Ukrainian Mathematical Bulletin

Volume 1, Issue 2, 2004 pp. 221-235.

Problem without Initial Conditions for a Degenerate Hyperbolic System of the First Order

Authors: S.P.Lavrenyuk and M.A.Oliskevich
Author institution: Tadeusz Kosciuszko Krakow Polytechnics,Poland, Ivan Franko Lviv National University, Lviv, Ukraine

Summary: We consider the problem without initial conditions for a hyperbolic system $$ u_t-\Lambda(x,t) u_x+A(x,t)u+G(x,t,u)=f(x,t) $$ in a domain $\{(x,t)\,:\,0 x a ,\,-\infty t T\},$ where $\Lambda$ and $A$ are square matrices and $u,$ $G,$ and $f$ are vectors. Moreover, for $x=0$ or $x=a$, $\Lambda$ is a diagonal matrix with degenerate elements. The existence and uniqueness of a weak solution of this problem are proved independently of the behavior of the data of the problem and its solution as $t\to -\infty.$

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