Fisher information, new functional inequalities and mechanics of a heated string

I will review the results of recent three articles (first two with P. Bies, the third one with J. Jendrej and C. Stinner) on mechanics of a heated string. The PDE system describing such a phenomenon is a particular case of 1D thermoelasticity system. So it is a nonlinear mixed type problem (coupled wave and heat equations). After the introduction of the problem, I will first show a new type functional leading to global-in-time existence and uniqueness of regular enough solutions. It makes use of a recent functional inequality. Next, I'm going to describe the asymptotic behavior of solutions (convergence to one particular steady state) and its physics interpretation. Last, but not least, I'll calculate the exponential rate of convergence of solutions to the steady state. At the end, I'll mention the so far known higher-dimensional results and some open problems.