Broadcasting is the widely accepted approach to disseminate information in a wireless environment and can be mainly categorized as push-based or pull-based (ondemand) approaches. One of the advantages of on-demand data broadcast is that it decreases the data access time since only the requested data are broadcast. Currently the majority of the research work focuses on how to reduce the data access time. In this paper we study the messages saving issue in on-demand broadcast environments and present a lazy demand request strategy, called LDR. Our experiment results indicate that the LDR approach not only significantly reduces the number of messages sent, but it can also reduce the average data access time of clients. In some cases the LDR reduces the average number of request messages by up to 50% and decreases the average access time by 20%.