No touchdown at points of small permittivity and nontrivial touchdown sets for the MEMS problem (Q2178052)

The authors consider a smooth and bounded domain \(\Omega \) in \(\mathbb{R}^{n}\), \(n\geq 1\), and the parabolic problem \(u_{t}-\Delta u=f(x)(1-u)^{p}\), posed in \((0,\infty )\times \Omega \), where \(p>1\) and \(f\) is non-negative and Hölder continuous on \(\overline{\Omega }\). The homogeneous Dirichlet boundary condition \(u=0\) is imposed on \((0,\infty )\times \partial\Omega \) and the solution starts from 0 at \(t=0\). When \(p=2\), this problem models the behavior of micro-electromechanical devices: \(u=u(t,x)\) represents the vertical deflection of the upper plate and \(f\) represents the dielectric permittivity of the material. This problem admits a unique maximal classical solution, whose maximal existence time is denoted by \(T=T_{f}\in (0,\infty ]\). The authors also recall that, under some largeness assumption on \(f\), the maximum of \(u\) reaches the value 1 at a finite time, which is called quenching phenomenon. The present paper focuses on the notion of touchdown point defined in the following way. A point \(x=x_{0}\in \overline{\Omega }\) is called a touchdown or quenching point if there exists a sequence \((t_{n},x_{n})\in (0,T)\times \Omega \) satisfying \(x_{n}\rightarrow x_{0}\), \(t_{n}\uparrow T\) and \(u(x_{n},t_{n})\rightarrow 1\) as \(n\rightarrow \infty \). The set of all such points is closed and is called the touchdown or quenching set and is denoted by \(\mathcal{T}=\mathcal{T}_{f}\). The first main result of the paper proves that, if \(T_{f}\leq M\), \(\left\Vert f\right\Vert _{\infty }\leq M\), \(f\geq r\chi _{B}\), with \(M,r>0\) and \(B\) is a ball of radius \(r\), there exists \(\gamma _{0}>0\) depending only on \(p,M,r\) such that, for any \(x_{0}\in \Omega \), if \(f(x_{0})<\gamma _{0}\delta ^{p+1}(x_{0})\), where \(\delta (x_{0})=\mathrm{dist}(x_{0},\partial \Omega )\), then \(x_{0}\notin \mathcal{T}_{f}\). The second main result proves the non-existence of touchdown points close to the boundary of \(\Omega \) under a quite similar smallness hypothesis on the permissivity \(f\). Assuming now that the permissivity \(f\) has an ``M''-shaped profile, that is, \(f\) is radially symmetric, nondecreasing in \(\left\vert x\right\vert \) on \([0,L]\) and nonincreasing in \(\left\vert x\right\vert \) on \([L,R]\) for some \(L\in (0,R)\), the authors deduce from the preceding results further characterizations of touchdown points or set. They also prove a stability result under smooth and small perturbations of the permissivity. If \(f\) is radially symmetric and nonincreasing, the authors prove a stability result for radially symmetric small perturbations of the permissivity. The authors then prove that the touchdown set can be concentrated near two arbitrarily given points. In the case when \(\Omega \) is a ball, they construct radially symmetric profiles for which the touchdown set is concentrated near two arbitrarily given \((n-1)\)-dimensional spheres. In their last main result, they prove the continuity of the touchdown time and an upper semi-continuity property of the touchdown set, with respect to the permissivity. For the proof of these results, the authors first establish estimates on the touchdown time. They then introduce the function \(J(t,x)=u_{t}-a(x)((1-u)^{-p}+1)\), where \(a\in C^{2}(\overline{\Omega })\) is a positive function to be chosen to obtain estimates on \(a\) in terms of the radius of a ball \(B\), but independent of its location. They also use parabolic maximum principle and comparison arguments among other tools.