Singular limit of some quasilinear wave equations with damping terms
Let \(\Omega\subset \mathbb{R}^3\) be a bounded domain with smooth boundary \(\partial\Omega\). The author considers the mixed problem for the quasilinear wave equation \[ \varepsilon u_{tt}- \Biggl(\alpha+ 2\beta \int_\Omega |\nabla u(t, y)|^2 dy\Biggr) \Delta u+ \delta u_t= \mu u^3,\quad t> 0,\quad x\in \Omega, \] \[ u(0, x)= u_{0\varepsilon}(x),\;\partial_t u(0, x)= u_{1\varepsilon}(x),\quad x\in \Omega,\quad u(t, x)|_{\partial\Omega}= 0,\quad t\geq 0. \] Here \(\varepsilon\) is a positive parameter with \(0< \varepsilon\leq 1\). \(\alpha> 0\), \(\beta\geq 0\), \(\delta> 0\) and \(\mu\) are given constants, \(u= u(t, x)\) is an unknown real-valued function on \([0, \infty)\times \Omega\). The author considers a relation between the mixed problem and the reduced problem of the parabolic equation for \(\varepsilon= 0\). It is shown that the solution \(u_\varepsilon\) of the mixed problem converges, uniformly on any finite time interval, to the solution of the parabolic equation in an appropriate Hilbert space as \(\varepsilon\to 0\). For the proof several \(\varepsilon\) weighted energy estimates are obtained in order to evaluate the difference norm of \(u_\varepsilon- u\).
scientific article; zbMATH DE number 844387