Let $B : H^1(\mathbb{R}^{N})\to H^1(\mathbb{R}^N)$ be a linear operator defined as follows \begin{align*} \begin{cases} D(B) := \{\varphi \in H^1(\mathbb{R}^N)\, | \, \Delta\varphi\in H^{1}(\mathbb{R}^N) \}\\ \forall\varphi\in D(B), B\varphi=\Delta\varphi \end{cases} \end{align*} Then, we know that $B$ is a dense and maximal dissipative operator in $L^2(\mathbb{R}^N)$ which generates a contraction semigroup $(S(t))_{t\geq0}$
Now, my question is as follows
Assume that $C: L^{\infty}(\mathbb{R}^{N})\to L^{\infty}(\mathbb{R}^{N})$. How do I define $D(C)$ so that $D(C)$ is dense in $L^{\infty}(\mathbb{R}^{N})$? I don't think I can apply the technique in $L^{2}(\mathbb{R}^{N})$ because it does not have a Hilbert Space structure.
I need $C$ to be a maximal dissipative operator with dense domain in $L^{\infty}(\mathbb{R}^{N})$ in order to use Hille-Yoshida Theorem so that I can apply it to obtain a local existence result.
Any help or hint or even good reference will be appreciated. I currently use "Introduction to Semilinear Evolution Equations" by Cazenave and Haraux.
Thank you very much!