Singular kernel problems in viscoelasticity and magneto-viscoelasticity

Problems in materials with memory are considered. In particular, the attention is focussed on the kernel of the integro-differential term to study the behaviour of ma- terials which can be modelled via an integro-differential equation with both regular as well as a singular kernel at t = 0. Some results obtained in recent works are presented. Specifically, problem in singular rigid thermodynamics [2,7] as well as in linear viscoelas- ticity [1,7] are considered. When, also the effect of magnetization is introduced, that is, the bahaviour of a magneto-elastic or magneto-viscoelastic body is studied the obtained problem is nonlinear, due to the magnetic term. Thus, respectively, anonlinear differen- tial systems follows coupling magnetization with linear elasticity ,[3,4], and a nonlinear integro-differential systems models the magneto-viscoelastic behaviour [5,6]. Existence and uniqueness results are proved in [1] in the case of a the 1-dimensional body, while only existence can be proved in the 3-dimensional problem.In [8], an existence result is proved when the kernel in the integro-differential term is assumed to be singular at t = 0. 1. S. Carillo, V. Valente and G. Vergara Caffarelli, A linear viscoelasticity problem with a singularmemory kernel: an existence and uniqueness result,Diff. and Integral Equations, 26, 9/10, (2013), 1115 – 1125. 2. S. Carillo., V. Valente and Vergara Caffarelli G. Heat conduction with memory: a singular kernel problem. E E C T, 3, 3, (2014), 399 – 410. 3. S. Carillo, V. Valente and G. Vergara Caffarelli, A result of existence and uniqueness for an integro-differential system in magneto-viscoelasticity, Applicable Analysis, (2010); 90, 12, (2011), 1791 – 1802. 4. M. Chipot, I. Shafrir, V. Valente and G. VergaraCaffarelli, A nonlocal problem arising in the study ofmagneto-elastic interactions. Boll. UMI Serie IX, I, (2008), 197 – 222 5. M. Chipot, I. Shafrir, V. Valente and G. Vergara Caffarelli, On ahyperbolic-parabolic system arising in magnetoelasticity. J.Math. Anal. Appl., 352, (2009), 120 – 131. 6. S. Carillo, V. Valente and G. Vergara Caffarelli, An existence theorem for the magneto-viscoelastic problem Discrete and Continuous Dynamical Systems Series S., 5, 3, (2012), 435 – 447. 7. S. Carillo, Singular kernel problems in materials with memory, Meccanica, 50, 3, (2015), 603 – 615. 8. S.Carillo,MichelChipot,V.ValenteandG.VergaraCaffarelli,Amagneto-viscoelasticity problem with a singular memory kernel,submitted (2016)