Asymptotic and quenching behaviors of semilinear parabolic systems with singular nonlinearities

Qi Wang; Yanyan Zhang

doi:10.3934/cpaa.2021199

Article Contents

2022, Volume 21, Issue 3: 797-816. Doi: 10.3934/cpaa.2021199

This issue Previous Article A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem Next Article On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations

Asymptotic and quenching behaviors of semilinear parabolic systems with singular nonlinearities

Qi Wang^1, and
Yanyan Zhang^2, ,

1.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2.
School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China

^*Corresponding author
^*Corresponding author

Received: July 31, 2021

Revised: September 30, 2021

Early access: December 2021

Published: March 2022

The corresponding author is sponsored by "Chenguang Program" supported by Shanghai Educational Development Foundation and Shanghai Municipal Education Commission [grant number: 13CG20]; NSFC [grant number: 11431005]; and STCSM [grant number: 18dz2271000]

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Abstract

In this paper, we consider a family of parabolic systems with singular nonlinearities. We study the classification of global existence and quenching of solutions according to parameters and initial data. Furthermore, the rate of the convergence of the global solutions to the minimal steady state is given. Due to the lack of variational characterization of the first eigenvalue to the linearized elliptic problem associated with our parabolic system, some new ideas and techniques are introduced.

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Mathematics Subject Classification: Primary: 35B40, 35K51, 35K58, 35A01, 35B44; Secondary: 53C35.

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Figure 1. The critical curve $ \Gamma $ in $ ( \lambda,\mu) $-plane

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