We consider the general degenerate parabolic equation :$$u_t-\Delta b\left(u\right)+div\tilde{F}\left(u\right)=f\text{in}Q=]0,T[\times {ℝ}^N,T\>0.$$We suppose that the flux $\tilde{F}$ is continuous, $b$ is nondecreasing continuous and both functions are not necessarily Lipschitz. We prove the existence of the renormalized solution of the associated Cauchy problem for $L^1$ initial data and source term. We establish the uniqueness of this type of solution under a structure condition $\tilde{F}\left(r\right)=F\left(b\left(r\right)\right)$ and an assumption on the modulus of continuity of $b$. The novelty of this work is that $\Omega ={ℝ}^N$, $u_0$, $f\in L^1$, $b$, $\tilde{F}$ are not Lipschitz...