Blow-up results for Inhomogeneous fourth-order nonlinear Schr\"odinger Equation

TL;DR

This paper studies the conditions under which solutions to an inhomogeneous fourth-order nonlinear Schrödinger equation blow up, focusing on energy levels and symmetry, using convexity and virial identities.

Contribution

It provides new blow-up criteria for inhomogeneous biharmonic Schrödinger equations based on energy and symmetry considerations, employing convexity methods.

Findings

Blow-up occurs for solutions with negative energy.

Solutions with positive energy below ground state can also blow up.

Radial symmetry influences blow-up behavior.

Abstract

In this paper, we investigate the blow-up phenomenon of the $H^2$ norm of solutions to the inhomogeneous biharmonic Schrodinger equation in two distinct scenarios. First, we consider the case of negative energy, analyzing separately the cases of radial and non-radial solutions. Then, we examine the positive energy case, where the energy is below that of the ground state and the ``kinetic'' energy exceeds the corresponding value for the ground state, again distinguishing between radial and non-radial solutions. Our approach is based on convexity methods, employing virial identities in the analysis.