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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.

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1 Answer 1

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 $$ S(t)u\in H^s,\forall s\geq0 $$

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