I am trying to solve the following problem: $$u_t + (-\Delta)^su = 0$$ in $\Omega \subset \mathbb{R}^N$ with $N > 2s$, where $s \in (0,1)$ and Dirichlet Boundary conditions.
Let my operator $A = (-\Delta)^s$. We have that $D(A) = \{u \in H^s(\Omega); Au \in L^2(\Omega)\}$ and $A : D(A) \subset L^2(\Omega) \to L^2(\Omega)$ is given by $$A u(x) = \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy.$$
The domain of $A$ is inspired by https://arxiv.org/pdf/1506.00210.pdf, pag 5, and references therein. Furthermore, $D(A)$ is dense in $L^2(\Omega)$.
I am looking to use semigroup methods. In particular I am trying to verify the conditions of Lumer-Phillips Theorem. The phase space is the usual fractional Sobolev space with the norm $\|u\|_{H^s(\Omega)} = \int_\Omega u^2(x)dx + \int_\Omega \int_\Omega \frac{|u(x) - u(y)|^2}{|x - y|^{N + 2s}}dydx$.
If I can show that $A$ satisfies the conditions of Lumer-Phillips I will have that A is the infinitesimal generator of a $C_0$-semigroup and so a solution will exist.
I still need to show the following:
(1) Re$\left<u, -Au\right>_{L^2(\Omega)} \leqslant 0$, for every $u \in D(A)$.
(2) There exists a $λ_0>0$ such that the range $R(A+λ_0I) = L^2(\Omega)$.
Here $\left<\cdot,\cdot\right>$ denotes the usual inner product.
Proof
First, we note that there exists a constant $C > 0$ such that $\left<Au,v\right>_{L^2(\Omega)} = C\left<u,v\right>_{H^s(\Omega)}$.
(1) $\left<u, -Au\right>_{L^2(\Omega)} = -C\left<u,u\right>_{H^s(\Omega)} \leqslant 0$. Therefore, (1) is satisfied.
(2) On the other hand, define bilinear form $a: H^s(\Omega) \times H^s(\Omega) \to \mathbb{R}$ by $a(u,v) = C\left<u,v\right>_{H^s(\Omega)} + \left<u,v\right>_{L^2(\Omega)}$. Note that $a$ is coercive and continuous. Thus, by the Lax-Milgram Theorem, there exists an unique $u \in H^s(\Omega)$ such that $a(u,v) = \left<f,v\right>_{L^2(\Omega)}$, for all $v \in H^s(\Omega)$. If $u \in D(A)$, it's over, because $a(u,v) = \left<Au + u, v\right>_{L^2(\Omega)}$. The problem occurs if $u \notin D(A)$.
In the classical heat equation this is not a problem because of the elliptic regularity theorem for the Dirichlet problem (see Theorem 9.5 of the book "Functional ANalysis, Sobolev Spaces and Partial Differential Equations" by Brezis).