The existence and uniqueness of a generalized solution $u=u(t,x)$ to the Cauchy problem for such kind of equations and for the more general one 
$$
u_t + \sum_{i=1}^n \frac{\partial}{\partial x_i}\varphi_i(t,x,u)+\psi(t,x,u)=0
$$
have been proved more than thirty years ago by Kruzhkov in his paper [1].
Kruzhkov builds $u$ in the class of bounded measurable function by using the method of vanishing viscosity introduced earlier by H. Hopf: the differentiability conditions required for the functions $\varphi(t,x,\cdot)$, $i=1,\ldots, n$ and $\psi(t,x,\cdot)$ are mild.<br>
Moreover, the first paragraph of the gives a nice historical introduction to this field of research, supported by the bibliography ash its end. 

**Reference**

[1] Stanislav Nikolaevich Kruzhkov, "First order quasilinear equations in several independent variables" (English, Russian original), Mathematics of the USSR, Sbornik, vol. 10 pp. 217-243 (1970), [DOI:SM1970v010n02ABEH002156](https://doi.org/10.1070/SM1970v010n02ABEH002156), [MR0267257](https://mathscinet.ams.org/mathscinet-getitem?mr=MR0267257), [Zbl 0215.16203](https://zbmath.org/0215.16203).