Existence and decay results for a von Karman equation with variable exponent nonlinearities

Abstract

In this article, we consider a von Karman equation with variable exponent nonlinearities wtt(x,t)+Δ2w(x,t)+|wt(x,t)|γ(x)−2wt(x,t)=[w(x,t),F(w(x,t))]+|w(x,t)|p(x)−2w(x,t) in a bounded domain normalΩ⊂ℝ2. We firstly discuss an existence result of solutions by utilizing Faedo‐Galerkin approximation technique. Then, we undertake an investigation of asymptotic stability to the solutions by making use of the multiplier method.

Timoshenko beams with variable‐exponent nonlinearity

“…Under appropriate conditions on the functions a, b, p, 𝜎, the local, global, and blow up solutions have been discussed. We also refer to earlier studies [9][10][11][12][13] for more existence and blow up results. For the stability, Messaoudi and Talahmeh 14 looked at…”

“…Under appropriate conditions on the functions $$$ a,b,p,\sigma, $$$ the local, global, and blow up solutions have been discussed. We also refer to earlier studies 9–13 for more existence and blow up results. For the stability, Messaoudi and Talahmeh 14 looked at $$$$ {u}_{tt}-\Delta u+\alpha {\left|{u}_t\right|}^{m(x)-1}{u}_t=0, $$$$ with $$$ \alpha \equiv 1 $$$ and $$$ m(x)\ge 2 $$$ and proved decay estimates for the solution under suitable assumptions on the variable exponent $$$ m $$$.…”