I have experience in this department, and I have to say its a very bad idea to use linoleum on the walls. I built a pen for my bottle lamb, and I thought i was very smart putting the linoleum on the walls too, thinking it would be easy to clean. After about 2 months the linoleum started falling and peeling off the walls. Linoleum is not made for walls as the sticky is not strong enough to go against gravity. The only thing I could suggest is to use nails and nail the 4 corners, but I imagine that might not work either it could puff up and still let the walls get exposed and gross.
Hope this helps
Hope this helps