Studying equations such as the heat equation and the wave equation in n$\le$3 dimensions makes sense to me as these are physical processes.
I can also justify studying PDEs in $\mathbb{R}^n$ because perhaps they might be governing things which have a higher dimension such as some kind of image manipulation processes over each dimension.
Can someone provide me some insight as to why we spend so much effort on generalizing the physical equations to higher dimensions when its often true that studying the problem in a higher dimension leads to more difficulty in the analysis.
Is it simply that these are our poster child problems in which we 'learned' to work with higher dimensional PDEs, or are there some deeper uses for these physical equations in higher dimensions.