The Cauchy problem for the Hartree type equation in modulation spaces. (January 2016) - Bhimani, Divyang G.
Abstract: We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F ( u ) = ( K ∗ | u | 2 ) u under a specified condition on potential K with Cauchy data in modulation spaces M p, q ( R d ) . We establish global well-posedness results in M 1, 1 ( R d ) when K ( x ) = λ | x | − γ ( λ ∈ R, 0 < γ < min 2, d / 2 ) ; in M p, q ( R d ) ( 1 ≤ q ≤ min p, p ′ where p ′ is the Hölder conjugate of p ∈ [ 1, 2 ] ) when K is in Fourier algebra F L 1 ( R d ), and local well-posedness result in M p, 1 ( R d ) ( 1 ≤ p ≤ ∞ ) when K ∈ M 1, ∞ ( R d ) .