Extension of non-linear Semigroup on $L^{p}(\Omega)$

Question

my question is the following: I have a family of (non-linear) operators semigroup $\left\{ T_\mu \right\} $ on $L^{2}(\Omega)$ and $L^{\infty}(\Omega)$

1. Is possible to get the semigroup $\left\{ T_\mu\right\}$ on $L^r(\Omega)$ for any $r\in [2, \infty]$?

2. Then, how can i get the same semigroup $\left\{ T_\mu \right\}$ on $L^{q}(\Omega)$ for $q\in [1, 2)$ by duality or interpolation?

Thank you very much.