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

TL;DR

This work analyzes the Cauchy problem $i\partial_t v+\Delta^2 v=-\epsilon |x|^b |v|^{q-1}v$ for the biharmonic Schrödinger equation with an unbounded inhomogeneous term, focusing on radial data. It develops a local well-posedness theory in the energy space $H^2_{rd}$ for $N\ge5$ and a local theory at lower regularity in $H^1_{rd}$ for $N\ge3$, leveraging Strichartz estimates, Hardy-type and Strauss-type radial decay, and a weighted Gagliardo--Nirenberg inequality to control the nonlinearity. It also proves a global theory for small data, including global existence and scattering in the energy space and a quantitative small-data global theory, under subcritical and critical scaling regimes. The results extend the understanding of inhomogeneous biharmonic NLS with $b>0$ under radial symmetry and provide a framework for asymptotic stability and scattering of high-intensity wave propagation in Kerr-type media.

Abstract

This paper studies a non-linear biharmonic Schödinger 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.

Theorems & Definitions (2)

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