Problems 15(1), L1–L4 (1999) · Zbl 0923.35154 [2] Bressan, A., Constantin, A.: Global conservative solutions of the Camassa–Holm equation. Arch. Ration. Mech. Anal. 183(2), 215–239 (2007) · Zbl 1105.76013 [3] Bressan, A., Constantin, A.: Global dissipative solutions of the Camassa–Holm equation. Anal. Appl. (Singap.) 5(1), 1–27 (2007) · Zbl 1139.35378 [4] Boutet de Monvel, A., Kostenko, A., Shepelsky, D., Teschl, G.: Long-time asymptotics for the Camassa–Holm equation. SIAM J. Math. Anal. 41(4), 1559–1588 (2009) · Zbl 1204.37073 [5] Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Ann. Inst. Fourier (Grenoble) 50(2), 321–362 (2000) · Zbl 0944.00010 [6] Constantin, A.: Finite propagation speed for the Camassa–Holm equation. J. Math. Phys. 46(2), 023506, 4 pp. (2005) · Zbl 1076.35109 [7] Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166(3), 523–535 (2006) · Zbl 1108.76013 [8] Constantin, A., Escher, J.: Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation. Commun. Pure Appl. Math. 51(5), 475–504 (1998) · Zbl 0934.35153 [9] Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998) · Zbl 0923.76025 [10] Constantin, A., Escher, J.: Particle trajectories in solitary water waves. Bull. Amer. Math. Soc. (N.S.) 44(3), 423–431 (2007) · Zbl 1126.76012 [11] Constantin, A., Gerdjikov, V., Ivanov, R.: Inverse scattering transform for the Camassa–Holm equation. Inverse Problems 22(6), 2197–2207 (2006) · Zbl 1105.37044 [12] Constantin, A., Ivanov, R.: On an integrable two-component Camassa–Holm shallow water system. Phys. Lett. A 372(48), 7129–7132 (2008) · Zbl 1227.76016 [13] Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2009) · Zbl 1169.76010 [14] Constantin, A., McKean, H.: A shallow water equation on the circle. Commun. Pure Appl. Math. 52(8), 949–982 (1999) · Zbl 0940.35177 [15] Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Commun. Math. Phys. 211(1), 45–61 (2000) · Zbl 1002.35101 [16] Constantin, A., Strauss, W.: Stability of peakons. Commun. Pure Appl. Math. 53(5), 603–610 (2000) · Zbl 1049.35149 [17] Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993) · Zbl 0936.35153 [18] Chen, M., Liu, S., Zhang, Y.: A two-component generalization of the Camassa–Holm equation and its solutions. Lett. Math. Phys. 75(1), 1–15 (2006) · Zbl 1105.35102 [19] Escher, J., Lechtenfeld, O., Yin, Z.: Well-posedness and blow-up phenomena for the 2-component Camassa–Holm equation. Discrete Contin. Dyn. Syst. 19(3), 493–513 (2007) · Zbl 1149.35307 [20] Falqui, G.: On a Camassa–Holm type equation with two dependent variables. J. Phys. A 39(2), 327–342 (2006) · Zbl 1084.37053 [21] Fuchssteiner, B., Fokas, A.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Phys. D 4(1), 47–66 (1981/82) · Zbl 1194.37114 [22] Guo, Z.: Blow-up and global solutions to a new integrable model with two components. J. Math. Anal. Appl. 372(1), 316–327 (2010) · Zbl 1205.35045 [23] Guo, Z., Zhou, Y.: On solutions to a two-component generalized Camassa–Holm equation. Stud. Appl. Math. 124(3), 307–322 (2010) · Zbl 1189.35255 [24] Henry, D.: Compactly supported solutions of the Camassa–Holm equation. J. Nonlin. Math. Phys. 12(3), 342–347 (2005) · Zbl 1086.35079 [25] Henry, D.: Infinite propagation speed for a two component Camassa–Holm equation. Discrete Contin. Dyn. Syst. Ser. B 12(3), 597–606 (2009) · Zbl 1180.35458 [26] Himonas, A., Misiolek, G., Ponce, G., Zhou, Y.: Persistence properties and unique continuation of solutions of the Camassa–Holm equation. Commun. Math. Phys. 271(2), 511–522 (2007) · Zbl 1142.35078 [27] Ivanov, R.: Water waves and integrability. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 365(1858), 2267–2280 (2007) · Zbl 1152.76322 [28] Johnson, R.: Camassa–Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002) · Zbl 1037.76006 [29] Mustafa, O.G.: On smooth traveling waves of an integrable two-component Camassa–Holm shallow water system. Wave Motion 46(6), 397–402 (2009) · Zbl 1231.76063 [30] Olver, P., Rosenau, P.: Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support. Phys. Rev. E (3) 53(2), 1900–1906 (1996) [31] Toland, J.: Stokes waves. Topol. Methods Nonlinear Anal. 7(1), 1–48 (1996) · Zbl 0897.35067 [32] Whitham, G.: Linear and Nonlinear Waves. Reprint of the 1974 Original, xviii+636 pp.. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. Wiley, New York (1999) [33] Xin, Z., Zhang, P.: On the weak solution to a shallow water equation. Commun. Pure Appl. Math. 53(11), 1411–1433 (2000) · Zbl 1048.35092 [34] Zhou, Y.: Wave breaking for a shallow water equation. Nonlinear Anal. 57(1), 137–152 (2004) · Zbl 1106.35070 [35] Zhou, Y.: Wave breaking for a periodic shallow water equation. J. Math. Anal. Appl. 290(2), 591–604 (2004) · Zbl 1042.35060 [36] Zhou, Y.: Stability of solitary waves for a rod equation. Chaos Solitons Fractals 21(4), 977–981 (2004) · Zbl 1046.35094 [37] Zhou, Y., Guo, Z.: Blow up and propagation speed of solutions to the DGH equation. Discrete Contin. Dyn. Syst., Ser. B 12(3), 657–670 (2009) · Zbl 1180.35473 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.