SCATTERING THRESHOLD FOR RADIAL BI-HARMONIC SCHRÖDINGER EQUATIONS

This paper examines the asymptotic behavior of energy solutions to a fourth-order Schrödinger 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.