arXiv subjects

Stefanie Sonner

Publications and source records attributed to Stefanie Sonner.

11 recordsLinked to original sources

arXiv2026

A panoramic view of exponential attractors

We state necessary and sufficient conditions for the existence of $T$-discrete exponential attractors for semigroups in complete metric spaces. These conditions are formulated in terms of a covering condition for iterates of the absorbing set under the time evolution of the semigroup and imply the existence and finite-dimensionality of the global attractor. We then review, generalize and compare existing construction methods for exponential attractors and show that they all imply the covering condition. Furthermore, we relate the results and concept of $T$-discrete exponential attractors to the classical notion of exponential attractors.

math.DS↗

arXiv2026

Attractors for Singular-Degenerate Porous Medium Type Equations Arising in Models for Biofilm Growth

We investigate the long-time behaviour of solutions of a class of singular-degenerate porous medium type equations in bounded domains with homogeneous Dirichlet boundary conditions. The existence of global attractors is shown under very general assumptions. Assuming, in addition, that solutions are globally Hölder continuous and the reaction terms satisfy a suitable sign condition in the vicinity of the degeneracy, we also prove the existence of an exponential attractor, which, in turn, yields the finite fractal dimension of the global attractor. Moreover, we extend the results for scalar equations to systems where the degenerate equation is coupled to a semilinear reaction-diffusion equation. The study of such systems is motivated by models for biofilm growth.

math.AP↗

arXiv2025

Mathematical models for the EP2 and EP4 signaling pathways and their crosstalk

The G-protein coupled receptors EP2 and EP4 both transduce the signal of the lipid messenger Prostaglandin E2 (PGE2). Changes in the cell response to PGE2 can have important effects on immunity and development of diseases, however, a thorough understanding of the EP2-EP4 receptors' signaling pathways is lacking. Experimental data show that receptor activity (indicated by cAMP expression) has a different kinetics depending on which receptor is triggered by PGE2 and that crosstalk exists between EP2 and EP4. To better understand the underlying mechanisms and be able to predict cell responses to PGE2, we develop novel mathematical models for the cAMP signaling pathways of EP2 and EP4 and their crosstalk. Ligand binding dynamics plays a crucial role for both, the single receptor activity and their crosstalk. The mathematical models can predict the qualitative cAMP levels observed experimentally and provide possible explanations for the differences and commonalities in the signaling behavior of EP2 and EP4. As inhibition of PGE2 signaling is gaining increasing attention in tumor immunology, these mathematical models could contribute to design effective anti-tumor therapies targeting EP2 and EP4.

q-bio.MN↗

arXiv2025

Pathwise mild solutions for superlinear stochastic evolution equations and their attractors

We investigate stochastic parabolic evolution equations with time-dependent random generators and locally Lipschitz continuous drift terms. Using pathwise mild solutions, we construct an infinite-dimensional stationary Ornstein-Uhlenbeck type process, which is shown to be tempered in suitable function spaces. This property, together with a bootstrapping argument based on the regularizing effect of parabolic evolution families, is then applied to prove the global well-posedness and the existence of a random attractor for reaction-diffusion equations with random non-autonomous generators and nonlinearities satisfying certain growth and dissipativity assumptions.

math.PR↗

arXiv2023

Cauchy problem for singular-degenerate porous medium type equations: well-posedness and Sobolev regularity

Motivated by models for biofilm growth, we consider Cauchy problems for quasilinear reaction diffusion equations where the diffusion coefficient has a porous medium type degeneracy as well as a singularity. We prove results on the well-posedness and Sobolev regularity of solutions. The proofs are based on m-accretive operator theory, kinetic formulations and Fourier analytic techniques.

math.AP↗

arXiv2023

Well-posedness and qualitative properties of quasilinear degenerate evolution systems

We analyze nonlinear degenerate coupled PDE-PDE and PDE-ODE systems that arise, for example, in the modelling of biofilm growth. One of the equations, describing the evolution of a biomass density, exhibits degenerate and singular diffusion. The other equations are either of advection-reaction-diffusion type or ordinary differential equations. Under very general assumptions, the existence of weak solutions is proven by considering regularized systems, deriving uniform bounds, and using fixed point arguments. Assuming additional structural assumptions we also prove the uniqueness of solutions. Global-in-time well-posedness is established for Dirichlet and mixed boundary conditions, whereas, only local well-posedness can be shown for homogeneous Neumann boundary conditions. Using a suitable barrier function and comparison theorems we formulate sufficient conditions for finite-time blow-up or uniform boundedness of solutions. Finally, we show that solutions of the degenerate parabolic equation inherit additional global spatial regularity if the diffusion coefficient has a power-law growth.

math.AP↗

arXiv2022

Travelling waves in a PDE-ODE coupled system with nonlinear diffusion

We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. Finally, numerical simulations are presented that affirm the theoretical predictions on the existence, stability, and parametric dependence of the TWs.

math.AP↗

arXiv2021

Well-posedness of singular-degenerate porous medium type equations and application to biofilm models

We show the well-posedness for a large class of degenerate parabolic equations with an additional singularity and mixed Dirichlet-Neumann boundary conditions on bounded Lipschitz domains. The proof is based on an $L^1$-contraction result. In addition, we analyze systems where degenerate equations are coupled to semilinear reaction diffusion equations. This setting includes mathematical models for biofilm growth which are the motivation for our analysis.

math.AP↗

arXiv2021

Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions. This approach is essential for our setting since the stochastic evolution equation cannot be transformed into a family of PDEs with random coefficients via the stationary Ornstein-Uhlenbeck process.

math.PR↗

arXiv2018

Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions

The stabilisation by noise on the boundary of the Chafee-Infante equation with dynamical boundary conditions subject to a multiplicative Itô noise is studied. In particular, we show that there exists a finite range of noise intensities that imply the exponential stability of the trivial steady state. This differs from previous works on the stabilisation by noise of parabolic PDEs, where the noise acts inside the domain and stabilisation typically occurs for an infinite range of noise intensities. To the best of our knowledge, this is the first result on the stabilisation of PDEs by boundary noise.

math.AP↗

arXiv2012

Validating Stochastic Models: Invariance Criteria for Systems of Stochastic Differential Equations and the Selection of a Stochastic Hodgkin-Huxley Type Model

In recent years, many difficulties appeared when taking into account the inherent stochastic behavior of neurons and voltage-dependent ion channels in Hodgking-Huxley type models. In particular, an open problem for a stochastic model of cerebellar granule cell excitability was to ensure that the values of the gating variables remain within the unit interval. In this paper, we provide an answer to this modeling issue and obtain a class of viable stochastic models. We select the stochastic models thanks to a general criterion for the flow invariance of rectangular subsets under systems of stochastic differential equations. We formulate explicit necessary and sufficient conditions, that are valid for both, Itô's and Stratonovich's interpretation of stochastic differential equations, improving a previous result obtained by A. Milian [A.Milian, Coll. Math. 1995] in the Itô case. These invariance criteria allow to validate stochastic models in many applications. To illustrate our results we present numerical simulations for a stochastic Hodgkin-Huxley model.

math.DS↗