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Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. (English) Zbl 0708.35017
The author considers degenerate parabolic equations of the type $(1)\quad u_ t-div(| \nabla u|^{p-2}\nabla u)=0,\quad p>2,\text{ in } D'(\Omega_ T);\quad u\in C(0,T;L^ 2(\Omega))\cap L^ p(0,T;W^{1,p}(\Omega))$ and of the type (2) $$u_ t-\Delta u^{m=0}$$, $$m>1$$, in $$D'(\Omega_ T)$$, $$u\in C(0,T;L^ 2(\Omega))$$, $$u^ m\in L^ 2(0,T;W^{1,2}(\Omega))$$, where $$\Omega$$ is an open set of $$R^ N$$, $$0<T<\infty$$, $$\Omega_ T\equiv \Omega \times (0,T]$$. For nonnegative weak solutions of the equation (1) the author gives an intrinsic Harnack type inequality: $u(x_ 0,t_ 0)\leq C_ 0\inf_{x\in B_ R(x_ 0)}u(x,t_ 0+\theta)$ (if $$u(x_ 0,t_ 0)>0$$, $$\theta =C_ 1R^ p/[u(x_ 0,t_ 0)]^{p-2}$$ and $$B_{2R}(x_ 0)\times (t_ 0-\theta,t+\theta)\subset \Omega_ T)$$ etc. For the equation (2), an analogous result is obtained.
Reviewer: Zheng Xingli

##### MSC:
 35B45 A priori estimates in context of PDEs 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations
##### Keywords:
Benedetto, E. di; Harnack type inequality
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##### References:
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