Why does it seem that there is little care for the problems of the West?
I do not know if your question is quite accurate. While it does seem that there is a lot of talk about the problems in other parts of the world, there is a good amount of emphasis on domestic issues and issues of the West. All you need to do and watch the news and you will hear a lot on the economy and jobs. You will also hear a lot about the EU and the problems in Greece. All of this discussion is focused on the West. Moreover, as you listen to the election coming up, you will hear much more.
With all this stated, I will admit that there much about the East and Middle East. There is good reason for this. First, much of the poverty is outside the West. So, when there is a disaster that overtakes these places, it is far worse than in the West. Second, from a political and social point of view, there is much instability in places outside the West. Think of what happened this summer in the Middle East and think of what is happening in Syria. These cultural events are happening outside the West and changing the world.