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

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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.