Qualitative properties of solutions to a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

Abstract In this paper, we investigate the initial-boundary value problem for a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. By introducing the potential well and its outside set, we prove the global existence of solutions in the case of subcritical initial energy, as well as the finite time blow-up of solutions in the case of negative, null, subcritical and critical initial energy, respectively. And we provide an upper bound estimate on the blow-up time in the case of negative and subcritical initial energy, respectively. Additionally, when the nonlinear source term takes a special form, we obtain the global existence of solutions in the case of critical initial energy and the finite time blow-up and an upper bound estimate on the blow-up time of solutions in the case of supercritical initial energy. In the case where the initial energy is not greater than a lower bound for the depth of the potential well, we discuss the asymptotic behavior of solutions by defining a family of potential wells.