Well-Posedness of Degenerate Integro-Differential Equations with Infinite Delay in Banach Spaces
Aparicio Cuello, Rafael Antonio
Valentin, Keyantuo (Consejero)
We are concerned with a class of degenerate integro-differential equations of second order in time in Banach spaces. We characterize their well-posedness using operator valued Fourier multipliers. These equations are important in several applied problems in physics and material science, especially for phenomena where memory effects are important. One such domain is viscoelasticity. We focus on the periodic case and we treat vector-valued Lebesgue, Besov and Trieblel-Lizorkin spaces. We note that in the Besov case, the results are applicable in particular to the scale of vector-valued H¨older spaces Cs, 0 < s < 1. The definition of well posedness we adopt is a modification of the one used so far in the special cases. Thus, our results have as corollaries those obtained by several authors for first and second order integro-differential equations in the non-degenerate context.