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Dynamic properties for nonlinear viscoelastic Kirchhoff-type equation with acoustic control boundary conditions II
Fushan Li,Shuai Xi,Ke Xu,Xiaominn Xue
Keywords:Kirchhoff-type equation, Acoustic boundary condition, Blow-up
Abstract:
      In this paper, we consider the nonlinear viscoelastic Kirchhoff-type equation \begin{align*} u_{tt}-M(\|\nabla u\|^{2}_{2})\Delta u+\int_0^t h(t-s)\Delta u(s)ds+a|u_t|^{m-2}u_t=|u|^{p-2}u \end{align*} with initial conditions and acoustic boundary conditions. Under suitable conditions on the initial data, the relaxation function $h(\cdot)$ and $M(\cdot)$, we prove that the solution blows up in finite time and give the upper bound of the blow-up time $T^*$.