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We prove the global existence of nonnegative variational solutions to the "drift diffusion" evolution equation ∂tu + div " u " 2D ∆ √ u √ u − f " « = 0 under variational boundary condition. Despite the lack of a maximum principle for fourth order equations, nonnegative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation, which is the gradient flow of the (perturbed) Fisher Information functional

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We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $1

Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold.

Abstract.A two-parameter family of Harnack type inequalities for non-negative solutions of a class of singular, quasilinear, homogeneous parabolic equations is established, and it is shown that such estimates imply the Hölder continuity of solutions. These classes of singular equations include p-Laplacean type equations in the sub-critical range 1 < p ≤ 2N N +1 and equations of the porous medium type in the sub-critical range 0 < m ≤ (N −2) + N .

We show that the gradient of solutions to degenerate parabolic equations of porous medium-type satisfies a reverse Hölder inequality in suitable intrinsic cylinders. We modify the by-now classical Gehring lemma by introducing an intrinsic Calderón-Zygmund covering argument, and we are able to prove local higher integrability of the gradient of a proper power of the solution u.

We show that the gradient of the m-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse Hölder inequality in suitable intrinsic cylinders. Relying on an intrinsic Calderón-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for m ∈ (n−2) + n+2 , 1 . Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for m ≥ 1 (see [GS16] in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.