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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 \(E_T\).

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
Full Text: DOI
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