Piotr Michał Bies

We study a simplified nonlinear thermoelasticity model on two- and three-dimensional tori. A novel functional involving the Fisher information associated with temperature is introduced, extending the previous one-dimensional approach from the first two authors [ SIAM J. Math. Anal., 55 (2023), pp. 7024–7038] to higher dimensions. Using this functional, we prove global/local existence of unique regular solutions for small/large initial data. Furthermore, we analyze the asymptotic behavior as time approaches infinity and show that the temperature stabilizes to a constant state, while the displacement naturally decomposes into two distinct components: a divergence-free part oscillating indefinitely according to a homogeneous wave equation and a curl-free part converging to zero. Analogous results for the Lamé operator are also stated.

We consider conditions for the decay in time of solutions of nonhomogenous hyperbolic equations. It is proven that solutions of the equations go to 0 in at infinity if and only if an equation's right‐hand side uniquely determines the initial conditions in a certain way. We also obtain that a hyperbolic equation has a unique solution that vanishes when .

We consider conditions for the decay in time of solutions of non-homogenous hyperbolic equations. It is proven that solutions of the equations go to 0 in L 2 at infinity if and only if an equation’s right-hand side uniquely determines the initial conditions in a certain way. We also obtain that a hyperbolic equation has a unique solution that fades when t →∞.

We construct a unique regular solution to the minimal nonlinear system of one-dimensional thermoelasticity. The obtained solution has a positive temperature. Our approach is based on an estimate, using the Fisher information, which seems completely new in this context. It is combined with a recent inequality in T. Cieślak, M. Fuest, K. Hajduk, and M. Sierżȩga, On Existence of Global Solutions for the 3 D Chemorepulsion System, preprint, arXiv:2303.09620 , 2023. and embeddings, which allow us to obtain a new energy estimate. The latter is used in a half-Galerkin procedure yielding global solutions. The uniqueness and further regularity of such solutions are obtained.

We construct unique regular solutions to the minimal nonlinear system of the 1d thermoelasticity. The obtained solution has a positive temperature. Our approach is based on an estimate, using the Fisher information, which seems completely new in this context. It is combined with a recent inequality in [2] and embeddings, which allows us to obtain a new energy estimate. The latter is used in a half-Galerkin procedure yielding global solutions. The uniqueness and further regularity of such solutions are obtained.