Global solvability of a mixed problem for a singular semilinear hyperbolic 1d system - V. M. Kyrylych; O. V. Peliushkevych (Математичні Студії)
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 $Π=\{(x,t)\colon 0<x<l<∞,\ 0<t<T<∞\}$ for the first order hyperbolic system of semi-linear equations of the form $$ \dfrac{\partial u}{\partial t}+Λ(x,t) \dfrac{\partial u}{\partial x}=f(x,t,u,v,w), \dfrac{\partial v}{\partial x}=g(x,t,u,v,w), \dfrac{\partial w}{\partial t }=h(x,t,u,v,w), $$ for a singular with orthogonal (degenerate) and non-orthogonal to the coordinate axes characte\-ristics and with nonlinear boundary conditions, where $Λ(x,t)=diag(λ_1(x,t),…,λ_k(x,t)),$ $u=(u_1,…,u_k),$ $v=(v_1,…,v_m),$ $w=(w_1,…,w_n),$ $f=(f_1,…,f_k),$ $g=(g_1,…,g_m),$ $h=(h_1,…,h_n)$ and besides $sign λ_i(0,t)=const≠ 0$, $sign λ_i(l,t)=const≠ 0 $ $\text{for all} \ t \in [0, T] $ and for all $i \in \{1,…,k\}$. 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.