I am dealing with the following Cauchy problem on $[0,T]\times \mathbb R$
\begin{cases} \partial_t u= \partial_x u+F(u(t)), \\ u(0,x)=\varphi(x) \end{cases} where $F$ is some Lipschitz continuous function, there are some local-existence results for this kind of problem.
I would like to write down the mild solution for this problem, so first thing I wrote this as an abstract Cauchy problem on some Banach space ($\star$) \begin{cases} u'(t)= A u(t)+F(u(t)), \\ u(0,x)=\varphi(x) \end{cases}
Then assuming that $A$ is the generator of a $C_0$-semigroup $(T(t))$ then the mild solution will be given by ($\star$)
$$u(t)=T(t)\varphi(x)+\int_0^t T(t-s)F(u(s))ds$$
Questions:
(1) When I write down my Cauchy problem in abstract form, which Banach space should I consider? Since $A$ is a differential operator I believe I should take $C^1$ or a Sobolev space $W^{1,1}$.
(2) $A$ is the infinitesimal generator of the translation semigroup, but is it a $C_0$-semigroup? Can we write a mild solution even if $A$ is not the generator of a $C_0$-semigroup? (namely if it is the generator of some semigroup with less regularity).
Thanks in advance.