Blow up phenomenon for a plate equation with logarithmic source term

We deal with the following plate equation with logarithmic non-linearity term: 𝑢tt+Δ2⁢𝑢+|𝑢𝑡|𝑚−2𝑢𝑡=|𝑢|𝑝−2𝑢⁢ln⁡|𝑢|𝑘,𝑥∈Ω,𝑡>0,where Ω⊂ℝ𝑛⁢(𝑛≥1) is a bounded domain with smooth boundary and k is some positive constant and 2≤𝑚<𝑝. In this paper, inspired by Kani and Messaoudi [Local existence and blow up of solutions to a logarithmic non-linear wave equation with delay. Appl Anal. 2019;99:530547], we present a preliminary lemma. Making use of this lemma, we obtain the blow up result without any upper restriction on p. To the best of our knowledge, this is the first result obtained without any upper bound on p for this type of equations. This process can be used in other equations with logarithmic term. In fact, for any 𝑝>𝑚≥2, we prove the blow up occurs in finite time in the cases of 𝐸⁡(0)<0 and 𝐸⁡(0)=0.