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On spae-time decay properties of solutions to hyperbolic-elliptic coupled systems. (English) Zbl 1021.35015

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.

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)
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