Nonlocal Source Term Research Articles

Mathematical models for cell migration: a non-local perspective.

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Asymptotic behaviour of the wave equation with nonlocal weak damping and anti-damping

In this paper we consider the initial-boundary value problem of the wave equation with nonlocal weak damping, nonlocal weak anti-damping and nonlinear source term. For this problem we establish the global well-posedness and the existence of global attractor under the assumption that the nonlinear source term is of subcritical growth.

Nonexistence of solutions for the quasilinear parabolic differential inequalities with singular potential term and nonlocal source

This paper is devoted to proving some new nonexistence theorems for a class of quasilinear parabolic differential inequalities with a singular potential term and nonlocal source term in the case of homogeneous and non-homogeneous by the test function method.

Generalized Fourier's law for nondiffusive thermal transport: Theory and experiment

Phonon heat conduction over length scales comparable to their mean free paths is a topic of considerable interest for basic science and thermal management technologies. Although the failure of Fourier's law beyond the diffusive regime is well understood, debate exists over the proper physical description of thermal transport in the ballistic to diffusive crossover. Here, we derive a generalized Fourier's law that links the heat flux and temperature fields, valid from ballistic to diffusive regimes and for general geometries, using the Peierls-Boltzmann transport equation within the relaxation time approximation. This generalized Fourier's law predicts that thermal conductivity not only becomes nonlocal at length scales smaller than phonon mean free paths, but also requires the inclusion of an inhomogeneous nonlocal source term that has been previously neglected. We demonstrate the validity of this generalized Fourier's law through direct comparison with time-domain thermoreflectance (TDTR) measurements in the nondiffusive regime without adjustable parameters. Furthermore, we show that interpreting experimental data without this generalized Fourier's law leads to inaccurate measurement of thermal transport properties.

Extinction and Decay Estimates of Solutions for a Non-Newton Polytropic Filtration System

This paper mainly deals with extinction and non-extinction of weak solutions for a non-Newton polytropic filtration system coupled with non-local source terms under homogeneous Dirichlet boundary condition by comparing the exponents of nonlinear terms. Particularly, precise decay estimates of extinction solutions in the whole dimensional space are also established.

A limiting obstacle type problem for the inhomogeneous p-fractional Laplacian

In this manuscript we study an inhomogeneous obstacle type problem involving a fractional p-Laplacian type operator. First, we focus our attention in establishing existence and uniform estimates for any family of solutions $$\{u_p\}_{p \ge 2}$$ which depend on the data of the problem and universal parameters. Next, we analyze the asymptotic behavior of such a family as $$p \rightarrow \infty $$ . At this point, we prove that $$\displaystyle \lim \nolimits _{p\rightarrow \infty } u_p(x) = u_{\infty }(x)$$ there exists (up to a subsequence), verifies a limiting obstacle type problem in the viscosity sense, and it is an s-Holder continuous function. We also present several explicit examples, as well as further features of the limit solutions and their free boundaries. In order to establish our results we overcome several technical difficulties and develop new strategies, which were not present in the literature for this type of problems. Finally, we remark that our results are new even for problems governed by fractional p-Laplacian operator, as well as they extend the previous ones by dealing with more general non-local operators, source terms and boundary data.

A blow-up result for nonlocal thin-film equation with positive initial energy

In this note, we consider a thin-film equation including a diffusion term, a fourth order term and a nonlocal source term under the periodic boundary conditions. In particular, a finite time blow-up result is established for the case of positive initial energy provided that \[ \frac{\pi^2}{a^2}\leq \frac{2}{p-1},\] where $a$ is the length of the interval and $p>1$ is the power of nonlinear force term. Also upper and lower blow-up times are estimated.

Bounds for the blow-up time of a porous medium equation with weighted nonlocal source and inner absorption terms

We investigate the blow-up phenomena for a porous medium equation with weighted nonlocal source and inner absorption terms subject to null Dirichlet boundary condition. Based on a modified differential inequality technique, we establish some sufficient conditions to guarantee the existence of non-global solutions to the model and also derive the upper bounds for the blow-up time. Moreover, the lower bounds for the blow-up time are obtained under some appropriate measure in the whole-dimensional space (Ngeq1).

Spiky ice and penitente tilting

Under certain conditions, at high altitude, the surface of snow develops spike-like structures known as penitentes. This is a rather counterintuitive phenomenon, which is a consequence of surface sublimation at a given point as a result of the incidence of light scattered by the surrounding region.Following the existing literature, we model the time evolution of the phenomenon described above as a 1D diffusion equation with a non-local source term, as it represents the light coming from all the line of sight defined for a point of the curve. For small initial perturbations in the surface, the system undergoes a thermodynamic instability which triggers the formation of spikes.For sunlight coming in at a given angle, numerical simulations account for a feature observed in the real system: penitentes get tilted in the direction of the sunlight.

Finite time blow-up for a thin-film equation with initial data at arbitrary energy level

In this paper, we consider the initial boundary value problem for a class of thin-film equations in Rn with a p-Laplace term and a nonlocal source term |u|q−2u−1|Ω|∫Ω|u|q−2udx. We prove that there exist weak solutions for the problem with arbitrarily initial energy that blow up in finite time. We also obtain the upper bounds for the blow-up time.

Uniform Blow-up Rate for Nonlocal Diffusion-like Equations with Nonlocal Nonlinear Source

We present new blow-up results for nonlocal reaction-diffusion equations with nonlocal nonlinearities. The nonlocal source terms we consider are of several types, and are relevant to various models in physics and engineering. They may involve an integral of an unknown function, either in space, in time, or both in space and time, or they may depend on localized values of the solution. We first show the existence and uniqueness of the solution to problem relying on contraction mapping fixed point theorem. Then, the comparison principles for problem are established through a standard method. Finally, for the radially symmetric and non-increasing initial data, we give a complete classification in terms of global and single point blow-up according to the parameters. Moreover, the blow-up rates are also determined in each case.

Extending the diffusion approximation to the boundary using an integrated diffusion model

The widely used diffusion approximation is inaccurate to describe the transport behaviors near surfaces and interfaces. To solve such stochastic processes, an integro-differential equation, such as the Boltzmann transport equation (BTE), is typically required. In this work, we show that it is possible to keep the simplicity of the diffusion approximation by introducing a nonlocal source term and a spatially varying diffusion coefficient. We apply the proposed integrated diffusion model (IDM) to a benchmark problem of heat conduction across a thin film to demonstrate its feasibility. We also validate the model when boundary reflections and uniform internal heat generation are present.

A quantum phase space formulation of radiative transfer

Abstract A recent transport model for partially coherent light is reexamined. In its original formulation, this model involves a transport equation for the one-photon Wigner function with nonlocal source and loss terms. We show here that under suitable approximations this equation can be reformulated in a compact form involving the Moyal star product and the related symmetric and antisymmetric brackets. This formulation provides a general framework for the establishment of opacity models accounting for radiation coherence. An investigation of anomalous dispersion in magnetic fusion plasma conditions is reported as an illustration of the model.

Extinction and decay estimates of solutions for a porous medium equation with nonlocal source and strong absorption

Abstract In this paper, we investigate extinction properties of the solutions for the initial Dirichlet boundary value problem of a porous medium equation with nonlocal source and strong absorption terms. We obtain some sufficient conditions for the extinction of nonnegative nontrivial weak solutions and the corresponding decay estimates which depend on the initial data, coefficients, and domains.

On a class of nonlocal degenerate reaction-diffusion equations with localized nonlinear diffusion term

We study a class of second order nonlocal degenerate semilinear reaction-diffusion equations with localized nonlinear diffusion term. Under a set of conditions on the localized nonlinear diffusivity and nonlinear nonlocal source term, we prove global existence and uniqueness result in the whole of some weighted Sobolev’s spaces. Furthermore, we prove nonexistence of smooth solution or blow-up of solution under some other set of conditions. Finally, we give illustrative examples for which our results apply.

Flow-controlled densification and anomalous dispersion of E. coli through a constriction

Dispersion and migration of bacteria under flow in tortuous and confined structures such as porous or fractured materials is related to a large spectrum of practical interest, but is still poorly understood. Here, we address the question of transport and dispersion of an E. coli suspension flowing through a micro-fluidic channel with a funnel-like constriction in its center. We show a counter-intuitive symmetry breaking of the bacterial concentration, which increases significantly past the funnel. This concentration enhancement persists over large distances from the funnel and disappears at large flow rate values. We map our results onto a one dimensional convection–diffusion equation predicting quantitatively the experimental results, without free parameters, when a conservative non-local source term is introduced. This last term, measured experimentally, represents a long range memory effect due to the unbalance of wall adsorption and desorption processes past the constriction. Our model experiment points out the generic importance of considering such constriction effects in the description of transport properties of active matter in porous media. It also opens the possibility to control the concentration of bacterial suspensions in micro-fluidic channels by simply tuning the flow intensity or direction.

Fluid model of dc glow discharge with nonlocal ionization source term

We developed and tested a simple hybrid model for a glow discharge, which incorporates nonlocal ionization by fast electrons into the fluid framework. Calculations have been performed for an argon gas. Comparison with the experimental data as well as with the hybrid (particle) and fluid modelling results demonstated good applicability of the proposed model.

Uniform blow-up rate for compressible reactive gas model

The Dirichlet initial-boundary value problem of a compressible reactive gas model equation with a nonlocal nonlinear source term is investigated. Under certain conditions, it can be proven that the blow-up rate is uniform in all compact subsets of the domain, and the blow-up rate is irrelative to the exponent of the diffusion term, however, relative to the exponent of the nonlocal nonlinear source.

Long-time behaviour of solutions of a non-linear diffusion problem with non-local source term

Long-time behaviour of solutions of a non-linear diffusion problem with non-local source term

On general balance laws with boundary

This paper is devoted to general balance laws (with a possibly non-local source term) with a non-characteristic boundary. Basic well posedness results are obtained. New uniqueness results for the solutions to conservation and/or balance laws with boundary are also provided.