On a singular epitaxial thin‐film growth equation involving logarithmic nonlinearity

Abstract

This paper is concerned with the well‐posedness and asymptotic behavior for a singular epitaxial thin‐film growth equation with logarithmic nonlinearity under the Navier boundary condition. Based on the technique of cut‐off and combining with Hardy–Sobolev inequality, the technique of Faedo–Galerkin, and multiplier, we establish the local solvability. Meantime, by virtue of the family of potential wells, we obtain the threshold between the existence and nonexistence of the global solution (including the critical case) and give the upper bound of lifespan and the estimate of blow‐up rate. Furthermore, the results of blow‐up with arbitrary initial energy and the lifespan are derived.