Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise

Fuzhi Li; Yangrong Li; Renhai Wang

doi:10.3934/dcds.2018158

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2018, Volume 38, Issue 7: 3663-3685. Doi: 10.3934/dcds.2018158

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Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: liyr@swu.edu.cn(Yangrong Li)
* Corresponding author: liyr@swu.edu.cn(Yangrong Li)

Received: November 2017

Published: July 2018

This work was supported by the Natural Science Foundation of China Grant 11571283.

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Abstract

This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the basic convergence. A bi-spatial attractor is obtained when the non-initial space is $p$-times Lebesgue space or Sobolev space. The measurability of the solution operator is proved, which leads to the measurability of the attractor in both state spaces. Finally, the upper semi-continuity of attractors under the $p$-norm is established when the narrow domain degenerates onto a lower dimensional set. Both methods of symbolical truncation and spectral decomposition provide all required uniform estimates in both Lebesgue and Sobolev spaces.

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Mathematics Subject Classification: Primary: 35B40; Secondary: 37L55, 60H15.

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