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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 L 1 -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 consider the first order differential problem:Under certain conditions on the functions f n , the problem (P n ) admits a unique solution u n ∈ W 1,1 ([0, 1], E).In this paper, we propose to study the limit behavior of sequences (u n ) n∈N and (u n ) n∈N , when the sequence (f n ) n∈N is subject to different growing conditions.