Blow-up at infinity of solutions to a class of pseudo-parabolic equations with logarithmic nonlinearity and singular potential

Abstract

We study the initial–boundary value problem for a class of pseudo-parabolic equations in a bounded domain Ω with smooth boundary ∂Ω and 0∈Ω⊂BR(0)⊂Rn(n≥1), where BR(0) is a ball centered at 0 with radius R. First, we simplify the scheme of potential well method, which helps us obtain the invariant set in an easier way. Combining logarithmic Sobolev inequality and Hardy inequality, we prove that the solutions to the pseudo-parabolic equations with logarithmic nonlinearity ulog|u| blow up at infinity when R≤1. However, there is no restriction on R if the singular potential is 1. Second, when the singular potential is 1 and the pseudo-parabolic term vanishes, the equations model the epitaxial growth of thin films. We also show that the solutions have better growth rate under weaker conditions, which improves the result in Han et al. (2019).