Energy and regularity inequalities for Volterra equations of parabolic type (Q1308512)

The author considers the equation \[ u(t)=\varphi-\int_{[0,t]}Lu(t-s) W(ds), \quad t \geq 0, \] where \(L\) is a nonnegative, selfadjoint operator on a Hilbert space and \(W\) satisfies the integral equation \[ aW \bigl( [0,t] \bigr)+\int_ 0^ tk(t-s)W\bigl( [0,s] \bigr) ds=t,\quad t \geq 0, \] for some constant \(a \geq 0\) and some nonnegative and nonincreasing function \(k\). The energy inequalities \[ \begin{multlined}\bigl\| e^{- LW([0,t])} \varphi \bigr\|^ 2 +2 \int_{[0,t]} \bigl\| \sqrt Le^{-LW([0,t-s])} \varphi \bigr\|^ 2W(ds) \leq\\ \| u(t) \|^ 2+2 \int_{[0,t]} \| \sqrt Lu(t-s) \varphi \|^ 2W(ds) \leq \| \varphi \|^ 2,\end{multlined} \] the author derives, become an energy equality for the evolution equation \[ u'(t)=-Lu(t)/a,\quad u(0)=\varphi, \] in the case where \(k \equiv 0\). The regularity inequalities are of the form \[ \| L^ \delta u(t)\| \leq \delta^ \delta e^{-\delta}f_ \delta(t)\| \varphi \|, \] where \(f_ \delta(t) \sim a^ \delta t^{-\delta}\) as \(t \downarrow 0\) when \(a>0\) and \(f_ \delta(t) \sim t^{-\alpha \delta}S(1/t)^ \delta \Gamma(1- \delta)/ \Gamma(1-\alpha \delta)\) as \(t \downarrow 0\) when \(a=0\) and \(k\) satisfies \(k(t) \sim t^{-\alpha}S(1/t)/ \Gamma(1-\alpha)\) where \(S\) is a slowly varying function.