Roughly speaking, we have good understanding of the solution to heat equation $u_t-\Delta u=0$, on bounded or unbounded domain. For example, we know the decay rate, we know it generates analytic semigroups, we know Fourier transform can be used to solve the whole space case, we know its connection to stochastic modelling, etc. $\Delta u$ has been interpreted as a diffusion mechanism.

What happens when we increase the 'order' of Laplacian? More precisely, for equation $u_t+(-\Delta)^su=0, s>1$, does anything change comparing to heat equation? Here $(-\Delta)^su$ is certainly defined by Fourier transform. By change I mean can we can say things like the solution decays faster; do we need to modify the equation in order for it to make sense, etc.

Similar question can be asked for wave equations.

Are there any papers or references in this aspect?

Thanks a lot!