Nonlocal Porous Medium Research Articles

Existence for a Nonlocal Porous Medium Equations of Kirchhoff Type with Logarithmic Nonlinearity

We study the Dirichlet problem for the nonlocal parabolic equation of the Kirchhoff type
 \[
 u_{t}-a\left(\|u\|_{L^{p}(\Omega)}^{p}\right)\sum\limits_{i=1}^{n}D_{i}\left(
 \left\vert u\right\vert ^{p-2}D_{i}u\right) +b(x,t) \left\vert u \right\vert ^{\alpha \left(
 x,t\right) -2}u\log|u|=f\left( x,t\right) \quad \text{in $Q_T=\Omega \times (0,T)$},
 \]
 where $p\geq2$, $T>0$, $\Omega \subset
 \mathbb{R}^{n}$, $n\geq 2$, is a smooth bounded domain. The coefficient $a(\cdot)$ is real-valued function defined on $\mathbb{R}_+$. It is shown that the problem has a weak solution under appropriate and general conditions on $a(\cdot)$, $\alpha(\cdot,\cdot)$ and $b(\cdot)$.

Wave propagation on a nonlocal porous medium with memory-dependent derivative and gravity

In this paper, a novel model in a nonlocal porous thermoelastic solid is formulated based on the dual-phase-lag model (DPL), the Lord–Shulman theory and coupled theory with a memory-dependent derivative. The Laplace–Fourier technique is used to solve the problem and to obtain the exact expressions of physical fields. Numerical calculation of temperature, displacement, change in the volume fraction and stress is carried out and displayed graphically. Comparisons are made with the results predicted in the absence and presence of the gravity field as well as a nonlocal parameter. Comparisons are also made with results for different memory Kernel.

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>

Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschläger’s approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo–Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.

On a class of nonlocal porous medium equations of Kirchhoff type

We study the Dirichlet problem for the degenerate parabolic equation of the Kirchhoff type \[ u_{t}-a\left(\ u\ _{L^{p}(\Omega)}^{p}\right)\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p-2}D_{i}u\right) +b\left( x,t,u\right)=f\left( x,t\right) \quad \text{in $Q_T=\Omega \times (0,T)$}, \] where $p\geq2$, $T>0$, $\Omega \subset \mathbb{R}^{n}$, $n\geq 2$, is a smooth bounded domain. The coefficient $a(\cdot)$ is real-valued function defined on $\mathbb{R}_+$ and $b(\cdot,\cdot,\tau)$ is a measurable function with variable nonlinearity in $\tau$. We prove existence of weak solutions of the considered problem under appropriate and general conditions on $a$ and $b$. Sufficient conditions for uniqueness are found and in the case $f\equiv0$ the decay rates for $\ u\ _{L^2(\Omega)}$ are obtained.

Reflection of thermo-elastic wave in semiconductor nanostructures nonlocal porous medium

The current work is an extension of the nonlocal elasticity theory to fractional order thermo-elasticity in semiconducting nanostructure medium with voids. The analysis is made on the reflection phenomena in context of three-phase-lag thermo-elastic model. It is observed that, four-coupled longitudinal waves and an independent shear vertical wave exist in the medium which is dispersive in nature. It is seen that longitudinal waves are damped, and shear wave is un-damped when angular frequency is less than the cut-off frequency. The voids, thermal and non-local parameter affect the dilatational waves whereas shear wave is only depending upon non-local parameter. It is found that reflection coefficients are affected by nonlocal and fractional order parameters. Reflection coefficients are calculated analytically and computed numerically for a material, silicon and discussed graphically in details. The results for local (classical) theory are obtained as a special case. The study may be useful in semiconductor nanostructure, geology and seismology in addition to semiconductor nanostructure devices.

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.

On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions

In this paper we analyze the porous medium equation \begin{document}$ \begin{equation} u_t = \Delta u^m + a\int_\Omega u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad {\rm{in}}\quad \Omega \times I,\;\;\;\;\;\;(◇) \end{equation} $\end{document} where \begin{document}$ \Omega $\end{document} is a bounded and smooth domain of \begin{document}$ \mathbb R^N $\end{document} , with \begin{document}$ N\geq 1 $\end{document} , and \begin{document}$ I = [0,t^*) $\end{document} is the maximal interval of existence for \begin{document}$ u $\end{document} . The constants \begin{document}$ a,b,c $\end{document} are positive, \begin{document}$ m,p,q $\end{document} proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of \begin{document}$ u $\end{document} . Under some hypotheses on the data, including intrinsic relations between \begin{document}$ m,p $\end{document} and \begin{document}$ q $\end{document} , and assuming that for some positive and sufficiently regular function \begin{document}$ u_0({\bf x}) $\end{document} the Initial Boundary Value Problem (IBVP) associated to (◇) possesses a positive classical solution \begin{document}$ u = u({\bf x},t) $\end{document} on \begin{document}$ \Omega \times I $\end{document} : \begin{document}$ \triangleright $\end{document} when \begin{document}$ p>q $\end{document} and in 2- and 3-dimensional domains, we determine a lower bound of \begin{document}$ t^* $\end{document} for those \begin{document}$ u $\end{document} becoming unbounded in \begin{document}$ L^{m(p-1)}(\Omega) $\end{document} at such \begin{document}$ t^* $\end{document} ; \begin{document}$ \triangleright $\end{document} when \begin{document}$ p and in \begin{document}$ N $\end{document} -dimensional settings, we establish a global existence criterion for \begin{document}$ u $\end{document} .

Uniqueness of entropy solutions to fractional conservation laws with “fully infinite” speed of propagation

Our goal is to study the uniqueness of bounded entropy solutions for a multidimensional conservation law including a non-Lipschitz convection term and a diffusion term of non-local porous medium type. The non-locality is given by a fractional power of the Laplace operator. For a wide class of nonlinearities, the L1-contraction principle is established, despite the fact that the “finite-infinite” speed of propagation (Alibaud (2007) [1]) cannot be exploited in our framework; existence is deduced with perturbation arguments. The method of proof, adapted from Andreianov and Maliki (2010) [9], requires a careful analysis of the action of the fractional laplacian on truncations of radial powers.

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.

Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation ∂tu−Lμ[φ(u)]=0. Here Lμ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function φ:R→R is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain stability, L1-contraction, and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.

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 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.