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This paper is concerned with the dissipative structure for the linear symmetric hyperbolic system with non-symmetric relaxation. If the relaxation matrix of the system has symmetric property, Shizuta and Kawashima in 1985 introduced the suitable stability condition called Classical Stability Condition in this paper, and Umeda, Kawashima and Shizuta in 1984 analyzed the dissipative structure of the standard type. On the other hand, Ueda, Duan and Kawashima in 2012 and 2018 focused on the system with non-symmetric relaxation, and got the partial result which is the extension of known results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, our purpose of this paper is to extend the stability theory introduced by Shizuta and Kawashima in 1985 and Umeda, Kawashima and Shizuta in 1984 for our general system. [ABSTRACT FROM AUTHOR]

We study the compressible viscous magnetohydrodynamic (MHD) system and investigate the large-time behavior of strong solutions near constant equilibrium (away from vacuum). In the 80s, Umeda et al. considered the dissipative mechanisms for a rather general class of symmetric hyperbolic–parabolic systems, which is given by Reλ(iξ)≤−Cη(ξ),η(ξ)=∣ξ∣21+∣ξ∣2(∣ξ∣≠0, C>0). Here, λ=λ(iξ) denotes the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of Lq-L2(q∈[1,2)) type for solutions to the MHD system. Now, by using Fourier analysis techniques, we present more precise description for the large-time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the Lp norm (the slightly stronger Ḃp,10 norm in fact) of global solutions with the critical regularity, decays like t−d(1/p−1/4) as t→∞. Our decay results hold in case of large highly oscillating initial velocity and magnetic fields, which improve Kawashima’s classical efforts. [ABSTRACT FROM AUTHOR]

The conformal identity for the solution of the Chern-Simons gauged wave equations is derived. As applications of it, we show some decay estimates and the nonexistence of nontrivial solutions. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for damped wave equations with a time-dependent propagation speed and dissipation. The model of interest is We assume . Then we propose a classification of dissipation terms in non-effective and effective. In each case we derive estimates for kinetic and elastic type energies by developing a suitable WKB analysis. Moreover, we show optimality of results by the aid of scale-invariant models. Finally, we explain by an example that in some estimates a loss of regularity appears. [ABSTRACT FROM AUTHOR]

We derive L∞-L1 decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces (M, g). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in ℝn. Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that n = M is odd. [ABSTRACT FROM AUTHOR]

We consider quasi-linear hyperbolic systems that describe the motion of viscoelastic materials. The decay property for this system exhibits a loss of regularity, and we establish a global existence theory together with sharp decay estimates, under a smallness condition and under sufficient regularity on the initial data. We combine together various tools of analysis, especially energy estimates in Fourier space, the time-weighted energy method, and the semi-group method. [ABSTRACT FROM AUTHOR]