Expansion of positivity to a class of doubly nonlinear parabolic equations

Abstract

We establish the expansion of positivity of the nonnegative, local, weak solutions to the class of doubly nonlinear parabolic equations $$\partial_t (u^{q}) -\operatorname{div}{(|D u|^{p-2} D u)}=0, \qquad\ p>1 \ \text{and} \ q>0$$ considering separately the two possible cases $q+1-p > 0$ and $q+1-p<0$. The proof relies on the procedure presented by DiBenedetto, Gianazza and Vespri for both the degenerate and the singular parabolic $p$-Laplacian equation.

Keywords

• doubly nonlinear parabolic equations
• expansion of positivity
• singular pde
• degenerate pde
• intrinsic scaling