Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type

This work is concerned with the gradient flow of absolutelyp-homogeneous convex functionals on a Hilbert space, which we show to exhibit finite (p<2 We give upper bounds for the finite extinction time and establish sharp convergence rates of the flow. Moreover, we study next order asymptotics and prove that asymptotic profiles of the solution are eigenfunctions of the subdifferential operator of the functional. To this end, we compare with solutions of an ordinary differential equation which describes the evolution of eigenfunction under the flow. Our work applies, for instance, to local and non-local versions of PDEs likep-Laplacian evolution equations, the porous medium equation, and fast diffusion equations, herewith generalizing many results from the literature to an abstract setting. We also demonstrate how our theory extends to general homogeneous evolution equations which are not necessarily a gradient flow. Here, we discover an interesting integrability condition which characterizes whether or not asymptotic profiles are eigenfunctions.