On a general family of nonautonomous elliptic and parabolic equations

Dong Ye; Feng Zhou

doi:10.1007/s00526-009-0262-1

On a general family of nonautonomous elliptic and parabolic equations

Dong Ye, Feng Zhou

School of Mathematical Sciences

Université de Lorraine

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

We consider the parabolic equation u_t - Δu = λf (x)g(u) as well as the corresponding elliptic problem, with a nonnegative profile f and a positive nondecreasing convex function g verifying lim_{u→ 1}-g(u) = ∞. Our study is motivated by a simplified Micro-Electromechanical Systems (MEMS) device model. We extend or improve many qualitative and quantitative results for the MEMS modeling to this very general setting, which help us to understand more about the influence of f on the pull-in voltage λ* and the quenching phenomenon. Especially, we show some new estimates for λ* and the quenching time T.

Original language	English
Pages (from-to)	259-274
Number of pages	16
Journal	Calculus of Variations and Partial Differential Equations
Volume	37
Issue number	1
DOIs	https://doi.org/10.1007/s00526-009-0262-1
State	Published - Nov 2009

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abstract = "We consider the parabolic equation ut - Δu = λf (x)g(u) as well as the corresponding elliptic problem, with a nonnegative profile f and a positive nondecreasing convex function g verifying limu→ 1-g(u) = ∞. Our study is motivated by a simplified Micro-Electromechanical Systems (MEMS) device model. We extend or improve many qualitative and quantitative results for the MEMS modeling to this very general setting, which help us to understand more about the influence of f on the pull-in voltage λ* and the quenching phenomenon. Especially, we show some new estimates for λ* and the quenching time T.",

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AB - We consider the parabolic equation ut - Δu = λf (x)g(u) as well as the corresponding elliptic problem, with a nonnegative profile f and a positive nondecreasing convex function g verifying limu→ 1-g(u) = ∞. Our study is motivated by a simplified Micro-Electromechanical Systems (MEMS) device model. We extend or improve many qualitative and quantitative results for the MEMS modeling to this very general setting, which help us to understand more about the influence of f on the pull-in voltage λ* and the quenching phenomenon. Especially, we show some new estimates for λ* and the quenching time T.

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On a general family of nonautonomous elliptic and parabolic equations

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