Ueda, Yoshihiro; Duan, Renjun; Kawashima, Shuichi Decay structure of two hyperbolic relaxation models with regularity loss. (English) Zbl 1397.35035 Kyoto J. Math. 57, No. 2, 235-292 (2017). The paper is devoted to the decay structure of solutions to the following Cauchy problem for linear hyperbolic systems with relaxation: \[ u_t + A_m u_x + L_m u=0, \,\,\,u(0,x)=u_0(x). \] Under the assumptions that \(A_m\) is real symmetric, that \(L_m\) is real symmetric and nonnegative definite with nontrivial kernel and that one kind of “compensation condition” is satisfied on the kernel of \(L_m\) one expects the following basic estimate of the Fourier image \(v:=\hat{u}\) of \(u\): \[ |v(t,\xi)|\leq C e^{-c \lambda(\xi)t}|v(0,\xi)|, \] where \(c\) and \(C\) are positive constants and \(\lambda(\xi):=\frac{\xi^2}{1+\xi^2}\).If \(L_m\) is only nonnegative definite with nontrivial kernel, then under an additional “compensation condition” to the above one one expects the basic estimate \[ |v(t,\xi)|\leq C e^{-c \lambda(\xi)t}|v(0,\xi)|, \] where \(c\) and \(C\) are positive constants and \(\lambda(\xi):=\frac{\xi^2}{(1+\xi^2)^2}\).Both of the above estimates for the Fourier image imply decay estimates with a regularity loss in the second case. The goal of this paper is to introduce two special models with a new dissipative structure and a loss of regularity. Both systems are uniformly dissipative of type \((p,q)\). This means, that the eigenvalue \(\eta=\eta(i\xi)\), that is a solution of \[ \text{det}\,(\eta I + i\xi A_m + L_m)=0, \] satisfies the estimate \[ \operatorname{Re} \eta(i\xi) \leq c \frac{|\xi|^{2p}}{(1+|\xi|^2)^q} \] for all \(\xi \in \mathbb{R}\), where \(c>0\) and \((p,q)\) is a pair of positive integers. The first model is uniformly dissipative of type \((m-3,m-2)\), the second model is uniformly dissipative of type \(((3m-10)/2,2(m-3))\). Then the authors prove the estimates \[ |v(t,\xi)|\leq C e^{-c \lambda_k(\xi)t}|v(0,\xi)| \] for \(k=1,2\) with \(\lambda_1(\xi)=\frac{\xi^{2(m-3)}}{(1+\xi^2)^{m-2}}\) for the first model and and \(\lambda_2(\xi)=\frac{\xi^{3m-10}}{(1+\xi^2)^{2(m-3)}}\) for the second model. The proofs base on the Fourier energy method. They require more “compensating conditions”. The crucial point is to justify if the choice of \((p,q)\) is optimal. Reviewer: Michael Reissig (Freiberg) Cited in 1 ReviewCited in 7 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs 35L45 Initial value problems for first-order hyperbolic systems Keywords:symmetric hyperbolic systems in 1d; energy method in Fourier space PDFBibTeX XMLCite \textit{Y. Ueda} et al., Kyoto J. Math. 57, No. 2, 235--292 (2017; Zbl 1397.35035) Full Text: DOI arXiv