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Semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation. (English) Zbl 1171.35073
Summary: We investigate totally linearly degenerate hyperbolic systems with relaxation. We aim to study their semilinear behavior, which means that the local smooth solutions cannot develop shocks, and the global existence is controlled by the supremum bound of the solution. In this paper we study two specific examples: the Suliciu-type and the Kerr-Debye-type models. For the Suliciu model, which arises from the numerical approximation of isentropic flows, the semilinear behavior is obtained using pointwise estimates of the gradient. For the Kerr-Debye systems, which arise in nonlinear optics, we show the semilinear behavior via energy methods. For the original Kerr-Debye model, thanks to the special form of the interaction terms, we can show the global existence of smooth solutions.

MSC:
35L65 Hyperbolic conservation laws
35L80 Degenerate hyperbolic equations
35L45 Initial value problems for first-order hyperbolic systems
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