Purchase 1 or 2 pieces of foamcore board - looks like stiff poster board, often can find them at Dollar Tree, etc. - and make a "pocket" to hold your board safely. I use duct tape around three sides of the 2 boards, leaving one open so I can slide the mat into the pocket.
The number of boards you buy will depend on the size of your cutting mat. I made these pockets in several sizes when I lived in Texas, where even placing a mat flat on the floor of the car trunk would allow them to warp from heat. Never had a problem in years after beginning to use these pockets.
An added benefit -- cover one side of the pocket (or both if you wish) with flannel or a piece of leftover batting and you have a portable design wall. I even slide my long ruler in beside the mat.
Jan in VA