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Anna Zhigun

Publications and source records attributed to Anna Zhigun.

17 recordsLinked to original sources
arXiv2024

Global existence of solutions to a nonlocal equation with degenerate anisotropic diffusion

Global existence of very weak solutions to a non-local diffusion-advection-reaction equation is established under no-flux boundary conditions in higher dimensions. The equation features degenerate myopic diffusion and nonlocal adhesion and is an extension of a mass-conserving model recently derived in arXiv:2308.05676. The admissible degeneracy of the diffusion tensor is characterised in terms of the upper box fractal dimension.

math.AP↗
arXiv2024

Global entropy solutions to a degenerate parabolic-parabolic chemotaxis system for flux-limited dispersal

Existence of global finite-time bounded entropy solutions to a parabolic-parabolic system proposed in [16] is established in bounded domains under no-flux boundary conditions for nonnegative bounded initial data. This modification of the classical Keller-Segel model features degenerate diffusion and chemotaxis that are both subject to flux-saturation. The approach is based on Schauder's fixed point theorem and calculus of functions of bounded variation.

math.AP↗
arXiv2024

Local well-posedness for a novel nonlocal model for cell-cell adhesion via receptor binding

Local well-posedness is established for a highly nonlocal nonlinear diffusion-adhesion system for bounded initial values with small support. Macroscopic systems of this kind were previously obtained by the authors through upscaling in [32] and can account for the effect of microscopic receptor binding dynamics in cell-cell adhesion. The system analysed here couples an integro-PDE featuring degenerate diffusion of the porous media type and nonlocal adhesion with a novel nonlinear integral equation. The approach is based on decoupling the system and using Banach's fixed point theorem to solve each of the two equations individually and subsequently the entire system. The main challenge of the implementation lies in selecting a suitable framework. One of the key results is the local well-posedness for the integral equation with a Radon measure as a parameter. The analysis of this equation utilizes the Kantorovich-Rubinstein norm, marking the first application of this norm in handling a nonlinear integral equation.

math.AP↗
arXiv2024

Modelling non-local cell-cell adhesion: a multiscale approach

Cell-cell adhesion plays a vital role in the development and maintenance of multicellular organisms. One of its functions is regulation of cell migration, such as occurs, e.g. during embryogenesis or in cancer. In this work, we develop a versatile multiscale approach to modelling a moving self-adhesive cell population that combines a careful microscopic description of a deterministic adhesion-driven motion component with an efficient mesoscopic representation of a stochastic velocity-jump process. This approach gives rise to mesoscopic models in the form of kinetic transport equations featuring multiple non-localities. Subsequent parabolic and hyperbolic scalings produce general classes of equations with non-local adhesion and myopic diffusion, a special case being the classical macroscopic model proposed in [4]. Our simulations show how the combination of the two motion effects can unfold. Cell-cell adhesion relies on the subcellular cell adhesion molecule binding. Our approach lends itself conveniently to capturing this microscopic effect. On the macroscale, this results in an additional non-linear integral equation of a novel type that is coupled to the cell density equation.

q-bio.TO↗
arXiv2023

Generalised global supersolutions with mass control for systems with taxis

The existence of generalised global supersolutions with a control upon the total mass is established for a wide family of parabolic-parabolic chemotaxis systems and general integrable initial data in any space dimension. It is verified that as long as a supersolution of this sort remains smooth, it coincides with the classical solution. At the same time, the proposed construction provides solvability beyond a blow-up time. The considered class of systems includes the basic form of the Keller-Segel model as well as the case of a chemorepellent.

math.AP↗
arXiv2022

Flux limitation mechanisms arising in multiscale modelling of cancer invasion

Tumour invasion is an essential stage of cancer progression. Its main drivers are diffusion and taxis, a directed movement along the gradient of a stimulus. Here we review models with flux limited diffusion and/or taxis which have applications in modelling of cell migration, particularly in cancer. Flux limitation ensures control upon propagation speeds, precluding unnaturally quick spread which is typical for traditional parabolic equations. We recall the main properties of models with flux limitation effects and discuss ways to construct them, concentrating on multiscale derivations from kinetic transport equations.

math.AP↗
arXiv2021

A novel derivation of rigorous macroscopic limits from a micro-meso description of signal-triggered cell migration in fibrous environments

In this work we upscale a prototypical kinetic transport equation which models a cell population moving in a fibrous environment with a chemo- or haptotactic signal influencing both the direction and the magnitude of the cell velocity. The presented approach to scaling does not rely on orthogonality and treats parabolic and hyperbolic scalings in a unified manner. It is shown that the steps of the formal limit procedures are mirrored by rigorous operations with finite measures provided that the measure-valued position-direction fiber distribution enjoys some spacial continuity.

math.AP↗
arXiv2021

Multiphase modelling of glioma pseudopalisading under acidosis

We propose a multiphase modeling approach to describe glioma pseudopalisade patterning under the influence of acidosis. The phases considered at the model onset are glioma, normal tissue, necrotic matter, and interstitial fluid in a void-free volume with acidity represented by proton concentration. We start from mass and momentum balance to characterize the respective volume fractions and deduce reaction-cross diffusion equations for the space-time evolution of glioma, normal tissue, and necrosis. These are supplemented with a reaction-diffusion equation for the acidity dynamics and lead to formation of patterns which are typical for high grade gliomas. Unlike previous works, our deduction also works in higher dimensions and involves less restrictions. We also investigate the existence of weak solutions to the obtained system of equations and perform numerical simulations to illustrate the solution behavior and the pattern occurrence.

q-bio.TO↗
arXiv2019

Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

math.AP↗
arXiv2019

Mathematical models for cell migration: a nonlocal perspective

We provide a review of recent advancements in nonlocal continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial nonlocality occurring in advection terms, used to characterise 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, nonlocal chemotaxis), but we also shortly address spatially nonlocal source terms. Following a brief introduction and motivation, we give a systematic classification of available PDE models with respect to the type of featured nonlocalities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. Keywords: cell-cell and cell-tissue adhesion; nonlocal and local chemotaxis; haptotaxis; classes of nonlocal models; integro-differential equations; mathematical challenges.

math.AP↗
arXiv2019

On a model for epidemic spread with interpopulation contact and repellent taxis

We study a PDE model for dynamics of susceptible-infected interactions. The dispersal of susceptibles is via diffusion and repellent taxis as they move away from the increasing density of infected. The diffusion of infected is a nonlinear, possibly degenerating term in nondivergence form. We prove the existence of so-called weak-strong solutions in 1D for a positive susceptible initial population. For dimension $N\geq 2$ and nonnegative susceptible initial density we show the existence of supersolutions. Numerical simulations are performed for different scenarios and illustrate the space-time behaviour of solutions.

math.AP↗
arXiv2018

Generalised supersolutions with mass control for the Keller-Segel system with logarithmic sensitivity

The existence of generalised global supersolutions with a control upon the total muss is established for the parabolic-parabolic Keller-Segel system with logarithmic sensitivity for any space dimension. It is verified that smooth supersolutions of this sort are actually classical solutions. Unlike the previously existing constructions, neither is the chemotactic sensitivity coefficient required to be small, nor is it necessary for the initial data to be radially symmetric. Keywords: chemotaxis; generalised supersolution; global existence; logarithmic sensitivity

math.AP↗
arXiv2017

On an exponential attractor for a class of PDEs with degenerate diffusion and chemotaxis

In this article we deal with a class of strongly coupled parabolic systems that encompasses two different effects: degenerate diffusion and chemotaxis. Such classes of equations arise in the mesoscale level modeling of biomass spreading mechanisms via chemotaxis. We show the existence of an exponential attractor and, hence, of a finite-dimensional global attractor under certain 'balance conditions' on the order of the degeneracy and the growth of the chemotactic function.

math.AP↗
arXiv2017

The Malliavin derivative and compactness: application to a degenerate PDE-SDE coupling

Compactness is one of the most versatile tools in the analysis of nonlinear PDEs and systems. Usually, compactness is established by means of some embedding theorem between functional spaces. Such theorems, in turn, rely on appropriate estimates for a function and its derivatives. While a similar result based on simultaneous estimates for the Malliavin and weak Sobolev derivatives is available for the Wiener-Sobolev spaces, it seems that it has not yet been widely used in the analysis of highly nonlinear parabolic problems with stochasticity. In the present work we apply this result in order to study compactness, existence of global solutions, and, as a by-product, the convergence of a semi-discretisation scheme for a prototypical degenerate PDE-SDE coupling.

math.AP↗
arXiv2016

Global existence for a degenerate haptotaxis model of tumor invasion under the go-or-grow dichotomy hypothesis

We propose and study a strongly coupled PDE-ODE-ODE system modeling cancer cell invasion through a tissue network under the go-or-grow hypothesis asserting that cancer cells can either move or proliferate. Hence our setting features two interacting cell populations with their mutual transitions and involves tissue-dependent degenerate diffusion and haptotaxis for the moving subpopulation. The proliferating cells and the tissue evolution are characterized by way of ODEs for the respective densities. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations in a two-dimensional setting.

math.AP↗
arXiv2015

Global existence for a degenerate haptotaxis model of cancer invasion

We propose and study a strongly coupled PDE-ODE system with tissue-dependent degenerate diffusion and haptotaxis that can serve as a model prototype for cancer cell invasion through the extracellular matrix. We prove the global existence of weak solutions and illustrate the model behaviour by numerical simulations for a two-dimensional setting.

math.AP↗