I'm working through some problems with contraction semigroup right now.
Show that the initial value problem for the transport equation: $$ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0; x \in \mathbb{R}, t>0 $$ $$ u(x, 0) = g(x) $$ defines a contraction semigroup on $L^P (\mathbb{R}) \forall 1 \leq p < \infty $ but not on $L^{\infty} (\mathbb{R})$.
I am kinda stuck at the very beginning, as in what to defined for the family of bounded linear operator here so that $S(0) u = u$ to start satisfying the definition of the semigroup.
For this PDE, assuming $c$ is a constant, the explicit solution is given by $u(x,t)=g(x-ct)$ thus your semigroup is merely $S(t)g=g(\cdot-ct)$. You mght use this property to address you problem.