A sharper decay estimate for the quasilinear wave equation with viscosity in two space dimensions (Q5933771)

This paper is devoted to a decay property of solutions of the quasilinear wave equation with a strong dissipation: NEWLINE\[NEWLINEu_{tt}- \text{div}\{\sigma(|\nabla u|^2)\nabla u\}- \Delta u_t= 0\quad\text{in }\Omega\times (0,\infty)NEWLINE\]NEWLINE with \(u(x,0)= u_0(x)\), \(u_t(x, 0)= u_1(x)\) and \(u|_{\partial\Omega}= 0\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) with a \(C^2\) class boundary \(\partial\Omega\), and \(\sigma\) is a nonlinear function line \(\sigma(w)= (\sqrt{1+ w^2})^{-1}\). Let NEWLINE\[NEWLINEE(t):= {1\over 2} \Biggl(\|u_t\|^2+ \int_\Omega \int^{|\nabla u(t)|^2}_0 \sigma(\eta) d\eta dx\Biggr).NEWLINE\]NEWLINE The author presents a sharper decay estimate of \(E(t)\), that is \(E(t)\leq C_0 e^{-\lambda t^{2/3}}\) with some \(\lambda> 0\).