On generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation in the class of locally summable functions

We construct a theory of locally summable generalized entropy solutions (g.e. solutions) of the Cauchy problem for a first-order non-homogeneous quasilinear equation with continuous flux vector satisfying a linear restriction on its growth. We prove the existence of greatest and least g.e. solutions, suggest sufficient conditions for uniqueness of g.e. solutions, prove several versions of the comparison principle, and obtain estimates for the $L^p$-norms of solution with respect to the space variables. We establish the uniqueness of g.e. solutions in the case when the input data are periodic functions of the space variables.