A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity

Abstract

We prove the existence and uniqueness of solution to a one-dimensional hyperbolic–parabolic system arising in the study of magneto-viscoelasticity. Specifically, the local existence and uniqueness result is proved on application of the fixed point theorem; then, a uniform a priori estimate of the solution is established. The latter, via a continuation method, allows as to obtain the global result. A crucial tool to achieve such a result is a technical lemma concerning the only viscoelastic contribution; it relies on the assumptions that the memory kernel is positive, monotonically non-increasing and convex.

Keywords:

• magneto-viscoelastic materials
• nonlinear hyperbolic–parabolic systems

AMS Subject Classifications::

• 74H20
• 35Q74
• 45K05

Acknowledgements

The partial financial support of Sapienza University of Rome through the Research Project (2009) ‘Modelli Differenziali in Matematica Applicata’, is gratefully acknowledged. The research of V. Valente was partially supported by the European Community through the RTN-Programme ‘Smart Systems’ (HPRN-CT-2002-00284). S. Carillo and G. Vergara Caffarelli acknowledge the partial support of the Italian GNFM-INdAM.