On the local behavior of non-negative solutions to a logarithmically singular equation (Q449283)

The authors study the local positivity of solutions to logarithmically singular diffusion equations. They extend recent intrinsic Harnack type proved for porous medium equations. More precisely it is shown that if at some time level \(t_0\in(0,T]\) and some point \(x_0\in E\) (where \(E\) is a space-time domain) the solution \(u(\cdot,t_0)\) is not identically zero in a neighborhood of \(x_0\), in a measure-theoretical sense, then it is strictly positive in a neighborhood of \((x_0, t_0)\) that is quantitatively determined. They are able to prove very interesting estimates for the porous medium equation \(u_t = \Delta (u^m)\) with \(m\) close to zero (\(m\) can be taken negative). To prove such a result, the authors use the so-called energy estimates, the DeGiorgi's approach and suitable \(L^1\)-\(L^1\) estimates.