On doubling properties for non-negative weak solutions of elliptic and parabolic PDE (Q1972835)

The paper studies quantitative properties for non-negative weak supersolutions for parabolic and elliptic PDE \[ \partial_t u-\operatorname {div}{\mathcal A}(t,x,u,\nabla u)+{\mathcal B} (t,x,u,\nabla u)=0\quad \text{in }(0,T)\times \Omega \] and \[ -\operatorname {div}{\mathcal A}(x,u,\nabla u)+{\mathcal B} (x,u,\nabla u)=0\quad \text{in }\Omega, \] where \(\Omega\subset{\mathbb R}^n\), \(n\geq 3.\) The doubling property of \(u^\delta\) with some small exponent \(1>\delta >0\) is established for these solutions instead of Harnack's inequality when the coefficients have strong singularity. Furthermore, the doubling property of \(u^q\) with large exponent \(2(n+2)/n\geq q>0\) for non-negative weak solutions \(u\) of the parabolic equation is shown.