Initial-boundary value problem in nonlinear hyperbolic thermoelasticity. Some applications in continuum mechanics
The aim of this paper is to present an elementary self-contained introduction to some important aspects of the theory of local (in time) solutions to the initial-boundary value problem for nonlinear hyperbolic equations of thermoelasticity theory. The relevant existence theorem is proved using the approach of Kato via semigroup theory for the associated linearized problem. Next, we prove an energy estimate in a suitably chosen Sobolev space for the solution of the linearized problem, using standard regularization arguments and energy methods. Finally, we show that the solution of our nonlinear problem can be obtained as the unique fixed point of a~contraction mapping in a suitable function space. The approach presented in this paper can be extended to other nonlinear systems of partial differential equations describing media in continuum mechanics.