from the interval $[0,T]$ to $X^\gamma$ with the maximum norm $\Vert U(\cdot)\Vert_{C([0,T],X^{\gamma})}=\sup_{\tau\in[0,T]} \Vert U(\tau)\Vert_{X^\gamma}$. We recall the well known result on existence and uniqueness of a solution to abstract parabolic equations in Banach spaces due to Henry [@Henry1981].

[@Henry1981 Section 1] \[semilinear\_general\_existence\_result\] Suppose that a densely defined closed linear operator $-A$ is a generator of an analytic semigroup $\left\{e^{-At},t\geq 0\right\}$ in a Banach space $X$, $U_{0}\in X^{\gamma}$ where $0\leq \gamma <1$. Assume $F:[0,T]\times X^{\gamma}\to X$ and $h:(0,T]\to X$ are Hölder continuous mappings in the $\tau$ variable, $\int_0^T \Vert h(\tau)\Vert_X {\mathrm{d}}x <\infty$, and $F$ is a Lipschitz continuous mapping in the $U$ variable. Then, there exists the unique solution $U\in C([0,T],X^{\gamma})$ of the following abstract semilinear evolution equation: $$\frac{\partial U}{\partial \tau}+A U=F(\tau, U) +h(\tau), \qquad U(0)=U_{0}.
\label{semilinear_problem}$$ Moreover, $\partial_\tau U(\tau) \in X, U(\tau)\in D(A)$ for any $\tau\in (0,T)$.

By a solution to (\[semilinear\_problem\]) we mean a function $U\in C([0,T],X^{\gamma})$ satisfying (\[semilinear\_problem\]) in the integral (mild) sense, i.e. $$U(\tau) = e^{-A \tau} U_0 + \int_0^\tau e^{-A (\tau-s)} (F(s, U(s)) + h(s) ) {\mathrm{d}}{s} \ \hbox{for any}\ \tau\in[0,T].$$ Recall that the key idea of the proof of Proposition \[semilinear\_general\_existence\_result\] is based on the Banach fixed point argument combined with the decay estimate $\Vert 