Research Article Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source Hazal Yüksekkaya, 1 Erhan Pișkin, 1 Salah Mahmoud Boulaaras , 2,3 Bahri Belkacem Cherif , 2,4 and Sulima Ahmed Zubair 2,5 1 Department of Mathematics, Dicle University, Diyarbakir, Turkey 2 Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia 3 Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, 31000 Oran, Algeria 4 Preparatory Institute for Engineering Studies in Sfax, Tunisia 5 Department of Mathematics, College of Education, Juba University, Sudan Correspondence should be addressed to Sulima Ahmed Zubair; sulimaa2021@gmail.com Received 12 May 2021; Accepted 16 June 2021; Published 28 June 2021 Academic Editor: Kamyar Hosseini Copyright © 2021 Hazal Yüksekkaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we study a plate equation with time delay in the velocity, frictional damping, and logarithmic source term. Firstly, we obtain the local and global existence of solutions by the logarithmic Sobolev inequality and the Faedo-Galerkin method. Moreover, we prove the stability and nonexistence results by the perturbed energy and potential well methods. 1. Introduction In this article, we consider a plate equation with frictional damping, delay, and logarithmic terms as follows: u tt + Δ 2 u + αu t t ðÞ + βu t x, t − τ ð Þ = u ln u jj γ for x, t ð Þ ∈ Ω × 0,∞ ð Þ, ux, t ð Þ = ∂ux, t ð Þ ∂υ =0 for x, t ð Þ ∈ ∂Ω × 0,∞ ð Þ, ux,0 ð Þ = u 0 x ðÞ, u t x,0 ð Þ = u 1 x ðÞ for x ∈ Ω, u t x, t ð Þ = j 0 x, t ð Þ for x, t ð Þ ∈ Ω × −τ,0 ð Þ, 8 > > > > > > < > > > > > > : ð1Þ where Ω ⊂ R N , N ≥ 1, is a bounded domain with smooth boundary ∂Ω. τ >0 denotes time delay, and α, β, and γ are real numbers that will be specified later. Generally, logarith- mic nonlinearity seems to be in supersymmetric field theories and in cosmological inflation. From quantum field theory, that kind of (ujuj p−2 ln juj k ) logarithmic source term seems to be in nuclear physics, inflation cosmology, geophysics, and optics (see [1, 2]). Time delays often appear in various problems, such as thermal, economic, biological, chemical, and physical phenomena. Recently, partial differential equations have become an active area with time delay (see [3, 4]). In 1986, Datko et al. [5] indicated that, in boundary control, a small delay effect is a source of instability. Gener- ally, a small delay can destabilize a system which is uniformly stable [6]. To stabilize hyperbolic systems with time delay, some control terms will be needed (see [7–9] and references therein). For the literature review, firstly, we begin with the studies of Bialynicki-Birula and Mycielski [10, 11]. The authors investigated the equation with the logarithmic term as follows: u tt − u xx + u − εu ln u jj 2 = 0, ð2Þ where the authors proved that, in any number of dimensions, wave equations including the logarithmic term have local- ized, stable, soliton-like solutions. In 1980, Cazenave and Haraux [12] studied the equation as follows: u tt − Δu = u ln u jj k , ð3Þ Hindawi Advances in Mathematical Physics Volume 2021, Article ID 8561626, 11 pages https://doi.org/10.1155/2021/8561626