Related papers: The fractional porous medium equation on manifolds…
This paper explores Barenblatt solutions of the time-fractional porous medium equation, characterized by a Caputo-type time derivative. Employing an integral equation approach, we rigorously prove the existence of these solutions and…
In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the…
We consider nonnegative solutions of the porous medium equation (PME) on a Cartan-Hadamard manifold whose negative curvature can be unbounded. We take compactly supported initial data because we are also interested in free boundaries. We…
In this article, we study the following parabolic equation involving the fractional Laplacian with singular nonlinearity \begin{equation*} \quad (P_{t}^s) \left\{ \begin{split} \quad u_t + (-\Delta)^s u &= u^{-q} + f(x,u), \;u >0\;…
Consider the non-linear stochastic fractional-diffusion equation \begin{eqnarray*} \left \{\begin{array}{lll} \frac{\partial}{\partial t}u(x,t)= -( \Delta)^{\alpha/2} u^m(x,t) + \sigma(u(x,t)) \dot{W}(x,t),\, x\in \mathbb{R}^d,t>0, u(x,0)=…
The aim of this paper is to study the singular solutions to fractional elliptic equations with absorption $$ \left\{\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+|u|^{p-1}u=0,\quad & \rm{in}\quad\Omega\setminus\{0\},\\[2mm]…
In this paper, we consider the following problem involving fractional Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u= |u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0 \,\, {\rm on}\, \,…
In this paper, we study the existence of solutions of the equation $(-\Delta)_1^s u=f$ in a bounded open set with Lipschitz boundary $\Omega\subset \Rn$, vanishing on $\Co \Omega$, for some given $s\in (0,1)$, and asymptotics as $p\to 1$ of…
We consider the large time behaviour of solutions to the porous medium equation with a Fisher-KPP type reaction term and nonnegative, compactly supported initial function in $L^\infty(\mathbb{R}^N)\setminus\{0\}$: \begin{equation}…
This article discusses a unified convergence analysis of the semilinear time-dependent equation $\partial_t u + (-1)^\mathrm{m}\Delta^{\mathrm{m}}u + u^3 - u = f$ with $\mathrm{m} \in \{1,2\}$ and homogeneous Dirichlet boundary conditions.…
In this paper, we establish the Freidlin-Wentzell type large deviation principles for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $\Delta (|u|^{m-1}u)$ is allowed. Our proof is based on…
We study the general nonlinear diffusion equation $u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)$ that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters $m>1$ and $0<s<1$, we…
We prove three sharp bounds for solutions to the porous medium equation posed on Riemannian manifolds, or for weighted versions of such equation. Firstly we prove a smoothing effect for solutions which is valid on any Cartan-Hadamard…
In this paper, we search for normalized solutions to a fractional, nonlinear, and possibly strongly sublinear Schr\"odinger equation $$(-\Delta)^s u + \mu u = g(u) \quad \hbox{in $\mathbb{R}^N$},$$ under the mass constraint…
This paper is concerned with the following fractional Schr\"{o}dinger equations involving critical exponents: \begin{eqnarray*} (-\Delta)^{\alpha}u+V(x)u=k(x)f(u)+\lambda|u|^{2_{\alpha}^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N},…
We prove existence of smooth solutions to linear degenerate parabolic equations on bounded domains assuming a structure condition of Fichera. We use this to give a proof of a smooth short time existence result for the porous medium equation…
We revisit the questions of density of smooth functions, and differential forms, in Sobolev spaces on Riemannian manifolds. We carefully show equivalence of weak covariant derivatives to weak partial derivatives.
In the first part of the article we develop a comparison method for positive solutions of the semilinear Dirichlet problem $\Delta u+f(u)=0$ on domains $\Omega\subset \mathcal M^n$ of a Riemannian manifold $(\mathcal{M}^n,g)$ with a Ricci…
In this work we prove that the time-fractional porous medium equation on the half-line with Dirichlet boundary condition has a unique compactly supported solution. The approach we make is based on a transformation of the fractional…
Let $u$ be a nonnegative, local, weak solution to the porous medium equation for $m\ge2$ in a space-time cylinder $\Omega_T$. Fix a point $(x_o,t_o)\in\Omega_T$: if the average \[…