Blow-up of solutions to a fourth-order parabolic equation with/without p-Laplician and general nonlinearity modeling epitaxial growth

This paper deals with a fourth-order parabolic equation with/without p-Laplician and general nonlinearity modeling epitaxial growth. By using the variational structure of the problem and differential inequalities, it is shown, under some conditions on the initial value, the solutions to the problem will blow up in finite time. Furthermore, the upper bound of the blow-up time for blowing-up solution is given. Moreover, the existence of a ground-state solution is obtained under approximate assumptions on the nonlinear term. The results of this paper extend and generalize some results got in the papers [G. A. Philippin, Blow-up phenomena for a class of fourth-order parabolic problems, Proceedings of the American Mathematical Society, 143(6): 2507-2513, 2015], [Y. Z. Han, A class of fourth-order parabolic equation with arbitrary initial energy, Nonlinear Anal. Real World Appl., 43: 451-466, 2018], and [J. Zhou, Global asymptotical behavior of solutions to a class of fourth order parabolic equation modeling epitaxial growth, Nonlinear Anal. Real World Appl., 48: 54-70, 2019].