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On the quenching behaviour of a semilinear wave equation modelling MEMS technology

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  • In this work we study the semilinear wave equation of the form \[ u_{tt}=u_{xx} + {\lambda}/{ (1-u)^2}, \] 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 $||u(\cdot,t)||_{\infty}\to 1$ as $t\to t^*- < \infty$, 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.
    Mathematics Subject Classification: Primary: 35L81, 35J60; Secondary: 74H35, 74G55, 74K15.


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