Evolution Equation

Question

Let $X=L^2(0,\pi)$. Define the operator $(A,D(A)$ by: $$D(A)=\{u\in H^2(0,\pi):u(0)=u'(\pi)=0\} ,\quad \quad Au=u''$$ Show that $A$ is the infinitesimal generator of a $C_0$ semigroup of contractions on $X$.

Answer 1

Adapt the proof from http://www.math.unipr.it/~lunardi/LectureNotes/I-Sem2005.pdf page 25.

Answer 2

Theorem (Lumer-Phillips). Let $X$ be an Hilbert space and $A : D(A)\subset X \to X$ a densely defined dissipative linear operator. If there exists $\lambda > 0$ such that $(\lambda I -A):D(A)\to X$ is surjective, then $A$ is the infinitesimal generator of a $C_0$ semigroup of contractions on $X$.

Reference: Pazy's book, page 14.

Notice that $X=L^2(0,\pi)$ is an Hilbert space, $D(A)$ is dense in $X$ (because $H_0^1(0,\pi)\subset D(A)$) and $A$ is linear. Furthermore, from integration by parts it follows that $$\langle Au,u\rangle_{L^2}=-\|u'(x)\|_{L^2}^2\leq 0$$ and thus $A$ is dissipative. Finally, from the Lax-Milgram Lemma (see Brezis book, pages 220-221) it follows that, for each $f\in X$, the equation $u-u''=f$ has an unique solution $$u\in H^2(0,\pi)\cap H_0^1(0,\pi)=D(A)$$ and thus $\lambda I-A$ is surjective for $\lambda =1$.

So, from Lumer-Philips Theorem we get the desired result.