On an initial-boundary value problem for a semilinear differential-algebraic system of partial differential equations of index $(1,0)$

We consider a mixed problem for some semilinear differential-algebraic system of partial differential equations of index $ (1,0) $ of the first order with a two-dimensional rectangular domain of definition. Using the method of characteristics and the method of successive approximations, the theorem of the existence and uniqueness of the classical solution of a mixed problem in the entire domain of definition is proved. It is shown that the solution and its first derivatives remain bounded in this region.