Tariel Kiguradze

Some Boundary Value Problems for Systems of Linear Partial Differential Equations of Hyperbolic Type

The linear hyperbolic system \begin{equation} \frac{\pa^2 u}{\pa x\pa y}=\cp_0(x,y)u+\cp_1(x,y)\frac{\pa u}{\pa x}+ \cp_2(x,y)\frac{\pa u}{\pa y}+q(x,y)\tag{$1$} \end{equation} is considered, where $\cp_0,\;\cp_1,\;\cp_2$ and $q$ are respectively the $\nn$ matrices and the $n$-dimensional vector whose components are measurable and essentially bounded functions in the rectangle $\cd_{ab}=[0,a]\tm[0,b]$ or in the strip $\cd_b=\linebreak =\br\tm[0,b]$. For system (1) problems with general functional boundary conditions are investigated in the rectangle $\cd_{ab}$ and problems on bounded, almost-periodic and periodic solutions in the strip $\cd_b$. Optimal in a certain sense conditions are established, guaranteeing the unique solvability of the problems and the stability of their solutions with respect to small perturbations of the coefficients of system (1) and of the boundary conditions.

Mathematics Subject Classification: 35L55

Key words and phrases: System of partial differential equations of hyperbolic type, boundary value problem in the rectangle, classical solution, absolutely continuous solution, generalized solution, periodic solution, almost-periodic solution, bounded solution.