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Blow-up solutions to Landau–Lifshitz–Maxwell systems. (English) Zbl 1222.35046

The authors discuss Maxwell’s equations with material laws \(D_{t}=E_{t}+\sigma E\), \(B=H+\beta Z\), where \(Z\) is a directional field, i.e. \(\left|Z\right|=1\) governed by the Landau-Lifshitz equation. The paper has two main goals. First the authors show local existence of smooth solutions. Secondly, it is shown that for certain initial data smooth local solutions of the coupled Landau-Lifshitz-Maxwell system show a finite time blow-up.

MSC:

35B44 Blow-up in context of PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35B65 Smoothness and regularity of solutions to PDEs
35K55 Nonlinear parabolic equations
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