On a construction of weak solutions to linear hyperbolic partial differential systems with the higher integrable gradients

By taking linear hyperbolic partial differential equations as an illustration, we make a trial of constructing weak solutions, with the higher integrable gradients in the sense of Gehring, to hyperbolic differential equations with initial and boundary conditions. We adopt Rothe's method and follow the calculation which has been expanded by Giaquinta and Struwe in dealing with parabolic equations. To establish the scheme we evaluate some local estimates for solutions to Rothe's approximations to hyperbolic differential equations. Bibl. 6 titles.