Iguchi, Tatsuo; Kawashima, Shuichi On spae-time decay properties of solutions to hyperbolic-elliptic coupled systems. (English) Zbl 1021.35015 Hiroshima Math. J. 32, No. 2, 229-308 (2002). The authors study the initial value problem for a certain class of hyperbolic-elliptic coupled systems, more precisely \[ \begin{cases} w_t+F(w,q)_x=0,\\ -q_{xx} +\widetilde Rq+\mu(w,q)G(w,q)_x=0,\\ w(x,0)=w_0(x),\end{cases} \] where \(w=w(x,t)\) and \(q=q(x,t)\) are unknown functions taking values in a domain \(\Omega\subseteq \mathbb{R}^m\) and \(\mathbb{R}^n\), respectively, \(x\in \mathbb{R}^1\), \(t\geq 0\), while \(F=F(w,q)\), \(G=G(w,q)\) and \(\mu(w,q)\) are given smooth mappings from \(\Omega\times \mathbb{R}^n\) into \(\mathbb{R}^m\), \(\mathbb{R}^n\) and \(\mathbb{R}^1_+=\{x\in \mathbb{R}^1\mid x>0\}\), respectively, and \(\widetilde R\) is a positive definite \(n\times n\) matrix of real constant entries. They prove that the solution is time asymptotically approximated by the superposition of diffusion waves constructed in terms of the selfsimilar solutions of the generalized Burgers equation. Reviewer: Messoud Efendiev (Berlin) Cited in 14 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35Q35 PDEs in connection with fluid mechanics 35L70 Second-order nonlinear hyperbolic equations 35M20 PDE of composite type (MSC2000) 35Q53 KdV equations (Korteweg-de Vries equations) Keywords:diffusion waves; selfsimilar solutions; generalized Burgers equation PDFBibTeX XMLCite \textit{T. Iguchi} and \textit{S. Kawashima}, Hiroshima Math. J. 32, No. 2, 229--308 (2002; Zbl 1021.35015) Full Text: DOI