Global solvability and decay of the energy for the nonhomogeneous Kirchhoff equation
The authors consider the following damped Kirchhoff equation in a bounded domain \(\Omega\subset\mathbb R^n\): \[ \begin{aligned} &u_{tt}-\phi(x,t)M \left(\int_\Omega|\nabla u(\xi,t)|^2 d\xi\right)\Delta u+g(x,t,u_t(x,t)),\\ &u|_{\partial\Omega}=0, \quad u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x) \end{aligned} \tag{1} \] where \(\phi\), \(M\) and \(g\) are given functions. They proved that, under natural assumptions on functions \(\phi\), \(M\) and \(g\) (\(M(\lambda)\geq M_0>0\), \(\partial_s g(x,t,s)\geq\alpha(1+|s|^\rho)\) for some positive \(\alpha\) and \(\rho\), \(g(x,t,0)=0\), etc.), problem (1) posseses a global solution if the initial data \((u_0(x),u_1(x))\) is small enough. Moreover, they also established the exponential energy decay of this solution: \[ \|u'(t)\|_{L^2(\Omega)}^2+\|u(t)\|_{H^1_0(\Omega)}^2\leq Ce^{-Kt}, \quad K>0,\;t\geq 0 . \]
nonhomogeneous damped Kirchhoff equation
scientific article; zbMATH DE number 1839900