I have to make exercise 6.2.6.1 form the notes of Lunardi.
Prove the following additional regularity properties of the solution to (6.12):
(i) if $u_0 \in BUC(\mathbb{R}^n,\mathbb{R}^m)$, then $u(t, x) \rightarrow u_0(x)$ as $t \rightarrow 0$, uniformly for $x$ in $\mathbb{R}^n$;
(ii) if for every $R > 0$ there is $K = K(R) > 0$ such that $$ |f(t, x, u) − f(s, y, v)|_{\mathbb{R}^m} \leq K((t − s)^{\theta} + |x − y|^{\theta}_{\mathbb{R}^n} + |u − v|_{\mathbb{R}^m}), $$ for $0 \leq s < t \leq T, x, y \in \mathbb{R}^n, u, v \in \mathbb{R}^m, |u|_{\mathbb{R}^m}, |v|_{\mathbb{R}^m} \leq R$, then all the second order derivatives $D_{ij}u$ are continuous in $I(u_0) \times \mathbb{R}^n$.
[Hint: $u'$ and $F(t, u)$ belong to $B([\epsilon, \tau(u_0)−\epsilon]; D_A(\theta/2,\infty))$, hence $u \in B([\epsilon, \tau(u_0)−\epsilon]; C^{2+\theta}_b(\mathbb{R}^N))$. To show Holder continuity of $D_{ij}u$ with respect to $t$, proceed as in Corollary 4.1.11].
6.12 is given by $$ \left\{ \begin{array}{rr} u_t(t,x)=D\Delta u(t,x) + f(t,x,u,u(t,x)), \qquad t>0, & x\in \mathbb{R}^n;\\ u(0,x)=u_0(x) & x\in\mathbb{R}^n, \end{array}\right. $$ where $u=(u_1,\dots,u_m)$ is unknown, and the regular function $f:[0,T]\times \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^m$, the bounded and continuous $u_0 : \mathbb{R}^n \rightarrow \mathbb{R}^m$ are given.
This material is all relative new to me. It would be very helpful if someone could give me some guidance in solving this problem. Right now I'm in the process of reading the notes and understanding Holder spaces.