The aim of the present work is to introduce the notion of finite tightness and then to obtain some results of compactness in the lack of L1 - boundedness.We use these results in order to obtain information about the limit of the solutions for a sequence of one dimensional heat equations.

We introduce two notions of tightness for a set of measurable functions - the finite-tightness and the Jordan finite-tightness with the aim to extend certain compactness results (as biting lemma or Saadoune- Valadier's theorem of stable compactness) to the unbounded case. These compactness conditions highlight their utility when we look for some alternatives to Rellich-Kondrachov theorem or relaxed lower semicontinuity of multiple integrals. Finite-tightness