Long-time dynamics of Kirchhoff equations with exponential nonlinearities

Abstract

Our aim in this paper is to study the initial boundary problem for the two-dimensional Kirchhoff type wave equation with an exponentially growing source term. We first prove that the Kirchhoff wave model is globally well-posed in (H01(Ω)∩L∞(Ω))×L2(Ω), which covers the case of degenerate stiffness coefficient, and then obtain that the semigroup generated by the problem has a global attractor in the corresponding phase space. We also point out that the above results are still true in the natural energy space H01(Ω)×L2(Ω).

Initial boundary value problem for a Kirchhoff wave model with strong nonlinear damping

“…Ma, Chen and Xie [19] considered the Kirchhoff‐type equation with strong damping: $$$$ \left\{\begin{array}{ll}{u}_{tt}-K\left({\left\Vert \nabla u\right\Vert}_2^2\right)\Delta u-\Delta {u}_t=f(u)+h(x),& x\in \Omega, t>0,\\ {}u\left(x,t\right)=0,& x\in \mathrm{\partial \Omega },t>0,\\ {}u\left(x,0\right)={u}_0(x),{u}_t\left(x,0\right)={u}_1(x),& x\in \Omega, \end{array}\right. $$$$ where $$$ \Omega \subset {\mathrm{\mathbb{R}}}^2 $$$ is a bounded domain with smooth boundary …”

Global existence and energy decay for a coupled system of Kirchhoff beam equations with weakly damping and logarithmic source

“…In the sequel, we mention some of them. The initial boundary problem for the two-dimensional Kirchhoff-type wave equation with an exponentially growing source term was presented in [25]. Existence, decay and blow up of solutions for the extensible beam equation with nonlinear damping and source terms was presented in [35].…”