Asymptotic and Quenching Behavior for a family of Parabolic System with General Singular Nonlinearities

TL;DR

This paper investigates the long-term behavior and quenching phenomena of a family of parabolic systems with singular nonlinearities, extending MEMS models and introducing new analytical techniques for convergence rate analysis.

Contribution

It generalizes MEMS systems to a broader class of parabolic systems with singular nonlinearities and develops novel methods to determine convergence rates without variational characterization.

Findings

Classification of global existence and quenching based on parameters and initial data

Derived convergence rates for the parabolic system

Introduced new analytical techniques for systems without variational eigenvalue characterization

Abstract

This study is concerned with a family of parabolic system with general singular nonlinearities, which is a generalization of MEMS system. To some extent, the classification of global existence and quenching according to parameters and initial data is given. Moreover, the convergence rate is also obtained. We point out that compared to single MEMS equation, new ideas and techniques are introduced in obtaining the convergence rate for system in our study. In fact, due to the lack of variational characterization for the first eigenvalue of the linearized elliptic system, the methods in obtaining convergence rate for single equation cannot work completely here.