Summary: Given a nonnegative function \(a\in C^1([0,1])\) vanishing only at \(0,1\) and \(b\in C([0,1])\), we study existence, qualitative properties, asymptotic behaviour and approximation of the solutions of the degenerate evolution problem \(u_t= (au_x)_x- bu\) in \(]0,1[\times ]0,+\infty[\), with \(u(x,0)= u_0(x)\) and boundary conditions \(a(x) u_x(x,t)\to 0\) as \(x\to 0,1\), \(t>0\).