We study the blow-up and stability of solutions of the equation ut+ux=uxx+λf(u)/(∫10f(u)dx)2 with certain initial and boundary conditions. When f is a decreasing function, we show that if ∫∞0f(s)ds<∞, then there exists a λ∗>0 such that for λ>λ∗, or for any 0<λ≤λ∗ but with initial data sufficiently large, the solutions blow up in finite time. If ∫∞0f(s)ds=∞, then the solutions are global in time. The stability of solutions in both cases is discussed. We also study the case of f being increasing.