Problem 16 of chapter 7 states
Use problem 15 to prove that if $u$ is the semigroup solution in $X=L^2(U)$ of
$$
\left\{
\begin{array}{rl}
u_t – \Delta u =0 & \text{in } U_T \\
u=0 & \text{on } \partial U \times [0,T] \\
u=g & \text{on } U \times \{t=0\},
\end{array}
\right.
$$
with $g \in C_c^{\infty}(U)$, then $u(\cdot,t) \in C^{\infty}(U)$
for each $0\leq t\leq T$.
Problem 15 states
Let $\{S(t)\}_{t \geq 0}$ be a contraction semigroup on X, with generator $A$.
Inductively define $D(A^k):= \{ u \in D(A^{k-1}) \textbf{ | } A^{k-1}u \in D(A)\}$ $(k=2,\dots)$. Show that if $u \in D(A^k)$ for some $k$, then
$S(t) u \in D(A^k)$ for each $t \geq 0$.
For problem 15 I didn't need the contraction property for this.
My question is how to use this in the Problem 16.
Any help would be appreciated.
Best Answer
Mmm, you can use 15 to be sure that $S(t)u\in L^2$. Then you may test $$ S(t)u=u+\int_0^t\Delta S(s)u ds $$ against $S(t)u$. Then you observe that $$ \int_0^t\|\nabla S(s)u\|_{L^2}^2ds<\infty $$ Then you have that $$ \|\nabla S(s)u\|_{L^2}^2ds<\infty\,a.e. $$ Pick one of these time points and redo everything. Then you finally get that $$ S(t)u\in H^s,\forall s\geq0 $$