Consider the linear fractional reaction diffusion equation:
$\begin{align} \frac{\partial u}{\partial t} + \alpha(-\Delta)^{s} u = 0, \\ \label{pr: fractionallineal2} u(x,0)=u_{0}(x) \end{align} $
I know this equation is well defined for $u \in H^s(\mathbb{R}^d) $ (Sobolev space $H^s(\mathbb{R}) = \big \{f \in L^2(\mathbb{R}) : \xi \mapsto\left(1+ |\xi|^2 \right )^{\frac{s}{2}}\mathscr{F}f(\xi) \in L^2(\mathbb{R}) \big \}$). I'm wandering if there is a process to extend this to $u \in C_u(\mathbb{R}^d)$ (bounded continuous functions). I know the integral equation admit this type of solutions but I would like to know what is the process to justify a solution in that set, in the differential equation. Any resources are welcomed.