Does the Bi-Laplacian generate an analytic semigroup?
Yes, the bi-Laplacian with Dirichlet boundary conditions generates an analytic semigroup on $L^p(\Omega), \; p\in (1,\infty)$. For $p=2$, you can show that the operator is self-adjoint. A general result was proved in Theorem 5.6, pp. 189 in
Hiroki Tanabe, Functional Analytic Methods for Partial Differential Equations, vol 204 of Monographs and Textbooks in Pure and Applied Mathematics, New York, 1997.