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.
Categories: math.AP