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We consider the Kerr–Debye model, describing the electromagnetic wave propagation in a nonlinear medium exhibiting a finite response time. This model is quasilinear hyperbolic and endowed with a dissipative entropy. We consider the Cauchy problem in the three-dimensional case and show that, if the initial data are sufficiently small, the solutions are global in time.
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Mohamed Kanso, Global existence of small solutions to the Kerr–Debye model for the three-dimensional Cauchy problem. Port. Math. 68 (2011), no. 4, pp. 389–409