Abstract integrodifferential equations and applications

In this work, we study the initial value problem associated with an abstract integrodifferential equation in interpolation scales. We prove local-in-time existence, uniqueness, continuation, and a blow-up alternative for regular mild solutions to the problem. Additionally, we apply this theory to the Navier-Stokes equations with hereditary viscosity, taking initial data in the scale of fractional power spaces associated with the Stokes operator. We also explore reaction-diffusion problems with memory, considering the effects of super-linear and gradient-type nonlinearities, and initial data in Lebesgue and Besov spaces, respectively.

💡 Research Summary

The paper develops a comprehensive functional‑analytic framework for abstract integrodifferential equations of the form

 u′(t)=∫₀ᵗ g(t−s) A u(s) ds + f(t,u(t)), u(0)=u₀,

where A is a densely defined closed operator on a Banach space X₀ such that –A is sectorial, and g is a locally integrable, Laplace‑transformable kernel. The authors work in a family of interpolation spaces {X_α}_α≥0 associated with the fractional powers of –A, which allows a precise measurement of spatial regularity.

Resolvent families.
Under three technical conditions on the kernel (B1–B3) concerning the meromorphic extension of 1/ĝ(λ) and its growth at infinity, they prove the existence of an analytic resolvent family {S(t)}_{t≥0} associated with (A,g). The family satisfies the smoothing estimates

 ‖S(t)‖{B(X₀)} ≤ M e^{ωt},
 ‖S(t)‖{B(X₀,X₁)} ≤ M t^{‑ζ_g},

with ζ_g>1 determined by the asymptotic behaviour of g. Moreover, for any 0≤θ≤γ≤1,

 ‖S(t)‖{B(X_γ,X{1+θ})} ≤ C t^{‑ζ_g(1+θ‑γ)}.

These estimates are the cornerstone for handling the nonlinear term.

Nonlinear analysis – non‑critical case.
If the nonlinearity f(t,·) maps X₁ into X_γ with γ>1‑1/ζ_g and is locally Lipschitz, the Banach fixed‑point theorem yields a unique mild solution u∈C(