$\alpha,\beta,T$ and the initial values and vary from line to line. $C_\ast$ is used to emphasize that the constant depends on the corresponding parameter $\ast$ which is one or more than one parameter.

Mixed SPDEs driven by fBm and Bm {#se-3}
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This section will prove an existence and uniqueness theorem for the mixed SPDEs driven by both fBm and Bm (\[spde-uniq\]). Let $V_\beta, \beta\geq 0$, denote the domain of the fractional power $(-A)^\beta$ equipped with the graph norm $|x|_{V_\beta}:=|(-A)^{\beta}x|, x\in V_\beta$. For shortness, denote, $|x|_\beta:=|x|_{V_\beta}.$ We recall here some properties of the analytic semigroup, which will be used later in our analysis. For $0 \leq   \gamma \leq \varsigma \leq 1$ and $\upsilon\in[0,1), \mu\in(0,1-\upsilon)$, there exists a constant $C>0$, such that for $0\leq s <t \leq T$, we have $$\begin{aligned}
\label{semi1}
|S_t|_{L(V_\gamma, V_{\varsigma})} &\leq C t^{-\varsigma+\gamma}e^{-\bar{\lambda}_1 t},\\
\label{semi2}
|S_{t-s}-\mathrm{id}|_{L(V_{\upsilon+\mu}, V_{\upsilon})} &\leq C (t-s)^{\mu}.\end{aligned}$$ We also note that from these inequalities, for $\varrho,\nu \in(0,1]$ and $0\leq \nu<\gamma+\varrho$, there exists $C>0$ such that for $0\leq q \leq r \leq s \leq t$, we derive $$\begin{aligned}
\label{semi3}
    |S_{t-r}-S_{t-q}|_{L\left(V_{\nu}, V_{\gamma}\right)} \leq C(r-q)^{\varrho}(t-r)^{-\varrho-\gamma+\nu},\end{aligned}$$ and $$\begin{aligned}
\label{semi4}
    |S_{t-r}-S_{s-r}-S_{t-q}+S_{s-q}|_{L(V,V)} \leq C(t-s)^{\varrho}(r-q)^{\nu}(s-r)^{-(\varrho+\nu)}.\end{aligned}$$ From now on, we use the symbol $\|\cdot\|$ to denote $|\cdot|_{L(V,V)}$ for shortness.

We assume that 