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Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation. (English) Zbl 1095.35039
The authors consider the initial value problem for a generalized Korteweg-de Vries equation in \(u=u(x,t)\), \(u_t+u_{xxx}+u^pu_x=0\), where \(p\) is a positive integer (\(p=1\) gives the Korteweg-de Vries equation itself, and \(p=2\) the modified Korteweg-de Vries equation, both completely integrable). In fact, they consider the case of complex-valued solutions (so the initial data will be drawn from a suitable class of analytic functions). The focus is on studying the asymptotics of the width of the strip of analyticity for large time \(t\). They obtain algebraically decreasing time-asymptotics for this width.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B40 Asymptotic behavior of solutions to PDEs
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[1] Bona, J.L.; Dougalis, V.A.; Karakashian, O.A.; McKinney, W.R., Conservative high-order numerical schemes for the generalized korteweg – de Vries equation, Philos. trans. roy. soc. London ser. A, 351, 107-164, (1995) · Zbl 0824.65095
[2] Bona, J.L.; Dougalis, V.A.; Karakashian, O.A.; McKinney, W.R., Numerical simulation of singular solutions of the generalized korteweg – de Vries equation, (), 17-29 · Zbl 0860.35112
[3] Bona, J.L.; Grujić, Z., Spatial analyticity for nonlinear waves, Math. models methods appl. sci., 13, 1-15, (2003)
[4] Bona, J.L.; Weissler, F.B., Similarity solutions of the generalized korteweg – de Vries equation, Math. proc. Cambridge philos. soc., 127, 323-351, (1999) · Zbl 0939.35164
[5] Bona, J.L.; Weissler, F.B., Blow-up of spatially periodic complex-valued solutions of nonlinear dispersive equations, Indiana univ. math. J., 50, 759-782, (2001) · Zbl 1330.35036
[6] Bourgain, J., Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Geom. funct. anal., 3, 107-156, (1993), 209-262 · Zbl 0787.35097
[7] de Bouard, A.; Hayashi, N.; Kato, K., Gevrey regularizing effect for the (generalized) korteweg – de Vries equation and nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 6, 673-715, (1995) · Zbl 0843.35098
[8] Foias, C.; Temam, R., Gevrey class regularity for the solutions of the navier – stokes equations, J. funct. anal., 87, 359-369, (1989) · Zbl 0702.35203
[9] Ginibre, J.; Tsutsumi, Y.; Velo, G., On the Cauchy problem for the Zakharov system, J. funct. anal., 151, 384-436, (1997) · Zbl 0894.35108
[10] Grujić, Z.; Kalisch, H., Local well-posedness of the generalized korteweg – de Vries equation in spaces of analytic functions, Differential integral equations, 15, 1325-1334, (2002) · Zbl 1031.35124
[11] Hayashi, N., Analyticity of solutions of the korteweg – de Vries equation, SIAM J. math. anal., 22, 1738-1743, (1991) · Zbl 0742.35056
[12] Hayashi, N., Solutions of the (generalized) korteweg – de Vries equation in the Bergman and szegö spaces on a sector, Duke math. J., 62, 575-591, (1991) · Zbl 0729.35119
[13] Kato, T., Quasilinear equations of evolution with applications to partial differential equations, (), 25-70
[14] Kato, T., On the korteweg – devries equation, Manuscripta math., 28, 89-99, (1979) · Zbl 0415.35070
[15] Kato, T.; Masuda, K., Nonlinear evolution equations and analyticity I, Ann. inst. H. Poincaré anal. non linéaire, 3, 455-467, (1986) · Zbl 0622.35066
[16] Kato, K.; Ogawa, T., Analyticity and smoothing effect for the korteweg – de Vries equation with a single point singularity, Math. ann., 316, 577-608, (2000) · Zbl 0956.35115
[17] Kenig, C.E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana univ. math. J., 40, 33-69, (1991) · Zbl 0738.35022
[18] Kenig, C.E.; Ponce, G.; Vega, L., On the Cauchy problem for the korteweg – devries equation in Sobolev spaces of negative indices, Duke math. J., 71, 1-20, (1993) · Zbl 0787.35090
[19] Martel, Y.; Merle, F., Blow up in finite time and dynamics of blow up solutions for the \(L_2\)-critical generalized KdV equation, J. amer. math. soc., 15, 617-664, (2002) · Zbl 0996.35064
[20] Staffilani, G., On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations, Duke math. J., 86, 109-142, (1997) · Zbl 0874.35114
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