SCATTERING THRESHOLD FOR RADIAL BI-HARMONIC SCHRODINGER EQUATIONS

This paper examines the asymptotic behavior of energy solutions to a fourth-order Schrodinger equation featuring a mixed nonlinearity. The aim is to analyze the competition between two source terms with opposite signs. Specifically, the paper demonstrates energy scattering versus finite-time blow-up of energy solutions in the inter-critical regime within a radial setting. The primary challenge arises from the absence of scaling invariance. For scattering, the method developed by Dodson-Murphy, which relies on Morawetz estimates and Tao's criteria, is employed. The assumption of spherical symmetry is essential for three reasons: first, because the rearrangement argument does not hold; second, to apply radial Strauss estimates; and third, due to the absence of a variance identity.