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

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