I know in rapidly dividing cells, the silencing of a gene by siRNA is lost because it gets diluted each time the cells divide...but in case of nondividing cells why it doesn't last till the cell die?
your siRNA will eventually be degraded and at a certain time point there will be no residual siRNA to repress the endogenous processing of your target transcript.
A rule of thumb is that peak activity is generally at about 24 hours post-transfection and start to markedly decrease beyond 48 hours. So depending on your cell line, the target protein, the knockdown efficiency and so on, it will be more or less stable with time.
You can always try to use a shRNA based plasmid system in which cell can be selected using an antibiotic and lead to the stable knockdown of your gene of interest. Using this framework, the cells that are in selection will need to express the shRNA plasmid to survive and thus will maintain a stable knockdown.
There are essential two possibilities to obtain stable silencing over time. 1) you stable transfect and put them under selection pressure or 2) you do repeated transient transfection.
The problem could be in the type of promoters that you used in the target gene. In general, promoters of genes that are constitutive usually give better results in terms of preserving the effect of silencing. What promoters did you use?