arXiv subjects

Nikos I. Kavallaris

Publications and source records attributed to Nikos I. Kavallaris.

15 recordsLinked to original sources

arXiv2026

Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics

We introduce a stochastic nonlocal reaction--diffusion model arising in tumour dynamics. Spatial dispersal is described by the fractional Laplacian, accounting for anomalous diffusion and long--range relocation events. The system is perturbed by multiplicative fractional Brownian motion (fBm) with Hurst parameter $H>1/2$, which we interpret as temporally correlated fluctuations in the tumour microenvironment and host response. We first establish well--posedness and identify parameter regimes leading to global--in--time solutions or finite--time blow--up under general multiplicative fractional noise. We then focus on linear multiplicative noise and, via a Doss--Sussmann transformation, derive sharper results: explicit lower and upper bounds for the blow--up time together with quantitative estimates of the blow--up probability, clarifying how noise intensity can accelerate progression or, on favourable paths, enhance suppression consistent with extinction (loss of viability). Finally, one--dimensional simulations illustrate the interplay between anomalous diffusion, fractional noise, and the nonlocal reaction mechanism in shaping the long--time dynamics.

math.AP↗

arXiv2026

Data-driven macroscopic dynamics of complex networks using Topological Data Analysis and the Equation-Free Method

In this work, we present a computational framework for exploring and analyzing the macroscopic dynamics of complex agent-based network models by integrating Topological Data Analysis with the Equation-Free Method. To demonstrate the effectiveness of our method, we apply it to Erdős--Rényi-type random networks. Central to our approach is a Topological Data Analysis-based filtration process driven by the density of activated network nodes (agents), from which we extract a coarse-grained macroscopic topological observable. This observable is defined via persistent Betti numbers, thus requiring significantly reduced data dimensionality while retaining essential topological features. Subsequently, within the Equation-Free Method framework, we show firstly that a \textit{lifting procedure} can be achieved using topological properties and secondly, a data-driven evolution law that governs the dynamics of this macroscopic variable. Finally, we perform a numerical bifurcation and stability analysis to investigate the global behavior and qualitative transitions of the emergent macroscopic dynamics.

math.DS↗

arXiv2025

A time-fractional Fisher-KPP equation for tumor growth: Analysis and numerical simulation

We study a time-fractional Fisher-KPP equation involving a Riemann-Liouville fractional derivative acting on the diffusion term, as derived by Angstmann and Henry (Entropy, 22:1035, 2020). The model captures memory effects in diffusive population dynamics and serves as a framework for tumor growth modeling. We first establish local well-posedness of weak solutions. The analysis combines a Galerkin approximation with a refined a priori estimate based on a Bihari-Henry-Gronwall inequality, addressing the nonlinear coupling between the fractional diffusion and the reaction term. For small initial data, we further prove global well-posedness and asymptotic stability. A numerical method based on a nonuniform convolution quadrature scheme is then proposed and validated. Simulations demonstrate distinct dynamical behaviors compared to conventional formulations, emphasizing the physical consistency of the present model in describing tumor progression.

math.AP↗

arXiv2025

Quenching time and probability estimates for a stochastic reaction-diffusion system with coupled inner singular absorption terms driven by mixed noises

This paper investigates a stochastic parabolic system under Robin boundary conditions, for which the deterministic counterpart exhibits finite quenching. The stochastic system incorporates mixed noise, combining standard one-dimensional Brownian motion and fractional Brownian motion. Under appropriate assumptions, we derive explicit lower and upper bounds for the quenching time of the solution and establish the global existence of a weak solution. Leveraging Malliavin calculus, we further obtain a quantifiable lower and upper bound on the quenching probability. To complement the theoretical analysis, we design a numerical scheme tailored to the system and present results that validate the analytical predictions, offering insights into the interplay between noise and quenching behaviour.

math.PR↗

arXiv2025

The Gierer-Meinhardt system in the entire space with non-local proliferation rates

In this work, we present a novel stationary Gierer-Meinhardt system incorporating non-local proliferation rates, defined as follows: $$ \begin{cases} \displaystyle -Δu+λu=\frac{J*u^p}{v^q}+ρ(x) &\quad\mbox{ in }\mathbb{R}^N\, , N\geq 1,\\[0.1in] \displaystyle -Δv+μv=\frac{J*u^m}{v^s} &\quad\mbox{ in }\mathbb{R}^N.\\[0.1in] \end{cases} $$ This system emerges in various contexts, such as biological morphogenesis, where two interacting chemicals, identified as an activator and an inhibitor, are described, and in ecological systems modelling the interaction between two species, classified as specialists and generalists. The non-local interspecies interactions are represented by the terms $J*u^p, J*u^m$ where the $*$-symbol denotes the convolution operation in $\mathbb{R}^N$ with a kernel $J\in C^1(\mathbb{R}^N\setminus\{0\})$. In the system, we assume that $0<ρ\in C^{0, γ}(\mathbb{R}^N)$ with $γ\in (0,1)$, while the parameters satisfy $λ, μ, q,m,s>0$ and $p>1$. Under various integrability conditions on the kernel $J$, we establish the existence and non-existence of classical positive solutions in the function space $C^{2, δ}_{loc}(\mathbb{R}^N).$ These results further highlight the influence of the non-local terms, particularly the proliferation rates, in the proposed model.

math.AP↗

arXiv2023

On the impact of noise on quennching for a nonlocal diffusion model driven by a mixture of Brownian and fractional Brownian motions

In this paper, we study a stochastic parabolic problem involving a nonlocal diffusion operator associated with nonlocal Robin-type boundary conditions. The stochastic dynamics under consideration are driven by a mixture of a classical Brownian and a fractional Brownian motions with Hurst index $H\in(\frac{1}{2}, 1).$ We first establish local in existence result of the considered model and then explore conditions under which the resulting SPDE exhibits finite-time quenching. Using the probability distribution of perpetual integral functional of Brownian motion as well as tail estimates of fractional Brownian motion we provide analytic estimates for certain statistics of interest, such as quenching times and the corresponding quenching probabilities. The existence of global in time solutions is also investigated and as a consequence a lower estimate of the quenching time is also derived. Our analytical results demonstrate the non-trivial impact of the considered noise on the dynamics of the system. Next, a connection of a special case of the examined model is drawn in the context of MEMS technology. Finally, a numerical investigation of the considered model for a fractional Laplacian diffusion and Dirichlet-type boundary conditions is delivered.

math.PR↗

arXiv2021

A study of a nonlocal problem with Robin boundary conditions arising from MEMS technology

In the current work we study a nonlocal parabolic problem with Robin boundary conditions. The problem arises from the study of an idealized electrically actuated MEMS (Micro-Electro-Mechanical System) device. Initially we study the steady-state problem in one dimension and we present some numerical results regarding its bifurcation diagram. Next the $N-$dimensional nonlocal problem steady-state problem is investigated and a Pohozaev-type identity is first obtained which then facilitates derivation of an estimate of the pull-in voltage for radially symmetric domains. Later the time-dependent problem is investigated and global-in-time as well as quenching results are obtained again for general and radially symmetric domains. We close the current work with a numerical investigation of the presented nonlocal model via an adaptive numerical method. Various numerical experiments are presented verifying the obtained analytical results.

math.AP↗

arXiv2021

Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer-Meinhardt system

In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem \begin{equation*} \left\{\begin{array}{rcl} \partial_t u &=& Δu - u + \displaystyle{\frac{u^p}{ \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_Ωu^r dr \right)^γ}}\quad\text{in}\quad Ω\times (0,T), \\[0.2cm] \frac{ \partial u}{ \partial ν} & = & 0 \text{ on } Γ= \partial Ω\times (0,T),\\ u(0) & = & u_0, \end{array} \right. \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial Ω;$ such problem is derived as the shadow limit of a singular Gierer-Meinhardt system, cf. \cite{KSN17, NKMI2018}. Under the Turing type condition $$ \frac{r}{p-1} < \frac{N}{2}, γr \ne p-1, $$ we construct a solution which blows up in finite time and only at an interior point $x_0$ of $Ω,$ i.e. $$ u(x_0, t) \sim (θ^*)^{-\frac{1}{p-1}} \left[κ(T-t)^{-\frac{1}{p-1}} \right], $$ where $$ θ^* := \lim_{t \to T} \left(\mathop{\,\rlap{-}\!\!\int}\nolimits_Ωu^r dr \right)^{- γ} \text{ and } κ= (p-1)^{-\frac{1}{p-1}}. $$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$ u(x,T) \sim ( θ^* )^{-\frac{1}{p-1}} \left[ \frac{(p-1)^2}{8p} \frac{|x-x_0|^2}{ |\ln|x-x_0||} \right]^{ -\frac{1}{p-1}} \text{ as } x \to 0, $$ and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in \cite{MZnon97} and \cite{DZM3AS19}.

math.AP↗

arXiv2021

Gradient inequality and convergence of the normalized Ricci flow

We study the problem of convergence of the normalized Ricci flow evolving on a compact manifold $Ω$ without boundary. In \cite{KS10, KS15} we derived, via PDE techniques, global-in-time existence of the classical solution and pre-compactness of the orbit. In this work we show its convergence to steady-states, using a gradient inequality of Łojasiewicz type. We have thus an alternative proof of \cite{ha}, but for general manifold $Ω$ and not only for unit sphere. As a byproduct of that approach we also derive the rate of convergence according to this steady-sate being either degenerate or non-degenerate as a critical point of a related energy functional.

math.DG↗

arXiv2020

Impacts of noise on quenching of some models arising in MEMS technology

In the current work we study a stochastic parabolic problem. The underlying problem is actually motivated by the study of an idealized electrically actuated MEMS (Micro-Electro-Mechanical System) device in the case of random fluctuations of the potential difference controlling the device. We first present the mathematical model and then we deduce some local existence results. Next for some particular versions of the model, regarding its boundary conditions, we derive quenching results as well as estimations of the probability for such singularity to occur. Additional numerical study of the problem in one dimension follows, investigating the problem further with respect to its quenching behaviour.

math.AP↗

arXiv2020

Diffusion-driven blow-up for a non-local Fisher-KPP type model

The purpose of the current paper is to unveil the key mechanism which is responsible for the occurrence of {\it Turing-type instability} for a non-local Fisher-KPP type model. In particular, we prove that the solution of the considered non-local Fisher-KPP equation in the neighbourhood of a constant stationary solution, is destabilized via a {\it diffusion-driven blow-up}. It is also shown that the observed {\it diffusion-driven blow-up} is complete, whilst its blow-up rate is completely classified. Finally, the detected {\it diffusion-driven instability} results in the formation of unstable blow-up patterns, which are also identified through the determination of the blow-up profile of the solution.

math.AP↗

arXiv2020

Finite-time blow-up of a non-local stochastic parabolic problem

The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.

math.AP↗

arXiv2017

On the dynamics of a non-local parabolic equation arising from the Gierer-Meinhardt system

The purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activator-inhibitor system known as a Gierer-Meinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reaction-diffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the Gierer-Meinhardt model is reduced to a single though non-local equation whose dynamics will be investigated. We mainly focus on the derivation of blow-up results for this non-local equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusion-driven blow-up, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns.

math.AP↗

arXiv2017

A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted vs. repelled by a single chemical substance. The production vs. destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model we investigate the variational structures, in particular we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy-Littlewood-Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

math.AP↗

arXiv2015

On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics

We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for \[ u_t = u Δu + u \int_Ω|\nabla u|^2 \] in bounded domains $Ω\subset\mathbb{R}^n$ and prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. We show that in this case the blow-up set coincides with $\overlineΩ$, i.e. the finite-time blow-up is global. Key words: Degenerate diffusion, non-local nonlinearity, blow-up, evolutionary games, infinite dimensional replicator dynamics

math.AP↗