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

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 aligned uₜ-∑ ᵢ₌₁N ( u(mᵢ-1)(pᵢ-1) \ |uxᵢ|pᵢ-2 \ uxᵢ ) xᵢ=0 , 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 pᵢ p i s and mᵢ 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.