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Harnack estimates for quasi-linear degenerate parabolic differential equations. (English) Zbl 1221.35213
The article is devoted to proving Harnack estimates for quasi-linear degenerate parabolic equations of the form $u_t - \text{div }\vec A(x,t,u,Du) = B(x,t,u,Du)$ given in the cylindrical domain $$E_t = E\times (0,T]$$ where $$E$$ is an open set of $$\mathbb{R}^N$$. Here the measurable functions $$\vec A:\, E_T\times \mathbb{R}^{N+1} \to \mathbb{R}^N$$ and $$B:\, E_T\times \mathbb{R}^{N+1} \to \mathbb{R}$$ subject to the constraints \begin{aligned} & \vec A(x,t,u,Du)\cdot Du \geq C_0| Du| ^p - C^p,\\ & | \vec A(x,t,u,Du)| \leq C_1| Du| ^{p-1} + C^{p-1}, \quad \text{a.e. in } E_t,\\ & | B(x,t,Du)| \leq C| Du| ^{p-1}, \end{aligned} where $$p > 2$$, $$C_0$$ and $$C_1$$ are given positive constants, and $$C$$ is a non-negative constant. For $$\varrho > 0$$ let $$K_{\varrho}$$ be the cube centered at the origin on $$\mathbb{R}^N$$ with edge $$2\varrho$$, and for $$y\in \mathbb{R}^N$$ let $$K_{\varrho}(y)$$ denote the homothetic cube centered at $$y$$. For $$\theta > 0$$ set $Q^-_{\varrho}(\theta) = K_{\varrho} \times (-\theta\varrho^p,0], \quad Q^+_{\varrho}(\theta) = K_{\varrho} \times (0,\theta\varrho^p],$ and for $$(y,s) \in \mathbb{R}^N\times \mathbb{R}$$, \begin{aligned} &(y,s) + Q^-_{\varrho}(\theta) = K_{\varrho}(y) \times (s-\theta\varrho^p,s], \\ & (y,s) + Q^+_{\varrho} (\theta) = K_{\varrho}(y) \times (s,s+\theta\varrho^p]. \end{aligned} Let $$u = C_{\text{loc}}(0,T;L^2_{\text{loc}}(E)\cap L^P_{\text{loc}}(0,T;W^{1,p}_{\text{loc}}(E))$$ be a continuous, non-negative weak solution of the above-mentioned equation, fix $$(x_0,t_0)\in E_T$$ such that $$u(x_0,t_0) > 0$$ and construct the cylinders $(x_0,y_0) + Q^{\pm}_{4\varrho}(\theta), \quad \theta = \left(\frac{c}{u(x_0,t_0)}\right)^{p-2},$ where $$c > 0$$. The main results (Intrinsic Harnack Inequality and Hölder continuity) read: There exist positive constants $$c$$ and $$\gamma$$ depending only upon the data, such that for all intrinsic cylinders $$(x_0,y_0) + Q^{\pm}_{4\varrho}(\theta)$$ contained in $$E_T$$, either $$u(x_0,t_0) \leq \gamma C\varrho$$ or $u(x_0,t_0) \leq \gamma \inf_{K_{\varrho}(x_0)}u(x,t_0 + \theta\varrho^p).$ Any locally bounded weak solutions of the equation, with no sign restrictions, is locally Hölder continuous in $$E_T$$.

##### MSC:
 35K65 Degenerate parabolic equations 35K59 Quasilinear parabolic equations 35B45 A priori estimates in context of PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35D30 Weak solutions to PDEs
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