Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded connected set having Lipschitz uniform boundary. Moreover let $d\in L^{\infty}(\Omega,\mathbb{R}^M),\ d_1(x),d_2(x),\dots, d_M(x)>d>0,\ \forall\ x\in\Omega$ and the linear operator: $$ A:D(A)\subseteq X\to X,\ A(y)=d\Delta y,\ \forall\ y\in D(A), $$ where $X\subseteq \{f:\Omega\to\mathbb{R}^M\ |\ f\text{ is a function}\}$ is a Banach (or reflexive Banach, or Hilbert space over $\mathbb{R}$ or $\mathbb{C}$). We can have $$ X= \begin{cases} L^2(\Omega;\mathbb{R}^M), & \text{or}\\ H^1(\Omega,\mathbb{R}^M), & \text{or}\\ L^p(\Omega;\mathbb{R}^M), & \text{or}\\ L^{\infty}(\Omega;\mathbb{R}^M), & \text{or}\\ W^{k,p}(\Omega;\mathbb{R}^M) & \text{or} \\\text{etc}\ldots\\ \end{cases} $$ and $$ D(A)=\left\{y\in X\ \Big| \ d\Delta y\in X,\ \alpha_iy_i+\beta_i\dfrac{\partial y_i}{\partial\nu}=0,\ \forall\ i\in\overline{1,M}\right\}.$$ Here $\alpha_i\in C(\partial\Omega),\beta_i\in\{0,1\},\ \forall\ i\in\overline{1,M}$.
My question is: what type of semigroup has $A$ as an infinitesimal generator? Is it a strongly continuous semigroup (or $C_0$ semigroup), or analytic semigroup, or sectorial, or compact... Are there any references that deal with this problem with a Robin type boundary condition in a clear way, or at least with a Neumann boundary condition?
Somebody told me that if we have Neumann Boundary conditions ($\alpha_i=0,\beta_i=1,\ \forall\ i\in \overline{1,M}$) then the semigroup $(T(t))_{t\geq 0}$ that generates $A$ is contractive and maps $L^1$ to $L^p$ with norm less than $Ct^{\frac{-N}{p'}}$, $1/p+1/p'=1$. What does that mean exactly, and is there a reference for that topic?
Thanks a lot.
