Results of Asymptotic Analysis of an Elliptic Equation

This chapter presents a detailed asymptotic analysis of radial solutions for a general elliptic equation within an open connected set, subject to Neumann boundary conditions. The focus is on the dynamics of thin film equations, where the controlling function g is assumed to be continuous and non-negative. The primary objective is to analyze radial rupture solutions and their associated energies, with the thickness of a thin fluid layer over a planar region represented by w. The evolution of such films is described by a fourth-order partial differential equation, which has been extensively studied for various physical phenomena, including thin jets in Hele-Shaw cells, fluid droplets, and solidification of hypercooled melts. The chapter explores the asymptotic behavior of point rupture solutions, particularly in the radial setting, and their consequences for the dynamics of thin film fluids. It delves into the connection between the pressure p and the solution, discovering a remarkable scaling relationship that leads to analogous relationships for critical points and constants. The work also conjectures that radial point rupture solutions will serve as the blow-up profile of the solution near any point rupture. The chapter provides a thorough analysis of the ordinary differential equation and the regular perturbation problem, stating and proving results of the asymptotic solution. It concludes with an exploration of the energy of radial solutions as the parameter changes, offering a profound insight into the interplay between the solution and its energies, and their limiting profiles.