A note on the fourth-order Schrodinger equation with spatially growing inhomogeneous source term

TL;DR

This paper develops local and global theories for a nonlinear fourth-order Schrödinger equation with an unbounded inhomogeneous source, addressing challenges posed by the loss of translation invariance.

Contribution

It introduces methods to handle unbounded inhomogeneous terms in the biharmonic Schrödinger equation, including Strauss type estimates under spherical symmetry.

Findings

Established local and global well-posedness in energy space.

Extended local theory to Sobolev spaces with lower regularity.

Handled unbounded inhomogeneous terms despite loss of translation invariance.

Abstract

This paper studies a non-linear biharmonic Sch\"odinger equation with an unbounded inhomogeneous term. The main goal is to develop a local theory but also a global theory for small data, in the energy space. Moreover, we develop a local theory in Sobolev spaces with lower regularity. The challenge is to deal with the inhomogeneous unbounded term, which broke down the space translation invariance. In order to handle the inhomogenous term, we use some Strauss type estimates, which require a spherically symmetric assumption.