Nonlocal Porous Medium Equation Research Articles

Homogenization of nonlinear nonlocal diffusion equation with periodic and stationary structure

<abstract><p>This paper is devoted to the homogenization of a class of nonlinear nonlocal parabolic equations with time dependent coefficients in a periodic and stationary structure. In the first part, we consider the homogenization problem with a periodic structure. Inspired by the idea of Akagi and Oka for local nonlinear homogenization, by a change of unknown function, we transform the nonlinear nonlocal term in space into a linear nonlocal scaled diffusive term, while the corresponding linear time derivative term becomes a nonlinear one. By constructing some corrector functions, for different time scales $ r $ and the nonlinear parameter $ p $, we obtain that the limit equation is a local nonlinear diffusion equation with coefficients depending on $ r $ and $ p $. In addition, we also consider the homogenization of the nonlocal porous medium equation with non negative initial values and get similar homogenization results. In the second part, we consider the previous problem in a stationary environment and get some similar homogenization results. The novelty of this paper is two folds. First, for the determination equation with a periodic structure, our study complements the results in literature for $ r = 2 $ and $ p = 1 $. Second, we consider the corresponding equation with a stationary structure.</p></abstract>

Existence of weak solutions to a continuity equation with space time nonlocal Darcy law

In this manuscript, we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator in two space dimensions for Global in time existence of weak solutions is shown by employing a time semi-discretization of the equations, an energy inequality and the Div-Curl lemma.

Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition

In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain (varOmegain mathbb{R}^{n}, ngeq 3) with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.

Longtime behavior and weak-strong uniqueness for a nonlocal porous media equation

In this manuscript we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator{∂tu=div(u∇p),∂tp=−(−Δ)sp+u2, in three space dimensions for 3/4≤s<1 and analyze the long time asymptotics. The proof is based on energy methods and leads to algebraic decay towards the stationary solution u=0 and ∇p=0 in the L2(R3)-norm. The decay rate depends on the exponent s. We also show weak-strong uniqueness of solutions and continuous dependence from the initial data. As a side product of our analysis we also show that existence of weak solutions, previously shown in [4] for 3/4≤s≤1, holds for 1/2<s≤1 if we consider our problem in the torus.

Long-time asymptotics for a 1D nonlocal porous medium equation with absorption or convection

In this paper, the long-time asymptotic behaviors of one-dimensional porous medium equations with a fractional pressure and absorption or convection are studied. In the parameter regimes when the nonlocal diffusion is dominant, the entropy method is adapted to derive the exponential convergence of relative entropy of solutions in similarity variables.

Stochastic models associated to a Nonlocal Porous Medium Equation

The nonlocal porous medium equation considered in this paper is a degenerate nonlinear evolution equation involving a space pseudo-differential operator of fractional order. This space-fractional equation admits an explicit, nonnegative, compactly supported weak solution representing a probability density function. In this paper we analyze the link between isotropic transport processes, or random flights, and the nonlocal porous medium equation. In particular, we focus our attention on the interpretation of the weak solution of the nonlinear diffusion equation by means of random flights.

Finite speed of propagation for a non-local porous medium equation

This note is concerned with proving the finite speed of propagation for some non-local porous medium equation by adapting arguments developed by Caffarelli and Vazquez (2010).

The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions

A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles --- the well-known solutions of the classical porous medium equation.

The second critical exponent for a nonlocal porous medium equation in RN

In this paper, we investigate the behavior of the positive solution of a nonlocal porous medium equation in R N , which was considered in Galaktionov and Levine (1998). A second critical exponent for the existence of global and non-global solutions is given.

Blow-up phenomena for the nonlinear nonlocal porous medium equation under Robin boundary condition

In this paper, we determine a lower bound for the blow-up time of the nonlinear nonlocal porous medium equation under Robin boundary condition if the solution blows up. The conditions which ensure that the blow-up does not occur are also presented.

Lower bounds estimate for the blow-up time of a nonlinear nonlocal porous medium equation

The lower bounds for the blow-up time of blow-up solutions to the nonlinear nolocal porous equationut=Δum+up∫Ωuqdxwith either null Dirichlet boundary condition or homogeneous Neumann boundary condition is given in this article by using a differential inequality technique.

Roles of Weight Functions to a Nonlocal Porous Medium Equation with Inner Absorption and Nonlocal Boundary Condition

This work is concerned with an initial boundary value problem for a nonlocal porous medium equation with inner absorption and weighted nonlocal boundary condition. We obtain the roles of weight function on whether determining the blowup of nonnegative solutions or not and establish the precise blow‐up rate estimates under some suitable condition.

Barenblatt profiles for a nonlocal porous medium equation

We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit self-similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the L p decay of weak solutions of the corresponding Cauchy problem.

Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation \({u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}\) with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and sn-1f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all \({x\in(-1,1)}\).

Localization of blow-up points for a nonlinear nonlocal porous medium equation

This paper deals with the porous medium equation with a nonlinearnonlocal source$u_t=\Delta u^m + au^p\int_\Omega u^q dx,\quad x\in \Omega, t>0$subject to homogeneous Dirichlet condition. We investigate theinfluence of the nonlocal source and local term on blow-upproperties for this system. It is proved that: (i) when $p\leq 1$,the nonlocal source plays a dominating role, i.e. the system hasglobal blow-up and the blow-up profile which is uniformly away fromthe boundary either at polynomial scale or logarithmic scale isobtained. (ii) When $p> m$, this system presents single blow-up pattern. In other words,the local term dominates the nonlocal term in the blow-up profile.This extends the work of Li and Xie in Appl. Math. Letter, 16(2003) 185--192.