We consider that the concept derived of the integration of pervasive, context-aware and grid computing is suitable for building the next generation of the grid applications which assist to nomadic and mobile users. To demonstrate our ideas we have implemented a pilot application, called GeneAl, using the API provided by EXEHDA middleware. EXEHDA is adaptive, service oriented and was conceived to support the execution of pervasive grid applications. The main concept embedded in middleware and application design is the context-awareness expressed by their adaptive behavior. This is also a key to provide functionality adapted to the constraints and unpredictability of the large-scale mobile environment. To achieve this objective, EXEHDA employs a lot of strategies in its services to allow the adaptation to the current state of the execution context, such as on-demand adaptive service loading and dynamic discovery and configuration. The middleware manages and implements the follow-me semantics for pervasive grid applications. In that sense, it provides services for distributed adaptive execution, context recognition, pervasive storage and access, anonymous and asynchronous communications.