Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source

Hazal Yüksekkaya,Bahri Belkacem Cherif,Salah Mahmoud Boulaaras,Erhan Pișkin,Sulima Ahmed Zubair,Kamyar Hosseini

doi:10.1155/2021/8561626

Abstract

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.

Highlights

In this article, we consider a plate equation with frictional damping, delay, and logarithmic terms as follows:8 >>>>>>< >>>>>>: utt + Δ2u + αutðtÞ + βutðx, t − τÞ uðx, tÞ =∂uðx, ∂υ uðx, 0Þ = u0ðxÞ, utðx, 0Þ = u1ðxÞ u ln jujγ utðx, tÞ = j0ðx, tÞ for ðx, tÞ ∈ Ω × ð0,∞Þ, for ðx, tÞ ∈ ∂Ω × ð0,∞Þ, for x ∈ Ω, for ðx, tÞ ∈ Ω × ð−τ, 0Þ, ð1Þ where Ω ⊂ RN, 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
Logarithmic nonlinearity seems to be in supersymmetric field theories and in cosmological inflation
That kind of logarithmic source term seems to be in nuclear physics, inflation cosmology, geophysics, and optics

Summary

Introduction

We consider a plate equation with frictional damping, delay, and logarithmic terms as follows:. In [15], Al-Gharabli and Messaoudi were concerned with the plate equation with the logarithmic term as follows: utt + Δ2u + u + hðutÞ = ku ln juj: ð5Þ They established the existence results by the Galerkin method and obtained the explicit and decay of solutions utilizing the multiplier method for equation (5). In [17], Messaoudi studied the equation as follows: utt + Δ2u + jutjm−2ut = jujp−2u, ð7Þ and obtained the existence results and obtained that, if m ≥ p, the solution is global and blows up in finite time if m < p. We studied the local existence, global existence, nonexistence, and stability results of plate equation (1) with delay and logarithmic terms, motivated by the above works.

Preliminaries

Existence

Nonexistence

Stability

Conclusions