There is a world somewhere between reality and fiction. Although ignored by many, it is very real and so are those living in it. This forum is about the natural world. Here, wild animals will be heard and respected. The forum offers a glimpse into an unknown world as well as a room with a view on the present and the future. Anyone able to speak on behalf of those living in the emerald forest and the deep blue sea is invited to join.
He will be justified to move southward, and live in the Canadian/Alaskan tundra with their smaller brown bear companions (or enemies, depending upon one's viewpoint): the Inland Tundra Grizzly. This is assuming that prey drops significantly and ice is reduced to 5% in the polar bear's native habitat.
Both will compete in terms of mates, food-abundant regions, and instinctive male rivalry. Competition for food type should not be a problem; polar bears are more carnivorous, and the grizzly is more suited to fruity shrubbery and carrion.
The problem is where to acquire their food. Both share similar territory (assuming the Arctic King moves south into the Brute King's headquarters), for example, if a polar bear hunts a small elk in a dominant brown bear's berry field, there will be problems. This situation would be similar to someone stealing a unfavorable food item from another's cart; even though the food item in question is unfavorable, the victim will most likely become angered.
Another problem with food: polar bears are best suited for aquatic animals, they rarely give chase to their musk oxen or caribou when predating. They implement ambush similar to their feline cousins (although, also like other bears and some felines, they can fight face-to-face with their prey.) So when a moose or large bison comes along in their path, they won't have that intuition or adaptability to know how to hunt those two animals.
Smaller prey are not an option; remember, we are talking about a 800-1200 pound bear with a greater nutritional load, slightly slower sprinting, and much quicker overheating than a 200-pound bison or moose calf.
Polar bears would face plenty of issues if the situation is described as above, but, like other bears, they will adapt quickly (albeit a tiny bit slower since they are specialists in cryo-environments and their specific physical needs.)