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Abstract
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The results of this talk are based on a joint work with Shuichi Kawashima (Kyushu University) and Renjun Duan (Chinese University of Hong Kong).In this talk, we consider the Cauchy problem for the first-order linear symmetrichyperbolic system of equations with relaxation. For this general linear degenerately dissipative system it is interesting to study its decay structure under additional conditions on the coefficient matrices and further investigate the corresponding time-decay property of solutions to the Cauchy problem. When the degenerate relaxation matrix is symmetric, Umeda-Kawashima-Shizuta proved the large-time asymptotic stability of solutions for a class of equations of hyperbolic-parabolic type with applications to both electromagneto-fluid dynamics and magnetohydrodynamics. The key idea that first introduced the so-called Kawashima-Shizuta condition is to design the compensating matrix to capture the dissipation of systems over the degenerate kernel space of the relaxation matrix. The typical feature of the time-decay property of solutions established in those work is that the high frequency part decays exponentially while the low frequency part decays polynomially with the rate of the heat kernel. Unfortunately, when the degenerate relaxation matrix is not symmetric, that results can not be applied any longer. In fact, this is the case for some concrete systems, for example, the Timoshenko system and the Euler-Maxwell system, where the linearized relaxation matrix indeed has a nonzero skew-symmetric part while it was still proved that solutions decay in time in some different way. Therefore, our purpose of this talk is to formulate some new structural conditions in order to extend the previous works to our general system when the relaxation matrix is not symmetric, which can include both the Timoshenko system and the Euler-Maxwell system.
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