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Journal of Evolution Equations

, Volume 18, Issue 3, pp 1427–1469 | Cite as

Strong solutions to a nonlinear stochastic Maxwell equation with a retarded material law

  • Luca HornungEmail author
Article

Abstract

We study the Cauchy problem for a semilinear stochastic Maxwell equation with Kerr-type nonlinearity and a retarded material law. We show existence and uniqueness of strong solutions using a refined Faedo–Galerkin method and spectral multiplier theorems for the Hodge–Laplacian. We also make use of a rescaling transformation that reduces the problem to an equation with additive noise to get an appropriate a priori estimate for the solution.

Keywords

Stochastic Maxwell equations Kerr-type nonlinearity Retarded material law Monotone coefficients Weak solutions Strong solutions Generalised Gaussian bounds Spectral multiplier theorems Hodge–Laplacian Rescaling transformation 

Mathematics Subject Classification

35Q61 35R60 60H15 34L05 32A70 60H30 76M35 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for AnalysisKarlsruhe Institute of TechnologyKarlsruheGermany

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