The Planar Coleman--Gurtin model with Beltrami conductivity

This article addresses the planar Coleman–Gurtin heat equation with memory on a bounded domain, with rough anisotropic diffusion $A_{\mu }$, typical of heterogeneous or composite media and encoded by a Beltrami coefficient $\mu \in L^{\mathrm{\infty }}(\mathrm{\Omega })$ satisfying $\mathrm{\parallel }\mu {\mathrm{\parallel }}_{\mathrm{\infty }}<1$. First, under no additional smoothness assumptions on $\mu$, solutions with $H_0^1(\mathrm{\Omega })$-based initial data enter a time-averaged $L^{\mathrm{\infty }}(\mathrm{\Omega })$ regime, and instantaneously regularize into the second-order graph space $D(A_{\mu })$. Assuming in addition $\mu \in W^{1,2}(\mathrm{\Omega })$, this regularization upgrades to $W^{2,p}(\mathrm{\Omega })$ for every $1<p<2$, and we construct regular global and exponential attractors of finite fractal dimension, for both the $L^2(\mathrm{\Omega })$ and $H_0^1(\mathrm{\Omega })$-based dynamics. The proof combines the instantaneous smoothing method of Chekroun, Di Plinio, Glatt-Holtz and Pata with maximal parabolic regularity for divergence-form operators with measurable coefficients, and with planar quasiconformal Beltrami estimates recently obtained in work by Green, Wick and the author.