Perturbation theory for homogeneous evolution equations

Daniel Hauer


In this paper, we develop a perturbation theory to show that if a homogeneous operator of order \(\alpha\neq1\) 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.

Keywords: Nonlinearsemigroups,regularityoftime-derivative,homogenousop- erators, p-Laplace-Beltrami operator, total variational flow on manifolds.

AMS Subject Classification: Primary 47H20; secondary 47h06, 47H14, 47J35, 35B65.

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Tuesday, March 24, 2020