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On the support of solutions to the generalized KdV equation. (English) Zbl 1001.35106

Summary: It is shown that if \(u\) is a solution of the initial value problem for the generalized Korteweg-de Vries equation \[ \partial_tu+\partial^3_xu+u^k\partial_x u=0,\quad (x,t)\in\mathbb{R}\times (t_1,t_2),\quad k\in\mathbb{Z}^+, \] such that there exists \(b\in\mathbb{R}\) with \(\text{supp} u(\cdot,t_j)\subseteq (b,\infty)\) (or \((-\infty,b))\), for \(j=1,2\) \((t_1\neq t_2)\), then \(u\equiv 0\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35G25 Initial value problems for nonlinear higher-order PDEs
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References:

[1] Bourgain, J., On the compactness of the support of solutions of dispersive equations, Internat. Math. Res. Notices, 9, 437-447 (1997) · Zbl 0882.35106
[2] Ginibre, J.; Tsutsum, Y., Uniqueness of solutions for the generalized Korteweg-de Vries equation, SIAM J. Math. Anal., 20, 1388-1425 (1989) · Zbl 0702.35224
[3] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Advances in Mathematics Supplementary Studies, Studies in Applied Math., 8, 93-128 (1983)
[4] Kenig, C. E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana University Math. J., 40, 33-69 (1991) · Zbl 0738.35022
[5] Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46, 527-620 (1993) · Zbl 0808.35128
[6] Kenig, C. E.; Ponce, G.; Vega, L., Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 122, 157-166 (1994) · Zbl 0810.35122
[7] Kenig, C. E.; Ruiz, A.; Sogge, C., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55, 329-347 (1987) · Zbl 0644.35012
[8] Kenig, C. E.; Sogge, C., A note on unique continuation for Schrödinger’s operator, Proc. Amer. Math. Soc., 103, 543-546 (1988) · Zbl 0661.35056
[9] Saut, J.-C.; Scheurer, B., Unique continuation for some evolution equations, J. Differential Equations, 66, 118-139 (1987) · Zbl 0631.35044
[10] Stein, E. M., Harmonic Analysis (1993), Princeton University Press
[11] Tarama S., Analytic solutions of the Korteweg-de Vries equation, preprint; Tarama S., Analytic solutions of the Korteweg-de Vries equation, preprint · Zbl 1078.35106
[12] Zhang, B.-Y., Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal., 23, 55-71 (1992) · Zbl 0746.35045
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