The Cauchy problem for nonlinear Schrödinger equations in modulation spaces. (August 2015) - Ru, Shaolei; Chen, Jiecheng
Abstract: We consider the nonlinear Schrödinger equation ( ∗ ) { i u t + Δ u ± f ( u ) = 0, f ( u ) = | u | 2 m u n, m, n ∈ N, ( x, t ) ∈ R N × R u ( 0, x ) = u 0 ( x ) . We give a representation of the unique solution of the Cauchy problem( ∗ ) existed in C ( 0, T ∗ ; M p, 1 ) . Moreover, by this representation, we obtain that there exists a constant B independent of p, N such that for any initial data ‖ u 0 ‖ M p ′, 1 < B ( N ( 1 / 2 − 1 / p ) − 1 ), 2 N N − 2 < p ⩽ 2 m + n + 1, N > 2, the Schrödinger equation( ∗ ) has a unique global solution u ∈ C ( − ∞, ∞ ; M p, 1 ) .