Related papers: Nonlinear porous medium flow with fractional poten…
We construct a finite element method for the numerical solution of a fractional porous medium equation on a bounded open Lipschitz polytopal domain $\Omega \subset \mathbb{R}^{d}$, where $d = 2$ or $3$. The pressure in the model is defined…
We establish local boundedness for solutions to fractional porous medium-type equations in the fast diffusion regime, under optimal tail assumptions.
In this short note we consider a nonlinear and spatially nonlocal PDE modelling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We…
We investigate quantitative properties of the nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + {\mathcal L} (u^m)=0$, posed in a bounded domain, $x\in\Omega\subset {\mathbb R}^N$ with $m>1$…
This work extends Perron's method for the porous medium equation in the slow diffusion case. The main result shows that nonnegative continuous boundary functions are resolutive in a general cylindrical domain.
In the slow diffusion case unbounded supersolutions of the porous medium equation are of two totally different types, depending on whether the pressure is locally integrable or not. This criterion and its consequences are discussed.
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 prove duality estimates for time-fractional and more general subdiffusion problems. An important example is given by subdiffusive porous medium type equations. Our estimates can be used to prove uniqueness of weak solutions to such…
In order to analyze numerically inverse problems several techniques based on linear and nonlinear stability analysis are presented. These techniques are illustrated on the problem of estimating mobilities and capillary pressure in…
This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of…
We discuss a class of diffusion-type partial differential equations on a bounded interval and discuss the possibility of replacing the boundary conditions by certain linear conditions on the moments of order 0 (the total mass) and of…
This study handles spatial three-dimensional solution of the nonlinear diffusion equation without particular initial conditions. The functional behavior of the equation and the concentration have been studied in new ways. An auxiliary…
We study existence and uniqueness of bounded solutions to a fractional nonlinear porous medium equation with a variable density, in one space dimension.
We study a porous medium-type equation whose pressure is given by a nonlocal L\'{e}vy operator associated to a symmetric jump L\'{e}vy kernel. The class of nonlocal operators under consideration appears as a generalization of the classical…
Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local…
In this paper, we investigate the speed of convergence and higher-order asymptotics of solutions to the porous medium equation posed in $\mathbf{R}^N$. Applying a nonlinear change of variables, we rewrite the equation as a diffusion on a…
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)=…
In this paper, we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time-dependent diffusion coefficient and an external force into account, which…
This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in…
We demonstrate the equivalence between the two popular models of nonlinear diffusion, the porous medium equation and the p-Laplacian equation. The equivalence is shown at the level of selfsimilar solutions.