The damage you’re asking about is caused by the acidic nature of dog urine and is usually more common with female dogs. To treat the brown spot, you will need to neutralize the soil and start over by applying Gypsum to the spots (about 1/2 cup per spot) and scratching up the soil to loosen it.  Then use Earl May’s Lawn Seed Repair Kit to re-seed. Follow the instructions on the package and you can expect a full rebound of the grass in those spots.