Harnack's inequality for doubly nonlinear equations of slow diffusion type
In this article the authors prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form \[\partial_t u- \operatorname{div} A(x, t, u, Du^m) = \operatorname{div} F,\] where the vector field \(A\) fulfills p-ellipticity and growth conditions. The authors treat the slow diffusion case in its full range, i.e. all exponents \(m > 0\) and \(p > 1\) with \(m(p- 1) > 1\) are included. Till now, some cases (for instance \(m(p - 1) > 1\) and \(1<p<2\)) were never considered. Lastly, the authors introduce a definition of weak solution involving \(u^m\), which is new even for the doubly degenerate case and the slow diffusion porous medium equation.
scientific article; zbMATH DE number 7391399
scientific article