$C_0$-semigroups generated by second order differential operators with general Wentzell boundary conditions

Abstract:

Let us consider the operator $\widetilde {A}u(x)=\phi (x,u’(x))u''(x),$ where $\phi$ is positive and continuous in $(0,1)\times \mathbf {R}$ and $\widetilde {A}$ is equipped with the so-called generalized Wentzell boundary condition which is of the form $a\widetilde {A} u+bu’+cu=0$ at each boundary point, where $(a,b,c)\neq (0,0,0).$ This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions. Under suitable assumptions on $\phi$, we prove that $\widetilde {A}$ generates a positive $C_{0}$-semigroup on $C[0,1]$ and, hence, many previous (linear or nonlinear) results are extended substantially.