Perturbation theory for homogeneous evolution equations

Daniel Hauer

Published 2020-03-14Version 1

In this paper, we develop a perturbation theory to show that if a homogeneous operator of order $\alpha\neq 1$ is perturbed by a Lipschitz continuous mapping then every mild solution of the first-order Cauchy problem governed by these operators is strong and the time-derivative satisfies a global regularity estimate. We employ this theory to derive global $L^q$-$L^{\infty}$-estimates of the time-derivative of the evolution problem governed by the $p$-Laplace-Beltrami operator and total variational flow operator respectively perturbed by a Lipschitz nonlinearity on a non-compact Riemannian manifold.