Qualitative properties of solutions to parabolic anisotropic equations: part I—expansion of positivity

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 $$\begin{aligned} u_t-\sum \limits _{i=1}^N \left( u^{(m_i-1)(p_i-1)} \ |u_{x_i}|^{p_i-2} \ u_{x_i} \right) _{x_i}=0 , \end{aligned}$$ u t - ∑ i = 1 N u ( m i - 1 ) ( p i - 1 ) | u x i | p i - 2 u x i x i = 0 , for a restricted range of $$p_i$$ p i s and $$m_i$$ 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.