We study a class of non-linear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to the case of one space dimension. We consider initial data for which the diffusion coefficients may vanish. We prove that, under this condition, those systems are locally well-posed in the class of Sobolev spaces of high enough regularity, but also that there exist smooth initial data for which the corresponding solutions blow up in finite time. We are able to put in evidence two different types of blow-up mechanism. In addition, the results are extended to the case of transport-diffusion systems, namely to the case when convection is taken into account.