Profile of a touch-down solution to a nonlocal MEMS model

Abstract

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on (0, T ) × Ω :where Ω is a bounded domain in R n and λ, γ > 0. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point a ∈ Ω. In particular, we give a description of the quenching behavior according to the following final profile1 3 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.

Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term

“…where 1 1−u may blow up in finite time (see Duong and Zaag [6] and the references therein). Specific difficulties arise in the study of blow-up for both equations (1.4) and (1.5), as one may see from the constructions of singular solutions in Tayachi and Zaag [26] (see also the note [27]) and Duong and Zaag [6].…”

Refined asymptotics for the blow-up solution of the complex Ginzburg–Landau equation in the subcritical case

“…Recently, this method has been applied to various nonvariational parabolic systems in [27] and [13][14][15][16], and to a logarithmically perturbed nonlinear equation in [7][8][9]26]. We also mention a result for a higher-order parabolic equation [17] and in [1,11] two more results for equations involving nonlocal terms.…”

“…C .p 1/.1 C ı 2 /.1 ıˇ/ 2 C .p 3/.1 ı 2 /.1 ıˇ/ 2 p 1 s /, T 1 D Á‚ and T D Á‚. j .y/ D j X kD0 .R j;k h k C z R j;k Q h k /: Expansion of  0 .s/‚.y/ We introduce ‚.y; s/ D i ' 0 .y; s/ C a.1 C i ı/ p s Á ;where ' 0 and a are defined as in (12) and(11), respectively. Using Taylor expansion we write ‚.y; s/ D i Ä C Ä.Á :…”

A study of a nonlocal problem with Robin boundary conditions arising from technology

“…The quenching behavior of the nonlocal Equation () associated with Dirichlet boundary ( ) has been treated in Kavallaris et al 20 and in references therein as well as in previous works 21–23 . Also, non‐local alterations of parabolic and hyperbolic problems arising in MEMS technology were tackled in previous works 5,7,20–22,24,25 . However, to the best of our knowledge, there are not similar studies available in the literature for the Robin problem (0 < β < + ∞ ,) so in the current work we study problem (1.1) and we extend some of the results given in Guo 1 for the local problem, but we also deliver a further investigation related to the steady‐state problem and the quenching behavior of the time‐dependent problem.…”

“… for some positive constant C * . For a more rigorous approach, which is out of the scope of the current work, one should follow similar arguments as in 24,46 to derive () where it is conjectured that .…”