The parabolic equations of the form \(\partial u/\partial t=\varphi(x,\nabla u)\Delta u\), where \(\varphi(x,\xi)>0\) on \(\Omega\times\mathbb{R}^ n\) \((\Omega\subset \mathbb{R}^ n)\) and \(\varphi(x,\xi)\to 0\) arbitrarily rapidly as \(x\) approaches the boundary \(\partial\Omega\). The problem of finding boundary conditions which characterize the one-dimensional and multidimensional diffusions are given by Feller and Wentzel, respectively. Using a semigroup version, this problem was studied by Ph. Clement and C. A. Timmermans [e.g.: Indagationes Math. 48, 379-387 (1986; Zbl 0618.47035)]. Using Wentzel-type boundary conditions, the authors proved the well-posedness of the parabolic equation under consideration. This is done with the aid of the Crandall-Liggett theorem.