In my talk I shall review recent results obtained in 2 papers,
First one in collaboration with M. Fuest, K.Hajduk and M. Sierzega and
a second one with P. Bies. I'm going to show a recently found functional
inequality, which seems promising when dealing with systems of PDEs via
the Fisher information method. In particular, I will show a result
obtained with P.Bies, stating global-in-time existence of unique regular
solutions to the system describing the evolution of the heated string.
It is a thermodynamically consistent combination of two very classical
equations: heat equation and string equation. Moreover, it's a
particular example of the basic problem of thermoelasticity. Still, till
our result the question of global existence was opened. Finally, at the
end I will mention a recent result applying our method to the 1D
combustion problem, a different area of mathematical physics.