## A quasi-linear, singular perturbation problem of hyperbolic type.(English)Zbl 0612.35007

The authors consider the singular perturbation problem $\epsilon L_ 2(u)+M_ 1(u)=h,\quad u(t=0)=f,\quad u_ t(t=0)=g,$ where $$L_ 2(u)=u_{tt}-c^ 2u_{xx}$$, $$M_ 1(u)=au_ t+bu_ x+du$$, a,b,c,d functions of (x,t,u), $$x\in {\mathbb{R}}$$, $$a,c>0$$. A formal approximation $\tilde u(x,t,\epsilon)=\sum^{N}_{m=0}\epsilon^ m w_ m(x,t)+\sum^{N+1}_{m=1}\epsilon^ mv_ m(x,t,\epsilon)- \epsilon^{N+1} v_{N+1}(x,0)$ is constructed for (x,t)$$\in D$$ for some $$D\subset {\mathbb{R}}^ 2$$. It is shown that $$\tilde u$$ approaches u of order $$\epsilon^{N+1}$$ if the characteristics of the unperturbed operator $$(\epsilon =0)$$ are time-like with respect to those of the perturbed one. The proof is based on Schauder’s fixed point theorem. This work generalizes that of R. Geel [”Singular perturbations of hyperbolic type”, Thesis, Univ. Amsterdam (1978; Zbl 0498.35001)].
Reviewer: R.Racke

### MSC:

 35B25 Singular perturbations in context of PDEs 35L70 Second-order nonlinear hyperbolic equations 35C20 Asymptotic expansions of solutions to PDEs

Zbl 0498.35001
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