Introduction to Nonlinear Dispersive Equations by Felipe Linares & Gustavo Ponce

Author:Felipe Linares & Gustavo Ponce
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(7.96)

(7.97)

(7.98)

and

(7.99)

Moreover, for any there exists such that the map from into the class defined by ( 7.96 )–( 7.99 ) with instead of T is smooth.

In addition, if with , the previous results hold with instead of s in the same time interval .

Proof

We define

(7.100)

For , , we define the operator:

(7.101)

We see that defines a contraction on .

Let . By using (7.74), (7.75), (7.76), and (7.91) in Lemma 7.13 we deduce that

(7.102)

Setting and such that

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