Qualitative properties of solutions to parabolic anisotropic equations: Part I -- Expansion of positivity

TL;DR

This paper establishes positivity expansion and oscillation reduction for solutions to a class of anisotropic parabolic equations, advancing understanding of their qualitative behavior under various conditions.

Contribution

It introduces new positivity and oscillation results for anisotropic parabolic equations with nonlinear and anisotropic diffusion, including singular and degenerate cases.

Findings

Expansion of positivity proven for anisotropic equations.

Oscillation reduction results for specific exponent ranges.

Analysis distinguishes between singular and degenerate cases.

Abstract

We prove expansion of positivity and reduction of the oscillation results to the local weak solutions to a doubly nonlinear anisotropic class of parabolic differential equations with bounded and measurable coefficients, whose prototype is \begin{equation*} u_t-\sum\limits_{i=1}^N \left( u^{(m_i-1)(p_i-1)} \ |u_{x_i}|^{p_i-2} \ u_{x_i} \right)_{x_i}=0 , \end{equation*} for a restricted range of $p_i$s and $m_i$s, that reflects their competition for the diffusion. The positivity expansion relies on an exponential shift and is presented separately for singular and degenerate cases. Finally we present a study of the local oscillation of the solution for some specific ranges of exponents, within the singular and degenerate cases.