Related papers: Perron's method for the porous medium equation
In this paper, we consider unsaturated poroelasticity, i.e., coupled hydro-mechanical processes in unsaturated porous media, modeled by a non-linear extension of Biot's quasi-static consolidation model. The coupled, elliptic-parabolic…
We consider a diffusion process on an evolving surface with a piecewise Lipschitz-continuous boundary from an energetic point of view. We employ an energetic variational approach with both surface divergence and transport theorems to derive…
A compressible laminar boundary layer developing over an isotropic porous substrate is investigated by asymptotic and numerical methods. The substrate is modeled as an array of cubes. The momentum and enthalpy balance equations are derived…
We consider the problem of the soliton propagation, in a slowly varying medium, for a generalized, variable-coefficients nonlinear Schr\"odinger equation. We prove existence and uniqueness of new soliton-like solutions for a large class of…
This study investigates three-dimensional, steady-state, and non-Newtonian flows within a very thin porous medium (VTPM). The medium is modeled as a domain confined between two parallel plates and perforated by solid cylinders that connect…
The paper deals with the homogenization of reaction-diffusion equations with large reaction terms in a multi-scale porous medium. We assume that the fractures and pores are equidistributed and that the coefficients of the equations are…
We study a generalized class of supersolutions, so-called supercaloric functions to the porous medium equation in the fast diffusion case. Supercaloric functions are defined as lower semicontinuous functions obeying a parabolic comparison…
We established the propagation equation of acoustical wave in media with the solid/porous media cylindrical boundary and obtained the analytic solution. We suggested the boundary condition on solid-porous media cylindrical boundary. Based…
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 the existence of a global solution to the Cauchy problem for a nonlinear reaction-diffusion system coupled with a system of ordinary differential equations. The system models the propagation of a combustion front in a porous medium…
We prove the existence and uniqueness of entropy solutions for nonlinear diffusion equations with nonlinear conservative gradient noise. As particular applications our results include stochastic porous media equations, as well as the…
In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the…
We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic $p$-Laplacian. Assuming that such solutions continuously vanish on some distinguished part of…
This is a study of a class of nonlocal nonlinear diffusion equations. We present a strong maximum principle for nonlocal time-dependent Dirichlet problems. Results are for bounded functions of space, rather than (semi)-continuous functions.…
In this paper, the long-time asymptotic behaviours of nonlocal porous medium equations with absorption or convection are studied. In the parameter regimes when the nonlocal diffusion is dominant, the entropy method is adapted in this…
Convergence of the solutions of nonhomogeneous linear singularly perturbed systems to that of the corresponding reduced singular system on the half-line [0, $\infty $) is considered. To include the situation on a neighborhood of initial…
We study the Dirichlet problem for non-homogeneous equations involving the fractional $p$-Laplacian. We apply Perron's method and prove Wiener's resolutivity theorem.
Consider the Boltzmann equation in a general non-convex domain with the diffuse boundary condition. We establish optimal BV estimates for such solutions. Our method consists of a new $W^{1,1}-$trace estimate for the diffuse boundary…
This papers deals with a construction and convergence analysis of a finite difference scheme for solving time-fractional porous medium equation. The governing equation exhibits both nonlocal and nonlinear behaviour making the numerical…
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.