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We consider the general parabolic equation : $u_t-\Delta b\left(u\right)+divF\left(u\right)=f$ in $Q=]0,T[\times {ℝ}^N,T\>0$ with ${\displaystyle u_0\in L^{\infty }\left({ℝ}^N\right),}$ $\text{for}a.et\in ]0,T[,f\left(t\right)\in L^{\infty }\left({ℝ}^N\right)$ and ${\displaystyle {\int }_0^T{∥f\left(t\right)∥}_{{\displaystyle L^{\infty }\left({ℝ}^N\right)}}dt\<\infty .}$ We prove the continuous dependence of the entropy solution with respect to $F,$ $b,$ $f$ and the initial data $u_0$ of the associated Cauchy problem. This type of solution was introduced and studied in [MT3]. We start by recalling the definition of weak solution and entropy solution. By applying an abstract result (Theorem 2.3), we get...

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 $\Omega ={ℝ}^N$, $u_0$,...