×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML arXiv
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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.