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### Continuous dependence of the entropy solution of general parabolic equation

Annales de la faculté des sciences de Toulouse Mathématiques

We consider the general parabolic equation : ${u}_{t}-\Delta b\left(u\right)+div\phantom{\rule{4pt}{0ex}}F\left(u\right)=f$ in $Q=\right]0,T\left[×{ℝ}^{N},\phantom{\rule{4pt}{0ex}}T>0$ with $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{u}_{0}\in {L}^{\infty }\left({ℝ}^{N}\right),$ $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}a.e\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in \right]0,T\left[,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}f\left(t\right)\in {L}^{\infty }\left({ℝ}^{N}\right)$ and ${\int }_{0}^{T}{∥f\left(t\right)∥}_{{L}^{\infty }\left({ℝ}^{N}\right)}dt<\infty .$ 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...

### Renormalized solution for nonlinear degenerate problems in the whole space

Annales de la faculté des sciences de Toulouse Mathématiques

We consider the general degenerate parabolic equation : ${u}_{t}-\Delta b\left(u\right)+div\phantom{\rule{4pt}{0ex}}\stackrel{˜}{F}\left(u\right)=f\phantom{\rule{28.45274pt}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}Q\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}\right]0,T\left[×{ℝ}^{N},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}T>0.$ We suppose that the flux $\stackrel{˜}{F}$ 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 $\stackrel{˜}{F}\left(r\right)=F\left(b\left(r\right)\right)$ and an assumption on the modulus of continuity of $b$. The novelty of this work is that $\Omega ={ℝ}^{N}$, ${u}_{0}$,...

### Dépendance continue de solutions généralisées locales

Annales de la Faculté des sciences de Toulouse : Mathématiques

### Solution généralisée locale d'une équation parabolique quasi linéaire dégénérée du second ordre

Annales de la Faculté des sciences de Toulouse : Mathématiques

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