In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale- and location-invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers.

We use a Leibniz--rule-type inequality for fractional derivatives to prove conditions under which a solution $u(x,t)$ of the k-generalized KdV equation is in the space $L^2(|x|^{2s}\,dx)$ for $s \in \mathbb R_{+}$.

We study the existence of a global attractor for a damped parametric nonlinear Schrödinger equation. We provide sufficient conditions for this attractor to have finite dimension.

We consider second-order evolution equations in an abstract setting with intermittently delayed/not-delayed damping and give sufficient conditions ensuring asymptotic and exponential stability results. Our abstract framework is then applied to the wave equation, the elasticity system, and the Petrovsky system. For the Petrovsky system with clamped boundary conditions, we further prove an internal observability estimate that was not available in the literature.

Here we study the nonnegative solutions of the viscous Hamilton--Jacobi equation \begin{equation*} u_{t}-\Delta u+|\nabla u|^{q}=0 \end{equation*} in $Q_{\Omega,T}=\Omega\times\left( 0,T\right) ,$ where $q>1,$ $T\in\left( 0,\infty\right] ,$ and $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N} $ containing $0,$ or $\Omega=\mathbb{R}^{N}$. We consider weak solutions with a possible singularity at the point $(x,t)=(0,0)$. We show that if $q\geq q_{\ast}=(N+2)/(N+1)$ the singularity is removable. For $1<q<q_{\ast}$, we prove the uniqueness of a very singular solution without condition as $% |x|\rightarrow\infty$; we also show the existence and uniqueness of a very singular solution of the Dirichlet problem in $Q_{\Omega,\infty},$ when $% \Omega$ is bounded. We give a complete description of the weak solutions in each case.

Variants of Kato's inequality are proved for general quasilinear elliptic operators $L$. As an outcome we show that, dealing with Liouville theorems for coercive equations of the type \begin{equation*} Lu = f(x,u,{\nabla_{\!\!L}} u) \quad on\ \Omega\subset{{\mathbb{R}}^N} , \end{equation*} where $f$ is such that $f(x,t,\xi) \,t\ge 0$, the assumption that the possible solutions are nonnegative involves no loss of generality. Related consequences such as comparison principles and a priori bounds on solutions are also presented. An underlying structure throughout this work is the framework of Carnot groups.