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Exponential decay of quasilinear Maxwell equations with interior conductivity

  • Irena Lasiecka
  • Michael Pokojovy
  • Roland SchnaubeltEmail author
Article
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Abstract

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of \(\mathbb {R}^{3}\) with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical \(L^{2}\)-Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time \(H^{3}\)-solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy, the lack of solenoidality, and the fact that only partial dissipation is imposed on the system. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate.

Keywords

Quasilinear Maxwell equations Boundary conditions of perfect conductor Inhomogeneous anisotropic material laws Global existence Exponential stability 

Mathematics Subject Classification

35Q61 35F61 35A01 35A02 35B40 35B65 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Irena Lasiecka
    • 1
    • 2
  • Michael Pokojovy
    • 3
  • Roland Schnaubelt
    • 4
    Email author
  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA
  2. 2.IBS, Polish Academy of SciencesWarsawPoland
  3. 3.Department of Mathematical SciencesUniversity of Texas at El PasoEl PasoUSA
  4. 4.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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