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If this is a true sunflower,If this is a true sunflower, Helianthus, and it is one of the annuals (especially a dwarf form), then it could be just that the season is over, or that the plant has reached mature height. Also, if you are watering the plants from overhead during hot sunny weather, the water can burn the leaves, causing them to brown; if this is the case, try watering at the base of the plant only. Sunflowers like moist, but not wet, soil.

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