Ukrainian Mathematical Bulletin

Volume 2, Issue 4, 2005 pp. 449-466.

System of Hyperbolic Variational Inequalities of the First Order

Authors: Nataliya I. Huzil' and Serhii P. Lavrenyuk
Author institution: Nataliya I. Huzil', Serhii P. Lavrenyuk, Ivan Franko Lviv National University, 1 Universytets'ka Str., 79602, Lviv, Ukraine, sp_lavreniuk@franko.lviv.ua

Summary: We investigate the inequality \begin{equation*} \int\limits_{Q_{\tau}}\biggl(u_t+\sum\limits_{i=1}^lA_i(x,t)u_{x_i}+ C(x,t)u+g(u)-f(x,t),v-u\biggr)dx\,dt\ge0, \end{equation*} $\tau\in (0,T],$ with the initial condition $ u(x,0)=u_0(x), $ where $(\cdot,\cdot)$ is the scalar product in the space $\mathbb{R}^m,$ $A_i$ and $C$ are square matrices of order $m$, $g=\mbox{\rm col}(g_1,\dots,g_m), $ $u=\mbox{\rm col}(u_1,\dots,u_m),$ $f=\mbox{\rm col}(f_1,\dots,f_m),$ and $v=\mbox{\rm col}(v_1,\dots,v_m),$ in a bounded domain $Q=\Omega\times (0,T).$

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