Global existence and asymptotic behavior for a viscoelastic Kirchhoff equation with a logarithmic nonlinearity, distributed delay and Balakrishnan-Taylor damping terms

Abdelbaki Choucha1,2, Salah Boulaaras3,∗ and Asma Alharbi3 1 Laboratory of Operator Theory and PDEs: Foundations and Applications, Department of Mathematics, Faculty of Exact Sciences, University of El Oued, Algeria 2 Department of Mathematics, Faculty of Sciences, Amar Teledji Laghouat University, Algeria 3 Department of Mathematics, College of Sciences and Arts, ArRass, Qassim University, Saudi Arabia

Physically, the relationship between the stress and strain history in the beam inspired by Boltzmann theory called viscoelastic damping term, where the kernel of the term of memory is the function h (See [8,13,[15][16][17][18][19][20][21][22]25]. In [3], Balakrishnan and Taylor they proposed a new model of damping called it the Balakrishnan-Taylor damping , as it relates to the span problem and the plate equation. For more depth, here are some papers that focused on the study of this damping [3,6,10,16,30].
The effect of the delay often appear in many applications and piratical problems and turns a lot of systems into different problems worth studying. Recently, the stability and the asymptotic behavior of evolution systems with time delay especially the distributed delay effect has been studied by many authors [1, 9, 12-14, 24, 25, 27-29, 31, 32, 34]. The great importance of the logarithmic nonlinearity in physics is that they appear in several issues and theories, including symmetry, cosmology, quantum mechanics, as well as nuclear physics. It is also used in many applications such as optical, nuclear and even subterranean physics. Many researchers also touched on this type of problem in several different issues, where the global existence of solutions, stability and blow-up of solutions were studied. For more information, the reader is referred to [4,5,7].
Based on all of the above, the combination of these terms of damping (Memory term, Balakrishnan-Taylor damping, logarithmic nonlinearity and the distributed delay terms ) in one particular problem with the addition of the distributed delay term ( constitutes a new problem worthy of study and research, different from the above that we will try to shed light on.
Our paper is divided into several sections: in the next section we lay down the hypotheses, concepts and lemmas we need. In the section 3, we state the global existence and in the section 4, we prove the general decay of solutions. Finally, we put a general conclusion.
Lemma 2. The energy functional E, defined by Proof. Taking the inner product of (1.7) 1 with u t , then integrating over Ω, we find (1.10) A calculation direct, gives by integration by parts, we find and we have and − ∇u(x, ).∇u(x, t) = 1 2 We use (1.2), we obtain By substitying (1.17) and (1.

Global existence
In this section, under smallness condition the global existence result is proved. Introducing the following functionals and Hence and and we define

Conclusions
The purpose of this work was to study the global existence of the solutions for a nonlinear viscoelastic Kirchhoff-type equation with a logarithmic nonlinearity, Balakrishnan-Taylor damping, dispersion and distributed delay terms, and by the energy method we prove an explicit and general decay rate result under suitable hypothesis. This type of problem is frequently found in some mathematical models in applied sciences.
In the next work, we will try to using the same method with same problem. But in added of other damping terms.