Singular perturbations for quasilinear hyperbolic equations (Q790361)

In this paper we deal with the following mixed problem for quasilinear hyperbolic equations \[ L_{\epsilon}u_{\epsilon}\equiv \epsilon \partial^ 2u_{\epsilon}/\partial t^ 2-\sum^{n}_{i,j=1}a_{ij} \partial^ 2u_ 2/\partial x_ i\partial x_ j \] \[ +b_ 0(t) \partial u_{\epsilon}/\partial t+\sum^{n}_{i=1}b_ i(x,t,u_{\epsilon}) \partial u_{\epsilon}/\partial x_ i+c(x,t,u_{\epsilon})=f(x,t); \] \[ u_{\epsilon}|_{t=0}=\phi(x);\quad \partial u_{\epsilon}/\partial t|_{t=0}=\psi(x);\quad u_{\epsilon}|_ F=\chi(x,t), \] where \(\epsilon\) is a positive and small parameter; \(x=(x_ 1,x_ 2,...,x_ n)\). \(b_ 0(t)\geq \alpha>0\), where \(\alpha\) is a given real number. The method of two-variable expansion is proposed for constructing the boundary layer term. The M order uniformly valid asymptotic solutions are obtained and their errors are estimated. Thus we conclude that if the reduced problem has a unique solution, then for sufficiently small \(\epsilon\), the mixed problem for quasilinear hyperbolic equations has a unique solution.