Hyperbolic problems with totally characteristic boundary

Abstract

We study first-order symmetrizable hyperbolic \(N\times N\) systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at \(x=0\) , these systems take the form $$\begin{aligned} \partial _t u + {{\mathcal {A}}}(t,x,y,xD_x,D_y) u = f(t,x,y), \quad (t,x,y)\in (0,T)\times {{\mathbb {R}}}_+\times {{\mathbb {R}}}^d, \end{aligned}$$ where \({{\mathcal {A}}}(t,x,y,xD_x,D_y)\) is a first-order differential operator with coefficients smooth up to \(x=0\) and the derivative with respect to x appears in the combination \(xD_x\) . No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator \(\partial _t + {{\mathcal {A}}}(t,x,y,xD_x,D_y)\) is well-posed in that scale. More specifically, solutions u exhibit formal asymptotic expansions of the form $$\begin{aligned} u(t,x,y) \sim \sum _{(p,k)} \frac{(-1)^k}{k!}x^{-p} \log ^k \!x \, u_{pk}(t,y) \quad \hbox { as}\ x\rightarrow +0 \end{aligned}$$ where \((p,k)\in {{\mathbb {C}}}\times {{\mathbb {N}}}_0\) and \(\Re p\rightarrow -\infty \) as \(|p|\rightarrow \infty \) , provided that the right-hand side f and the initial data \(u|_{t=0}\) admit asymptotic expansions as \(x \rightarrow +0\) of a similar form, with the singular exponents p and their multiplicities unchanged. In fact, the coefficients \(u_{pk}\) are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients  \(u_{pk}\) solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator \(\partial _t+{{\mathcal {A}}}(t,x,y,xD_x,D_y)\) is well-posed in the scale of standard Sobolev spaces \(H^s((0,T)\times {{\mathbb {R}}}_+^{1+d})\) .