Analytic Function Spaces and their Applications to Nonlinear Evolution Equations

Abstract

In this note we present a survey of recent progress on analyticity of solutions to nonlinear Schrödinger equations and generalized Korteweg-de Vries equation. We also state some applications of analytic function spaces to these equations. Nonlinear Schrödinger equations considered in this note are written

$$\left\{ {\begin{array}{*{20}{c}} {i{\partial _t}u + \frac{1}{2}\Delta u = N(u),(t,x) \in R \times {R^n}} \\ {u(0,x) = {u_0}(x),x \in {R^n}} \end{array}} \right.$$

(1.1)

where nonlinear terms N(u) will be defined in each theorem below and n denotes the spatial dimension. The generalized Korteweg-de Vries equation is written

$$\left\{ {\begin{array}{*{20}{c}} {{\partial _t}u + \frac{1}{3}\partial _{{x_1}}^3u + {\partial _{{x_1}}}({{\left| u \right|}^{p - 1}}u) = 0,(t,{x_1}) \in R \times R} \\ {U(0,{x_1}) = {u_0}({x_1}),{x_1} \in R} \end{array}} \right.$$

(1.2)

where p ∈ N. Local and global in time of solutions to these equations were studied extensively by many authors in the usual Sobolev spaces (see, e.g.,[8], [22],[23], [41], [42], [44], [51] and references cited these papers). In order to state recent progress on (1.1) and (1.2) precisely, we need some function spaces and notations Function spaces and notation. We use the usual Lebesgue space

$${L^p} = \left\{ {\phi \in S';{{\left\| \phi \right\|}_p} < + \infty } \right\}.$$

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