The Cauchy Problem for Non-linear Higher Order Hartree Type Equation in Modulation Spaces

Abstract

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity $$F(u) = (K \star |u|^{2k})u$$ under a specified condition on potential K with Cauchy data in modulation spaces $$M^{p,q}(\mathbb {R}^n)$$ . We establish global well-posedness results in $$M^{1,1}(\mathbb {R}^n)$$ , when $$K(x)= \frac{\lambda }{|x|^{\nu }} ~ (\lambda \in \mathbb {R}, ~ 0< \nu < min\{2, \frac{n}{2}\})$$ , for $$k < \frac{n+2-\nu }{n};$$ and local well-posedness results in $$M^{1,1}(\mathbb {R}^n)$$ , when $$K(x)= \frac{\lambda }{|x|^{\nu }} ~ (\lambda \in \mathbb {R}, ~ 0< \nu < n),$$ for $$k \in \mathbb {N};$$ in $$M^{p,q}(\mathbb {R}^n)$$ with $$1 \le p\le 4,~ 1\le q \le \frac{2^{2k-2}}{2^{2k-2}-1},~k \in \mathbb {N}$$ , when $$K \in M^{\infty , 1}(\mathbb {R}^n)$$ . Moreover, we also consider the Cauchy problem for the non-linear higher order Hartree equations on modulation spaces $$M^{p,1}(\mathbb {R}^n),$$ when $$K \in M^{1, \infty }(\mathbb {R}^n)$$ and show the existence of a unique global solution by using integrability of time decay factors of Strichartz estimates. As a consequence, we are able to deal with wider classes of a nonlinearity and a solution space.