That's burn from excess nitrogen in the dogs urine. Give the patches a very good flush of water to wash the nitrogen away and the grass should come up again. If not, reseed the areas.

In the U.S., garden centers carry bags of seed and mulch, or seed and fluff, specifically for repairing dog damage. I would be surprised if they aren't available in the UK, if a good soaking doesn't do the job.

For instant results (depending if you have this in the UK) there is a grass colorant which you can spray on to turn the brown spots into green.

You'll probably find that when the grass re-grows it will be much lusher, taller and greener than the rest. That's what we've always found. I don't believe that dog pee does any harm though those who like a bowling green lawn will disagree!

Can't believe that anyone would want to spray dye on their grass!

It's medium common here, when the Bermudagrass goes dormant for winter, and people don't want to plant ryegrass for the winter.