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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 · 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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