Besides the concrete or blacktop, there is a base of compacted gravel that extends several inches away from the edge providing support for the edge of the driveway so it does not crumble. In the spring and early summer there is enough soil moisture to support growth. Later in the summer, the gravel blocks water movement and the grasses go dormant or die. If there is a drought tolerant grass available in your area then you might consider that grass for those areas. As the summer progresses watering much more often may help.
