The seminar will be based on results obtained in collaboration with Stefano Bianchini. In the first part of the talk I will explain what I mean by viscous profiles for mixed hyperbolic parabolic systems. In the second part I will discuss the reasons why the analysis of these profiles is related to the study of the singular ordinary differential equation
dU/dt=f(U)/z(U)+g(U).
In the previous expression, U is a vector valued function which depends on the scalar variable t. The function z is real valued and the equation is singular because z can attain the value 0.
Finally, in the last part of the seminar I will discuss some results concerning the solutions of this singular equation that belong to a small enough neighbourhood of a point U satisfying z(U)=0 and f(U)=g(U)=ec 0.