One can easily check for any $\beta, \gamma \in (0,1)$, $k_\beta * k_\gamma = k_{\beta+\gamma}$. So in particular, we have $$\label{e:1.10}
k_{\beta}*k_{1-\beta} (t)\equiv 1 .$$ Suppose $u(x, t)$ is a solution of (\[e:1.8\]). In view of the definition of Caputo derivative , equation is understood by its integral version $$k_{1-\beta}*(u(t,x)-u(0, x))=\int^t_0 \Delta u(s,x)ds+k_{1-\gamma}*\int^t_0
g(s)dW_s.$$ By taking convolution with $k_{\beta}$ on both sides, we get from and $$\int^t_0
\left(u(s,x)-u(0,x)\right)\,ds=\left(I^{\beta}*\int^\cdot_0 \Delta
u(s,x)ds\right)(t)+\left(I^{\beta+1-\gamma}* \int^\cdot_0 g(s,x)dW_s\right)(t).$$ Since the first two terms are differentiable in $t$, the last term above should be differentiable in $t$. Recall that $$I^{a}: C^{b}\to C^{a+b},$$ and $\int^t_0 g(s)dW_s \in C^{1/2-\varepsilon}$ for any $\varepsilon>0$. Thus we must have $$\beta+1-\gamma>1/2,$$ which is equivalent to .

The main results of this paper are Theorems \[thm divtype\] and \[main\], on the unique solvability of SPDEs and , and $L_2$-estimates of their solutions. For SPDE , we establish in Theorem \[main\] the unique solvability in the space $L_2(\Omega\times[0,T], H^{\sigma}_2)$ for any $\sigma\in {\mathbb{R}}$ under appropriate differentiability assumption on $x$-variable of the coefficients. On the other hand, the unique solvability of SPDE in the space $L_2(\Omega\times[0,T], H^{1}_2)$ is obtained in Theorem \[thm divtype\] under the merely measurability condition of the coefficients $a^{ij}$.

The rest of the 