Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

(ω,c)-Periodic Mild Solutions to Non-Autonomous Abstract Differential Equations

We investigate the semi-linear, non-autonomous, first-order abstract differential equation x′(t)=A(t)x(t)+f(t,x(t),φ[α(t,x(t))]),t∈R. We obtain results on existence and uniqueness of (ω,c)-periodic (second-kind periodic) mild solutions, assuming that A(t) satisfies the so-called Acquistapace–Terreni conditions and the homogeneous associated problem has an integrable dichotomy. A new composition theorem and further regularity theorems are given.

Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments

In this paper, we propose the study of an integral equation, with deviating arguments, of the typey(t)=ω(t)-∫0∞‍f(t,s,y(γ1(s)),…,y(γN(s)))ds,t≥0,in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at∞asω(t). A similar equation, but requiring a little less restrictive hypotheses, isy(t)=ω(t)-∫0∞‍q(t,s)F(s,y(γ1(s)),…,y(γN(s)))ds,t≥0.In the case ofq(t,s)=(t-s)+, its solutions with asymptotic behavior given byω(t)yield solutions of the second order nonlinear abstract differential equationy''(t)-ω''(t)+F(t,y(γ1(t)),…,y(γN(t)))=0,with the same asymptotic behavior at∞asω(t).