Global Existence, Finite Time Blow-Up, and Vacuum Isolating Phenomenon for a Class of Thin-Film Equation

Abstract

We consider a thin-film equation modeling the epitaxial growth of nanoscale thin films. By exploiting the boundary conditions and the variational structure of the equation, we look for conditions on initial data that ensure the solution exists globally or blows up in finite time. Moreover, for global solution, we establish the exponential decays of solutions and energy functional, and give the concrete decay rate. As for blow-up solution, we prove that the solution grows exponentially and obtain the behavior of energy functional as t tends to the maximal existence time. Under the low initial energy, we get further two necessary and sufficient conditions for the solution existing globally and blowing up in finite time, respectively. A new sufficient condition such that the solution exists globally is obtained; we point out that this initial condition is independent to initial energy. Finally, we discuss the vacuum isolating phenomena of the solution.