New results on the Cauchy problem for parabolic systems and equations with strongly nonlinear sources

We consider both degenerate and uniformly parabolic systems and equations, containing a forcing term (a ``source'') depending on the solution itself. The source is such that the solution may become unbounded in a finite time, even if the initial data are bounded. In this connection we investigate the problem of the existence of non negative solutions defined for all positive times. Moreover, even the problem of the existence of local in time solutions is not trivial, owing to the effect of nonlinear sources of this kind. In fact (as a marked difference with the corresponding homogeneous problems), local solutions may exist only under certain restrictions on the local regularity of the initial data.