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Decay structure for symmetric hyperbolic systems with non-symmetric relaxation and its application. (English) Zbl 1273.35051

The authors consider the following Cauchy problem of a first-order linear symmetric system of hyperbolic equations with relaxation \[ A^0u_t + \sum_{j=1}^n A^j u_{x_j} + L u =0 \quad \text{on} \quad (0, \infty) \times \mathbb{R}^n, \] with \(u(0,x) =u_0(x) \in \mathbb{R}^{m}\), \(A_j\) and \(L\) are \(m \times m\) real constant matrices where \( A_j\) are symmetric, \(A_0\) is positive definite, and \(L\) is nonnegative definite with a nontrivial kernel.When the degenerate relaxation matrix \(L\) is symmetric, the dissipative structure of the system is completely characterized by the Kawashima-Shizuta stability condition, and one can obtain the asymptotic stability result along with the explicit time-decay rate under that stability condition. In the present paper, the authors formulate a new structural condition, that includes the Kawashima-Shizuta one, and analyze the weak dissipative structure for general systems with non-symmetric relaxation.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
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