Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems (Q988142)

There is considered the singularly perturbed problem with unknown vector \(u^\varepsilon(t,x) \in \mathbb R^N\), \(\partial_tu^\varepsilon+A(t,x,u^\varepsilon)\partial_xu^\varepsilon =\varepsilon B(t,x,u^\varepsilon)\partial^2_x u^\varepsilon\), \(x>0\), \(0<t<T\), \(u^\varepsilon|_{t=0}=u^\varepsilon_0(x)\), \(u^\varepsilon|_{x=0}=0\), where \(B\) is a symmetric, positive definite matrix, \(A(t,x,u^\varepsilon)|_{x=0}=0\) for all \(t\), \(u^\varepsilon\). Using asymptotic expansion of the initial function \(u^\varepsilon_0(x)\) with respect to \(\varepsilon\), the authors approximate the solution \(u^\varepsilon\) by the solutions of suitable problems. With the help of weighted estimates with powers of \(x\) weights, the convergence of the solution \(u^\varepsilon\) (as \(\varepsilon\to 0\)) to the solution \(u\) of the unperturbed problem is proved, i.e., of the degenerate hyperbolic problem \(\partial_tu+A(t,x,u)\partial_xu =0\), \(x>0\), \(0<t<T\), \(u|_{t=0}=u^0_0(x)\), where \(u^0_0(x)=u^\varepsilon_0(x)|_{\varepsilon=0}\).