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Katerina Nik

Publications and source records attributed to Katerina Nik.

16 recordsLinked to original sources
arXiv2026

Completion of DNA replication is constrained by the spatiotemporal organisation of origin firing

DNA replication requires the coordination of origin firing and fork progression to ensure the entire genome is timely duplicated before cell division. Yet origin firing is stochastic, giving rise to the classical random completion problem of how probabilistic local events can nevertheless ensure reliable genome duplication. Although several biological mechanisms have been proposed to resolve this problem, a quantitative account of how heterogeneous initiation and fork speed govern the persistence of the final unreplicated regions is still lacking. To address this gap, we introduce a population-level kinetic framework that extends KJMA nucleation-and-growth models by tracking unreplicated intervals over size, genomic position and time. We establish well-posedness of the resulting mean-field system and global existence for compatible data. Notably, by introducing a local initiation mass function, we quantify how the density and spatial organisation of origin firing constrain replication completion, yielding novel and sharp upper bounds on both the worst-locus unreplicated fraction and locuswise near-completion time. These results provide a rigorous and computable foundation for mapping vulnerabilities in replication completion and relating persistent unreplicated regions to replication stress and genome instability.

math.AP↗
arXiv2026

Stability of Equilibria in a Biofilm Reactor Model with Wall Attachment and Thermodynamic Growth Inhibition

The dynamics of a mathematical model for a chemostat-type reactor is investigated. The model describes the temporal evolution of suspended and wall-attached bacterial populations, with the latter represented as a one-dimensional biofilm, subject to a non-reproducing growth-limiting substrate and a reaction product formed through bacterial utilization of the substrate. In particular, it is shown that, in the regime where the trivial (washout) equilibrium is unstable, there exists a unique nontrivial equilibrium that is locally asymptotically stable. Under slightly stronger assumptions, uniform persistence and global asymptotic stability of the nontrivial equilibrium are established.

math.AP↗
arXiv2026

Dynamics of Coral-Macroalgae Interactions under Crowding

We study a planar ODE model for the benthic competition between coral, macroalgae, and algal turf on a reef, extending the classical model of Mumby, Hastings, and Edwards by a nonlinear, density-dependent coral mortality that accounts for crowding. The strength of crowding is set by an exponent $\delta>0$ that reshapes the coral nullcline and enriches the bifurcation structure of the system. We establish positive invariance of the biologically relevant region and the absence of periodic orbits, classify the three boundary equilibria together with their local stability, and reduce the coexistence problem to a single scalar equation whose shape, in particular its concavity, controls the number and local stability of the interior equilibria. The grazing intensity $g$ organizes the dynamics through two thresholds $g_0<g_1$ determining the stability of the coral- and macroalgae-dominated states, and a further threshold $g^\star$ at which two interior equilibria collide. We prove that the system undergoes transcritical bifurcations at the boundary equilibria and a saddle-node bifurcation of interior equilibria, and we discuss the implications for coral reef resilience and hysteresis. We complement these results with numerical simulations that illustrate the bifurcation sequence across the grazing regimes.

math.DS↗
arXiv2026

Strong solutions and stability for a thin-film equation of shear-thinning fluids with contact line in partial wetting

We consider a power-law thin-film equation for strongly shear-thinning fluids. Weak solutions to this equation have been constructed more than twenty years ago by Ansini and Giacomelli. Here, we pass over to analyzing strong solutions with nonzero contact angle (partial-wetting regime), and place emphasis on studying the behavior of solutions near points where the film height vanishes (the contact-line region) by considering perturbations of a linear profile. The leading-order equation in von-Mises coordinates shows similarities with the evolution equation for the $p$-Laplace, though being of fourth order. Using a time discretization, we reduce the leading-order problem to finding a variational solution, and pass to the limit in the discretization scheme on suitably estimating higher-order nonlinear terms in conjunction with compactness arguments. This proves existence and asymptotic stability of strong solutions that are perturbations of the linear profile, and yields control on the contact-line velocity on carefully tracking singular terms in our estimates. While we believe that the transformed equation shows mathematical features the analysis of which stands on its own merit, it also physically corroborates shear thinning behavior as an alternative in resolving the no-slip paradox, as opposed to more standard approaches like introducing slip at the liquid-solid interface.

math.AP↗
arXiv2026

Analysis of a Biofilm Model in a Continuously Stirred Tank Reactor with Wall Attachment

We investigate a mathematical model for a bacterial population in a continuously stirred tank reactor with wall attachment. The model couples a free-boundary value problem for substrate diffusion in the one-dimensional biofilm with a system of nonlinear ODEs for biofilm thickness, suspended biomass, and free substrate concentration. We establish global well-posedness and analyze the long-term dynamics. In particular, we characterize the local and global stability of the trivial (washout) equilibrium, prove the existence of a nontrivial equilibrium, and, under additional structural assumptions, establish its uniqueness and derive conditions for its local stability.

math.AP↗
arXiv2024

Bernis estimates for higher-dimensional doubly-degenerate non-Newtonian thin-film equations

For the doubly-degenerate parabolic non-Newtonian thin-film equation $$ u_t + \text{div}\bigl(u^n |\nabla Δu|^{p-2} \nabla Δu\bigr) = 0, $$ we derive (local versions) of Bernis estimates of the form $$ \int_Ω u^{n-2p} |\nabla u|^{3p}\, dx + \int_Ω u^{n-\frac{p}{2}} |Δu|^{\frac{3p}{2}}\, dx \leq c(n,p,d) \int_Ω u^n|\nabla Δu|^p\, dx, $$ for functions $u \in W^2_p(Ω)$ with Neumann boundary condition, where $2 \leq p < \frac{19}{3}$ and $n$ lies in a certain range. Here, $Ω\subset \mathbb{R}^d$ is a smooth convex domain with $d < 3p$. A particularly important consequence is the estimate $$ \int_Ω |\nabla Δ(u^{\frac{n+p}{p}})|^p\, dx \leq c(n,p,d) \int_Ω u^n|\nabla Δu|^p\, dx. $$ The methods used in this article follow the approach of [Grü01] for the Newtonian case, while addressing the specific challenges posed by the nonlinear higher-order term $|\nabla Δu|^{p-2} \nabla Δu$ and the additional degeneracy. The derived estimates are key to establishing further qualitative results, such as the existence of weak solutions, finite propagation of support, and the appearance of a waiting-time phenomenon.

math.AP↗
arXiv2024

On an obstacle problem for the Brakke flow with a generalized right-angle boundary condition

We study Brakke's mean curvature flow with obstacles and with a right-angle boundary condition. Assuming that the obstacles have $C^{1,1}$-boundaries we prove that a weak solution exists globally in time. To show the existence we apply the phase-field method and thus investigate the singular limit of the Allen-Cahn equation with forcing term and homogeneous Neumann bounday condition. We also construct sub- and supersolutions that correspond to the obstacles.

math.AP↗
arXiv2024

An existence result for accretive growth in elastic solids

We investigate a model for the accretive growth of an elastic solid. The reference configuration of the body is accreted in its normal direction, with space- and deformation-dependent accretion rate. The time-dependent reference configuration is identified via the level sets of the unique viscosity solution of a suitable generalized eikonal equation. After proving the global-in-time well-posedness of the quasistatic equilibrium under prescribed growth, we prove the existence of a local-in-time solution for the coupled equilibrium-growth problem, where both mechanical displacement and time-evolving set are unknown. A distinctive challenge is the limited regularity of the growing body, which calls for proving a new uniform Korn inequality.

math.AP↗
arXiv2022

Long-time behaviour and stability for quasilinear doubly degenerate parabolic equations of higher order

We study the long-time behaviour of solutions to quasilinear doubly degenerate parabolic problems of fourth order. The equations model for instance the dynamic behaviour of a non-Newtonian thin-film flow on a flat impermeable bottom and with zero contact angle. We consider a shear-rate dependent fluid the rheology of which is described by a constitutive power-law or Ellis-law for the fluid viscosity. In all three cases, positive constants (i.e. positive flat films) are the only positive steady-state solutions. Moreover, we can give a detailed picture of the long-time behaviour of solutions with respect to the $H^1(Ω)$-norm. In the case of shear-thickening power-law fluids, one observes that solutions which are initially close to a steady state, converge to equilibrium in finite time. In the shear-thinning power-law case, we find that steady states are polynomially stable in the sense that, as time tends to infinity, solutions which are initially close to a steady state, converge to equilibrium at rate $1/t^{1/β}$ for some $β> 0$. Finally, in the case of an Ellis-fluid, steady states are exponentially stable in $H^1(Ω)$.

math.AP↗
arXiv2022

A direct construction of a full family of Whitham solitary waves

Starting with the periodic waves earlier constructed for the gravity Whitham equation, we parameterise the solution curves through relative wave height, and use a limiting argument to obtain a full family of solitary waves. The resulting branch starts from the zero solution, traverses unique points in the wave speed-wave height space, and reaches a singular highest wave at $φ(0) = \fracμ{2}$. The construction is based on uniform estimates improved from earlier work on periodic waves for the same equation, together with limiting arguments and a Galilean transform to exclude vanishing waves and waves levelling off at negative surface depth. In fact, the periodic waves can be proved to converge locally uniformly to a wave with negative tails, which is then transformed to the desired branch of solutions. The paper also contains some proof concerning uniqueness and continuity for signed solutions (improved touching lemma).

math.AP↗
arXiv2021

Existence results for a morphoelastic model

We present some existence results for three-dimensional quasistatic morphoelasticity. The state of the growing body is described by its deformation and the underlying growth tensor and is ruled by the interplay of hyperelastic energy minimization and growth dynamics. By introducing a regularization in the model, we prove that solutions can be obtained as limits of time-discrete solutions, built by means of an exponential-update scheme. By further allowing the dependence of growth dynamics on an additional scalar field, to be interpreted as a nutrient or inhibitor, we formulate an optimal control problem and prove existence of optimal controls and states. Eventually, we tackle the existence of coupled morphoelastic and nutrient solutions, when the latter is allowed to diffuse and interact with the growing body.

math.AP↗
arXiv2021

On a Free Boundary Model for Three-Dimensional MEMS with a Hinged Top Plate I: Stationary Case

A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter $λ$ which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of $λ$. Nonexistence of stationary solutions is obtained when $λ$ is large enough.

math.AP↗
arXiv2021

On a Free Boundary Model for Three-Dimensional MEMS with a Hinged Top Plate II: Parabolic Case

A parabolic free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a fourth-order semilinear parabolic equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the device. The strength of the coupling is tuned by a parameter $λ$ which is proportional to the square of the applied voltage difference. It is proven that the model is locally well-posed in time and that, for $λ$ sufficiently small, solutions exist globally in time. In addition, touchdown of the top plate on the ground plate is shown to be the only possible finite time singularity.

math.AP↗
arXiv2020

Energy minimizers for an asymptotic MEMS model with heterogeneous dielectric properties

A model for a MEMS device, consisting of a fixed bottom plate and an elastic plate, is studied. It was derived in a previous work as a reinforced limit when the thickness of the insulating layer covering the bottom plate tends to zero. This asymptotic model inherits the dielectric properties of the insulating layer. It involves the electrostatic potential in the device and the deformation of the elastic plate defining the geometry of the device. The electrostatic potential is given by an elliptic equation with mixed boundary conditions in the possibly non-Lipschitz region between the two plates. The deformation of the elastic plate is supposed to be a critical point of an energy functional which, in turn, depends on the electrostatic potential due to the force exerted by the latter on the elastic plate. The energy functional is shown to have a minimizer giving the geometry of the device. Moreover, the corresponding Euler-Lagrange equation is computed and the maximal regularity of the electrostatic potential is established.

math.AP↗