On a quasilinear parabolic-hyperbolic system arising in MEMS modeling (Q6624154)

Let \((a,b,\theta_1,\theta_2)\) be positive parameters and \(\sigma\ge 0\). The local well-posedness of the initial boundary value problem\N\begin{align*}\N\partial_t (wu) & = \mathrm{div}\left( w^3 u \nabla u \right), \qquad t>0, \ x\in\Omega, \\\N\partial_t^2 w + \sigma \partial_t w & = \Delta w - aw^{-2} + b(u-1), \qquad t>0, \ x\in\Omega, \\\N(u,w) & = (\theta_1,\theta_2), \qquad t>0, \ x\in\partial\Omega,\N\end{align*}\Nis established when \(\Omega\) is a bounded domain of \(\mathbb{R}^n\), \(n\in \{1,2\}\), and the initial conditions \((u,w,\partial_t w)(0) = (u_0,w_0,w_0')\) satisfy \(u_0\in H^{r+2}(\Omega)\) for some \(r>0\), \(w_0\in H^2(\Omega)\cap C^1(\bar{\Omega})\), \(w_0'\in H^1(\Omega)\) with \(u_0>0\) and \(w_0>0\) in \(\Omega\) and \(u_0-\theta_1 = w_0-\theta_2 = w_0'=0\) on \(\partial\Omega\). In addition, the solution can be continued as long as \(w(t)\) and \(u(t)\) remain positive in \(\bar{\Omega}\) and the \(C^1\)-norm of \((u,v)(t)\) does not blow up.\N\NThe above system describes the dynamics of an electrostatically actuated microelectromechanical system in the small aspect ratio limit, where the narrow gap of width \(w>0\) separating the two plates of the device is filled with a rarefied gas with pressure \(u\ge 0\). It couples a quasilinear parabolic equation which degenerates as either \(u\) or \(w\) vanish with a possibly damped wave equation featuring a reaction term involving a singularity when \(w\) vanishes. The above system is actually included in a more general framework for which well-posedness is derived by combining the semigroup theory for quasilinear parabolic and semilinear hyperbolic equations.