Well-posedness of degenerate integro-differential equations in function spaces

Abstract

We use operator-valued Fourier multipliers to obtain characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. We treat periodic vector-valued Lebesgue, Besov and Trieblel-Lizorkin spaces. We observe that in the Besov space context, the results are applicable to the more familiar scale of periodic vector-valued H\"older spaces. The equation under consideration are important in several applied problems in physics and material science, in particular for phenomena where memory effects are important. Several examples are presented to illustrate the results.

Keywords

• Well-posedness
• maximal regularity
• R-boundedness
• operator-valued Fourier multiplier
• Lebesgue-Bochner spaces
• Besov spaces
• Triebel-Lizorkin spaces
• Holder spaces