Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 21, pp. 1-14.

Title: Solutions of Kirchhoff plate equations with internal damping and logarithmic nonlinearity
       
Authors: Ducival Pereira (State Univ. of Para, Belem, PA, Brazil)
         Sebastiao Cordeiro (Federal Univ. of Para, Abaetetuba, PA, Brazil)
         Carlos Raposo  (Federal Univ.of Sao Joao del-Rei, MG, Brazil)
         Celsa Maranhao (Federal Univ. of Para, Belem, PA, Brazil)

Abstract:
 In this article we study the existence of weak solutions
 for the nonlinear initial boundary value problem of the Kirchhoff equation
 $$\displaylines{
 u_{tt}+\Delta^2 u + M(\|\nabla u\|^2)(-\Delta u) +  u_{t}
  =  u \ln |u|^2,\text{ in }\Omega\times (0,T), \cr
 u(x,0) = u_0(x), \quad u_{t}(x,0)=u_1(x),\quad x \in \Omega, \cr
 u(x,t) = \frac{\partial u}{\partial \eta}(x,t)=0, \quad x \in \partial \Omega,\; t\geq 0,
 }$$
 where &Omega; is a bounded domain in R<sup>2</sup> with smooth boundary
 $\partial\Omega$, T>0 is a fixed but arbitrary real number,
 M(s) is a continuous function on $[0,+\infty)$ and &eta; is the unit outward
 normal on $\partial\Omega$.
 Our results are obtained using the Galerkin method, compactness approach,
 potential well corresponding to the logarithmic nonlinearity, and the energy estimates
 due to Nakao.

Submitted May 19, 2020. Published March 29, 2021.
Math Subject Classifications: 35L15, 35L70, 35B40, 35A01
Key Words: Extensible beam; existence of solutions; asymptotic behavior; logarithmic source term.