Abstract: We separately stated BV solutions for the Cauchy problems of the first order quasilinear equation and the first order quasilinear equation with source term with measure as initial value. Firstly, we displayed the general theory of above two Cauchy problems, such as existence, uniqueness and asymptotic behavior of solutions, etc. Then we displayed someresults based on it, i.e., existence, uniqueness and other properties of BV solutions for the cauchy problems of above two equations with σ-finite Borel measure as initial condition。

Key words: existence, uniqueness, Cauchy problem, quasilinear equation