Harnack-type estimates and extinction in finite time for a class of anisotropic porous medium type equations

In this work we are interested in the study of a class of anisotropic porous medium-type equations whose prototype is $$u_t =\sum_{i=1}^N \left( m_i u^{m_i-1} u_{x_i} \right)_{x_i} \ , \qquad 0<m_1 \leq \cdots \leq m_N <1 \ ,$$ for which we derive several estimates, namely two Harnack-type inequalities; and, when considering the associated Dirichlet problem, we determine the finite time of extinction and thereby present a decay rate of extinction.Comment: 31 page

Liouville rigidity and time-extrinsic Harnack estimates for an anisotropic slow diffusion

We prove that ancient non-negative solutions to a fully anisotropic prototype evolution equation are constant if they satisfy a condition of finite speed of propagation and if they are both one-sided bounded, and bounded in space at a single time level. A similar statement is valid when the bound is given at a single space point. As a general paradigm, H\"older estimates provide the basics for rigidity. Finally, we show that recent intrinsic Harnack estimates can be improved to a Harnack inequality valid for non-intrinsic times. Locally, they are equivalent.Comment: 15 page

Evolution equations with nonlocal initial conditions and superlinear growth

We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case we are able to prove the existence of at least one periodic solution on the half line.Comment: 18 page

A new short proof of regularity for local weak solutions for a certain class of singular parabolic equations

We shall establish the interior H\"older continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are \begin{equation} u_t= \nabla \cdot \bigg( |\nabla u|^{p-2} \nabla u \bigg), \quad \text{ for } \quad 1<p<2, \end{equation} and \begin{equation} u_{t}- \nabla \cdot ( u^{m-1} | \nabla u |^{p-2} \nabla u ) =0 , \quad \text{for} \quad m+p>3-\frac{p}{N}, \end{equation} via a new and simplified proof using recent techniques on expansion of positivity and $L^{1}$-Harnack estimates.Comment: 13 pages long, references 25 title

Una breve nota su disuguaglianze integrali di Harnack per operatori parabolici non lineari singolari

In this brief note we introduce Harnack-type inequalities, which are typical in the context of singular nonlinear parabolic operators, and describe their state of art in the context of anisotropic operators.In questa nota breve presentiamo alcune disuguaglianze integrali di Harnack che sono tipiche di operatori parabolici nonlineari singolari, e descriviamo il loro stato dell'arte nel contesto di operatori singolari che presentano anisotropie

Fine boundary continuity for degenerate double-phase diffusion

We study the boundary behavior of solutions to parabolic double-phase equations through the celebrated Wiener's sufficiency criterion. The analysis is conducted for cylindrical domains and the regularity up to the lateral boundary is shown in terms of either its $p$ or $q$ capacity, depending on whether the phase vanishes at the boundary or not. Eventually we obtain a fine boundary estimate that, when considering uniform geometric conditions as density or fatness, leads us to the boundary H\"older continuity of solutions. In particular, the double-phase elicits new questions on the definition of an adapted capacity.Comment: 33 pages, 2 figures, appendix at the end of the pape