We examine nonlinear scale-spaces in the general form ut = P (u(t)), where P is a bounded nonlinear operator. We seek solutions with separation of variables in space and time u(x, t) = a(t)f (x), where f is the initial condition. We term these as shape-preserving flows and provide necessary and sufficient conditions for their existence. We show that homogeneous operators admit the above conditions. It turns out that the initial condition must admit a nonlinear eigenvalue problem, with respect to the operator P , P (f) = λf , where λ is the eigenvalue. In this case we can formulate a closed form solution for any P which is homogeneous of positive degree. Consequently, we can determine if a finite extinction time exists. We show that in all cases the extinction time is inversely proportional to the eigenvalue λ. Following the above analysis, we generalize the total-variation and one-homogeneous transforms to a homogeneous spectral representation. The notions of spectrum, generalized Parseval's theorem and filtering are defined. We apply these formulations to the p−Laplace operator for 1 < p < 2.