Sunflowers in Arizona enjoy DEEP watering once a week--or, when the soil is dry below the first joint of your finger. For most varieties, you should expect them to grow for about 90 days and then the bloom will last a week or so. They will slowly dry out until it's a dried head on a brown stalk. At that point, you can take out some seeds to plant next year, harvest the seeds for eating, or let the birds enjoy the seeds. As far as the stalk goes: remove it and hack it up into the compost pile -- or leave it and use for a stake or trellis!