Global solvability of a mixed problem for a singular semilinear hyperbolic 1d system

Abstract

Using the method of characteristics and the Banach fixed point theorem (for the Bielecki metric), in the paper it is established the existence and uniqueness of a global (continuous) solution of the mixed problem in the rectangle $\Pi=\{(x,t)\colon 0<x<l<\infty,\ 0<t<T<\infty\}$ for the first order hyperbolic system of semi-linear equations of the form $$ \dfrac{\partial u}{\partial t}+\Lambda(x,t) \dfrac{\partial u}{\partial x}=f(x,t,u,v,w),  \dfrac{\partial v}{\partial x}=g(x,t,u,v,w), \dfrac{\partial w}{\partial t }=h(x,t,u,v,w), $$ for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characte\-ristics and with nonlinear boundary conditions, where $\Lambda(x,t)=\mathrm{diag}(\lambda_1(x,t),\ldots,\lambda_k(x,t)),$ $u=(u_1,\ldots,u_k),$ $v=(v_1,\ldots,v_m),$ $w=(w_1,\ldots,w_n),$ $f=(f_1,\ldots,f_k),$ $g=(g_1,\ldots,g_m),$ $h=(h_1,\ldots,h_n)$ and besides $\textrm{sign } \lambda_i(0,t)=\mathrm{const}\neq 0$, $\textrm{sign } \lambda_i(l,t)=\mathrm{const}\neq 0 $ $\text{for all} \ t \in [0, T] $ and  for all $i \in \{1,\ldots,k\}$.  The presence of non-orthogonal and degenerate characteristics of the hyperbolic system for physical reasons indicates that part of the oscillatory disturbances in the medium propagates with a finite speed, and part with an unlimited one. Such a singularity (degeneracy of characteristics) of the hyperbolic system allows mathematical interpretation of many physical processes, or act as auxiliary equations in the analysis of multidimensional problems.