On the Quenching Behaviour of a Semilinear Wave Equation Modelling MEMS Technology

In this work we study the semilinear wave equation of the form
utt = uxx + lambda/(1 − u)2;
with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any lambda > 0: Then we focus on the singular behaviour of the solution, which occurs through
finite-time quenching, i.e. when ||u(·; t)||1 → 1 as t →t*− < ∞, investigating both conditions for quenching and the quenching profile of u: To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.