Sharp large time asymptotic behavior for the multi-dimensional thermoelastic systems of type II and type III
Wenhui Chen, Ryo Ikehata
TL;DR
This work analyzes sharp, large-time behavior of the multi-dimensional thermoelastic systems of Green–Naghdi type II and type III in the whole space, without relying on thermal displacement transforms. By performing Fourier-space reductions and introducing double diffusion wave (type III) and double wave (type II) profiles, the authors derive precise upper and lower bounds, identify critical dimensions, and describe when solutions exhibit growth, decay, or blow-up. Key contributions include the novel double diffusion wave framework, the detailed frequency-space analysis across short, middle, and large frequencies, and the explicit characterization of u and θ via the profiles φ, ψ (and their type II analogs), with data in L1 or L1∩L2 spotlighting optimal rates. The results illuminate the competition between elastic waves and hyperbolic thermal effects, explaining dimensional thresholds and providing sharp asymptotics that extend prior energy-based decay results to the solutions themselves, with implications for understanding hyperbolic heat conduction in thermoelastic media.
Abstract
In this paper, we study large time asymptotic behavior of the elastic displacement $u$ and the temperature difference $θ$ for the thermoelastic systems of type II and type III in the whole space $\mathbb{R}^n$ without using the thermal displacement transformation. For the type III model with hyperbolic thermal effect, we derive optimal growth/decay estimates and the novel double diffusion waves profiles for the solutions $u,θ$ with the $L^1$ integrable initial data as large time, which improve the results in [32,26,31,17]. This hyperbolic thermal law produces a stronger singularity than the classical Fourier law in thermoelastic systems. For the type II model lacking of dissipation mechanism, we obtain optimal growth estimates and the new double waves profiles for the solutions $u,θ$ as large time in low dimensions. Particularly, these results on the type II/III models show the infinite time blowup phenomena for $u$ if $n\leqslant 4$ and $θ$ if $n\leqslant 2$ in the $L^2$ norm due to the hyperbolic influence. We also clarify competitions between the elastic waves effect and the hyperbolic thermal effect for the thermoelastic systems via the critical dimensions.