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i'm trying to find the Green's function for the modified heat equation $u_t = t * grad(u) $ in n dimensions using scaling arguments. I know how to do this for the regular heat equation, by switching to polar coordinates and noticing that the equation and initial conditions are invariant under the scaling r -> L*r, t-> L^2*t, G -> L^n*G and then letting L = 1/sqrt(t), everything reduces to an ODE which we solve.

I'm trying to do something similar here, i figured it would be easy but the only scaling that I can think of which preserves the initial conditions is r -> r and t -> L*t but this doesn't work, it gives a really dumb ODE since the left hand side goes to zero. I already solved this using Fourier series, the asnwer is basically integrating initial conditions times a Gaussianesque creature and a 1/t^n so i know what to look for, but for now i'm stuck :P

any suggestions? :)