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

Van Harten, A.; Van Hassel, R. R.

SIAM J. Math. Anal. 16, 1258-1267 (1985).

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

Keywords:

matched asymptotic expansions; singular perturbation; formal approximation; Schauder’s fixed point theorem

Citations:

Zbl 0498.35001