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Large time asymptotics for partially dissipative hyperbolic systems. (English) Zbl 1237.35017
This paper is concerned with the long time asymptotics for partially dissipative hyperbolic systems. First, linear systems are analyzed and then existence of global solutions for nonlinear systems is discussed. The role of theoretical control tools is examined, and the proof of decay using Lyapunov functionals is developed. A general statement giving a decomposition for the solutions is established. The existence of global (in time) smooth solutions for the nonlinear systems around a constant equilibrium is studied. As an illustration, a class of nonlinear hyperbolic systems with relaxation is considered, and its uniform well posedness with respect to the relaxation parameter is established.

MSC:
35B40 Asymptotic behavior of solutions to PDEs
35L60 First-order nonlinear hyperbolic equations
35L45 Initial value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
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