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


35B45 A priori estimates in context of PDEs
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
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