On the Cauchy problem for higher-order nonlinear dispersive equations
نویسنده
چکیده
We study the higher-order nonlinear dispersive equation ∂tu+ ∂ 2j+1 x u = ∑ 0≤j1+j2≤2j aj1,j2∂ j1 x u∂ j2 x u, x, t ∈ R. where u is a real(or complex-) valued function. We show that the associated initial value problem is well posed in weighted Besov and Sobolev spaces for small initial data. We also prove ill-posedness results when a0,k 6= 0 for some k > j, in the sense that this equation cannot have its flow map C at the origin in H(R), for any s ∈ R. The same technique leads to similar ill-posedness results for other higherorder nonlinear dispersive equation as higher-order Benjamin-Ono and intermediate long wave equations.
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