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

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems - Series A following peer review. The definitive publisher-authenticated version 2015, 35(3), pp. 1009-1037 is available online at: http://dx.doi.org/10.3934/dcds.2015.35.100

Hyperbolic quenching problem with damping in the micro-electro mechanical system device

[[abstract]]We study the initial boundary value problem for the damped hyperbolic equation arising in the micro-electro mechanical system device with local or nonlocal singular nonlinearity. For both cases, we provide some criteria for quenching and global existence of the solution. We also derive the existence of the quenching curve for the corresponding Cauchy problem with local source[[notice]]補正完畢[[journaltype]]國外[[incitationindex]]SCI[[ispeerreviewed]]Y[[countrycodes]]US

The Role of Critical Exponents in Blowup Theorems

In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation $u_t = \Delta u + u^p$ in $R^N$ with $p \u3e 1$ and nonnegative initial values. Fujita showed that if $1 \u3c p \u3c 1 + {2 / N}$, then the initial value problem had no nontrivial global solutions while if $p \u3e 1 + {2 / N}$, there were nontrivial global solutions. This paper discusses similar results for other geometries and other equations including a nonlinear wave equation and a nonlinear Schrödinger equation

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

From Wiley via Jisc Publications RouterHistory: received 2020-08-03, rev-recd 2021-02-18, accepted 2021-02-22, pub-electronic 2021-05-04Article version: VoRPublication status: PublishedIn the current work, we study a nonlocal parabolic problem with Robin boundary conditions. The problem arises from the study of an idealized electrically actuated MEMS (micro‐electro‐mechanical system) device, when the ends of the device are attached or pinned to a cantilever. Initially, the steady‐state problem is investigated estimates of the pull‐in voltage are derived. In particular, a Pohožaev's type identity is also obtained, which then facilitates the derivation of an estimate of the pull‐in voltage for radially symmetric N‐dimensional domains. Next a detailed study of the time‐dependent problem is delivered and global‐in‐time as well as quenching results are obtained for generic and radially symmetric domains. The current work closes with a numerical investigation of the presented nonlocal model via an adaptive numerical method. Various numerical experiments are presented, verifying the previously derived analytical results as well as providing new insights on the qualitative behavior of the studied nonlocal model

Dynamics of Waves and Patterns (hybrid meeting)

The dynamics of waves and patterns play a significant role in the sciences, especially in fluid mechanics, material science, neuroscience and ecology. The mathematical treatment interconnects several areas, ranging from evolution equations and functional analysis to dynamical systems, geometry, topology, and stochastic as well as numerical analysis. This workshop has specifically focussed on dynamic stability on extended domains, bifurcations of waves and patterns, effects of stochastic driving, and spatio-temporal inhomogenities. During the workshop, multiple new directions, collaborations, and very interesting scientific conversations arose across the entire field