Related papers: De Giorgi type results for equations with nonlocal…

The paper deals with second order parabolic equations on bounded domains with Dirichlet conditions in arbitrary Euclidean spaces. Their interest comes from being models for describing reaction-diffusion processes in several frameworks. A…

We prove existence and nonexistence results concerning elliptic problems whose basic model is \begin{equation*} \begin{cases} \displaystyle-\Delta u+\mu(x)\frac{|\nabla u|^2}{(u+\delta)^\gamma}= \lambda u^p, &x\in \Omega, \\ u> 0, &x\in…

We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class…

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form $$-div (|x|^{a} D u ) = f(x,u), ~ u > 0,\, \mbox{ in } \Omega,$$ where $N \geq 3$, $\Omega$ is an open domain in…

In this paper, we consider the fractional heat equation $u_{t}=\triangle^{\alpha/2}u+f(u)$ with Dirichlet boundary conditions on the ball $B_{R}\subset \mathbb{R}^{d}$, where $\triangle^{\alpha/2}$ is the fractional Laplacian,…

We present an $L_q(L_{p})$-theory for the equation $$ \partial_{t}^{\alpha}u=\phi(\Delta) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $\alpha\in (0,1)$, $\partial_{t}^{\alpha}$ is the Caputo fractional…

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate…

We investigate Liouville-type results, existence, uniqueness and symmetry to the solution of nonlinear nonlocal elliptic equations of the form \[ Lu = |x|^{\gamma}\,H(u)\,G(\nabla u), \qquad x\in\R^n, \] where $L$ is a symmetric,…

We consider the semilinear heat equation \begin{eqnarray*} \partial_t u = \Delta u + |u|^{p-1} u \ln ^{\alpha}( u^2 +2), \end{eqnarray*} in the whole space $\mathbb{R}^n$, where $p > 1$ and $ \alpha \in \mathbb{R}$. Unlike the standard case…

In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$ $$ {\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) + \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) , $$ where…

We extend the De Giorgi-Nash-Moser theory to a class of nonlocal hypoelliptic equations arising naturally in kinetic theory, in which a first-order transport operator is coupled with an elliptic nonlocal operator involving fractional…

This paper is concerned with the local and global properties of nonnegative solutions for semilinear heat equation $u_t-\Delta u=u^p+M|\nabla u|^q$ in $\Omega\times I\subset \R^N\times \R$, where $M>0$, and $p,q>1$. We first establish the…

In this paper existence and nonexistence results of positive radial solutions of a Dirichlet $m$-Laplacian problem with different weights and a diffusion term inside the divergence of the form $\big(a(|x|)+g(u)\big)^{-\gamma}$, with…

A nonlocal gravity model, which does not assume the existence of a new dimensional parameter in the action and includes a function $f(\Box^{-1} R)$, with $\Box$ the d'Alembertian operator, is considered. The model is proven to have de…

We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}^{\sigma,\mu}[\varphi(u)]=f \quad\quad\text{in}\quad\quad…

We consider the Hardy-H\'enon system $-\Delta u =|x|^a v^p$, $-\Delta v =|x|^b u^q$ with $p,q>0$ and $a,b\in {\mathbb R}$ and we are concerned in particular with the Liouville property, i.e. the nonexistence of positive solutions in the…

We study the Lane-Emden conjecture, which asserts the non-existence of non-trivial, non-negative solutions to the Lane-Emden system \[ -\Delta u = v^p, \quad -\Delta v = u^q, \quad x \in \mathbb{R}^n\] in the subcritical regime. By…

A semilinear heat equation $u_{t}=\Delta u+f(u)$ with nonnegative initial data in a subset of $L^{1}(\Omega)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $\Omega\subseteq \R^{n}$. A simple technique for…

We consider Liouville-type theorems for the following H\'{e}non-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture…

In this paper, we study the following class of nonlinear equations: $$ -\Delta u+V(x) u = \left[|x|^{-\mu}*(Q(x)F(u))\right]Q(x)f(u),\quad x\in\mathbb{R}^2, $$ where $V$ and $Q$ are continuous potentials, which can be unbounded or vanishing…