In the book "Partial Differential Equations" by Evans we are trying to solve the initial/boundary problem
\begin{cases} \partial_tu+Lu=0 , (x,t)\in U_T\\ u=0 , (x,t)\in \partial U\times [0,T]\\ u=g , (x,0)\in U\times \{0\} \end{cases} where $L$ has the divergence structure, the coefficients don't depend on $t$ and satisfies the strong ellipticity conditions. Now the author is able to see that the operator $A:=-L$ is the generator of a contraction Semigroup $\{S(t)\}_{t\geq 0}$.
Now that we have this Semigroup I wanted to see how we actually solve the PDE. I would think the idea would be to consider $u(t):=S(t)g$. With this I was able to see that $\partial_t u+Lu=0$ and that $u=g$ on $U\times \{0\}$. However I am not sure why we would have that $u=0$ in $\partial U\times [0,T]$, we can show that $u\in H_{0}^1(\Omega)\cap H^2(\Omega)$, but unless the function $S(t)g$ is continuous we don't really have that equality . And all of this assuming that $g\in H^1_0(\Omega)\cap H^2(\Omega)$
Also what do we know about regularity of the solution ? All I was able to figure out was that if $g\in H_0^1(\Omega)\cap H^2(\Omega)$ then $S(t)g\in H^1_0(\Omega)\cap H^2(\Omega)$.
Does anyone know if we can use this for $g\in L^2(\Omega)$, if we have some regularity results and how I can check that $u=0$ on $\partial U\times [0,T]$?
Any help is appreciated, thanks in advance.
The regularity property is due to the analytic semigroup which in particular satisfies the smoothing effect: $$g\in X=L^2(\Omega) \Longrightarrow \forall t>0, \quad S(t)g \in D(A)=H^1_0 \cap H^2.$$