The authors consider the problem \(u_ t=(\varphi(u)\psi(u_ x))_ x\) for \(x\in\mathbb{R}\), \(t>0\), \(u(x,0)=u_ 0(x)\) for \(x\in\mathbb{R}\) where \(\varphi:\mathbb{R}\to\mathbb{R}^ +\) is smooth and strictly positive, and \(\psi:\mathbb{R}\to\mathbb{R}\) is a smooth, odd function such that \(\psi'>0\) in \(\mathbb{R}\) and \(\lim_{s\to\infty}\psi(s)\equiv\psi_ \infty<\infty\). The initial function \(u_ 0:\mathbb{R}\to\mathbb{R}\) is bounded and strictly increasing. This problem is solved by the parabolic regularization method and the question is discussed to what extent the solution behaves qualitatively like solutions of the first-order conservation law \(u_ t=\psi_ \infty\cdot(\varphi(u))_ x\).