We investigate the inhomogeneous nonlinear Schrödinger equation with partial harmonic confinement. First, we present a global well-posedness result for small data in the intercritical regime. Second, we obtain a threshold of global existence versus finite-time blow-up in the mass-critical regime. Finally, we prove the $L^2$ concentration of the mass-critical non-global solution with minimal mass. The challenge is to address the fact that the standard scale invariance is broken by the partial confinement. We use the associated ground state without potential in order to describe the threshold of global versus non-global existence of solutions.

This study explores the threshold of global existence and exponential decay versus finite-time blow-up for solutions to an inhomogeneous nonlinear bi-harmonic heat problem. The novelty is to consider the inhomogeneous source term. The method uses some standard stable sets under the flow of the fourth-order parabolic problem, due to Payne-Sattynger.

This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.