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Grass, particularly the warm season grass St. Augustine, can grow as much as 3/4" a day in shade. The reason is that the grass needs leaf material exposed to light in order to photosynthesize. Freshly mowed grass planted in shady locations grows faster than the same grass planted in sunny locations, because the grass' response to the reduced surface area in such locations is to increase it. However, this can come at the expense of spreading, so grass grown in shade is almost always more sparse. Mow your shady lawn very high and let the grass have the light it needs.