Although there is no precise boundary between dialects and languages, dialects can be defined as different speech systems of the same language which are mutually intelligible. Here we propose a mathematical model to investigate the dynamics of survival of a local dialect when competing against a national standard language. We assume individuals may be able to use different dialects, in response to different situations. For example, in public situations, e.g. in a formal workplace, they may use the national language, while in private situations, e.g. in family or close friend meetings, they may use a local dialect. We consider the choice of a dialect is guided by, mainly, two forces: (i) attractiveness of the local/standard language; (ii) willingness to speak the same dialect (conformity factor) inside a private group. Our results show that conformity usually enhances the effect dictated by the difference in attractiveness between the local dialect and the standard language if conformity works favoring only those states in which all speakers use the same dialect (unanimity pressure model). However, conformity may have no effect at all if it works in proportion to the fraction among peers (peer pressure model).