CO-semigroups generated by second order differential operators with general wentzell boundary conditions

Let us consider the operator Ãu(x) = φ(x,u′(x))u″(x), where φ is positive and continuous in (0,1) × R and à is equipped with the so-called generalized Wentzell boundary condition which is of the form aÃu+bu′+cu = 0 at each boundary point, where (a,b,c) ≠ (0,0,0). This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions. Under suitable assumptions on φ, we prove that à generates a positive Cosemigroup on C[0, 1] and, hence, many previous (linear or nonlinear) results are extended substantially. © 2000 American Mathematical Society