Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions

Klemens Fellner; Stefanie Sonner; Bao Quoc Tang; Do Duc Thuan

doi:10.3934/dcdsb.2019050

Article Contents

2019, Volume 24, Issue 8: 4055-4078. Doi: 10.3934/dcdsb.2019050

This issue Previous Article Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation Next Article Detecting coupling directions with transcript mutual information: A comparative study

Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions

1.
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
2.
Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
3.
Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
4.
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai Co Viet No. 1, Hanoi, Vietnam

^* Corresponding author: Stefanie Sonner
^* Corresponding author: Stefanie Sonner

Received: February 28, 2018

Revised: July 31, 2018

Early access: February 2019

Published: July 2019

Our research was supported by the ASEAN-European Academic University Network (ASEA-UNINET), and partially supported by NAWI Graz, IGDK 1754, and NAFOSTED project 101.01-2017.302

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Abstract

The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative Itô noise is studied. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the trivial steady state. This differs from previous works on the stabilisation by noise of parabolic PDEs, where the noise acts inside the domain and stabilisation typically occurs for an infinite range of noise intensities. To the best of our knowledge, this is the first result on the stabilisation of PDEs by boundary noise.

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Mathematics Subject Classification: Primary: 34H15, 35K57; Secondary: 35P15, 35R60.

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