A class of thin-film equations with logarithmic nonlinearity

Abstract

The main purpose of this paper is to study an initial-boundary value problem for a thin-film equation with logarithmic nonlinearity. Local existence of weak solutions is obtained by using Galerkin's approximation. By using the modified logarithmic Sobolev inequality and the potential well method, we obtain the existence of global solutions and solutions that blow up at infinity under some suitable conditions. The decay estimates of the global solutions are also derived.

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

“…Recently, there are some researches on the models of the fourth-order thin-film epitaxial growth with logarithmic nonlinearities in [13,14]. Mathematically, the problem with logarithmic nonlinearity is more difficult compared with power-like source, since the logarithmic nonlinearity does not satisfy monotonicity and may change signs.…”

“…Mathematically, the problem with logarithmic nonlinearity is more difficult compared with power-like source, since the logarithmic nonlinearity does not satisfy monotonicity and may change signs. For example, Han et al [13] considered the fourth-order parabolic equation with logarithmic nonlinearity: u t + Δ 2 u = u log |u|, (x, t) ∈ Ω × (0, +∞).…”