Renovating Old Lawn

Hello

My back garden lawn is quite small and in poor condition. During the winter it becomes very water logged causing a lot of the grass to die and become patchy.

So my plan is rake out the thatch, use a hollow tine aerator to take out cores and improve the drainage and then over seed.

However around the edges there are some very dry dead looking patches, please see the photo. Do these also just need aerating or is there something more serious going on?