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Smoothing properties and retarded estimates for some dispersive evolution equations. (English) Zbl 0762.35008
Dispersive partial differential equations of the type \[ \partial_ tu- Lu=f \] with skew adjoint operator \(L\) acting in some Hilbert space, for instance, \(L^ 2(\mathbb{R}^ n)\), are considered. The Cauchy problem with initial data \(u(t=0)=u_ 0\) is formally solved by \[ u(t)=U(t)u_ 0+\int^ t_ 0d\tau U(t-\tau)f(\tau), \] therefore an essential role in the treatment of the problem is played by the properties of the operators \[ u_ 0\to U(t)u_ 0,\quad f\to\int^ t_ 0d\tau U(t-\tau)f(\tau). \] “Smoothing” means that less derivatives occur on \(u_ 0\) and \(f\) than on their images. “Retardation” refers to the specific form of the second operator.
In this paper, the mentioned operators are treated in terms of abstract Banach space theory. The obtained estimates are applied to the Cauchy problem for the generalized Benjamin-Ono equation that corresponds to \(L=i\Phi(-i\partial/\partial x)\), \(f=(\partial/\partial x)v'(u)\), \(\Phi(\xi)=\xi|\xi|^{2\mu}\), \(v(0)=v'(0)=0\).

35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
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