Existence and regularity results for a class of singular parabolic problems with \(L^1\) data

In this paper we prove existence and regularity results for a class of parabolic problems with irregular initial data and lower order terms singular with respect to the solution. We prove that, even if the initial datum is not bounded but only in \(L^1(\Omega )\) $L^1(\mathrm{\Omega })$ , there exists a solution that “instantly” becomes bounded. Moreover we study the behavior in time of these solutions showing that this class of problems admits global solutions which all have the same behavior in time independently of the size of the initial data.