Conference Agenda

Collective behaviour plays a central role in many biological, physical and social systems. A common modelling approach consists in describing these situations by interacting particle systems. Moreover, many systems exhibit an underlying network structure which describes the coupling between the particles and the corresponding strength. Often, this network structure is not fixed for all time but also evolves dynamically together with the particle system.
This talk addresses such adaptively coupled interacting systems when the number of particles is very large. More precisely, we start with a system of finitely many particles which is coupled to an evolution equation for the underlying network. Relying on the concept of graph convergence, we prove that for a large class of couplings the system can be approximated by a mean-field equation in the infinite particle limit.

In the last decade, the study of double phase problems (involving (p,q) - Laplace) has attracted great interest due to their ability to model complex phenomena and materials with heterogeneous properties, e.g. composites made of two different materials. From a mathematical point of view, double phase elliptic and parabolic equations, even for the constant exponents p and q (isotropic case), attract great interest due to their so-called nonstandard growth conditions, for which most of the well-established methods of nonlinear analysis are no longer applicable. In the anisotropic and variable exponents settings, the situation becomes even more complex. This creates a strong motivation to develop new analytical tools tailored to these equations. Numerous publications in the last years highlight the active development of double phase problems, see [Boegelein, Duzaar, Marcellini, Scheven, JMPA, 2022; Boegelein, Strunk, Annali di Mat. Pura ed Appl., 2024; Crespo-Blanco, Gasinski, Harjulehto, Winkert, JDE, 2022; De Filippis, G. Mingione, ARMA, 2023; Buryachenko, Skrypnik, Potential An., 2022, J. Math.Sci., 2017].

In this talk we study a coupled double phase (p(x), q(x)) - Laplace thermistor model, that describes the electrothermal behavior of semiconductor devices, taking different parallel charge transport mechanisms into account. Such systems model materials conducting both heat and electrical current and for which the electrical conductivity in the definition of current density can strongly depend on the temperature. Devices of this type are called thermistors, see, for instance, [Cimatti, Quant. Appl. Math., 1989]. As for coupled systems involving double phase equations, and, in particular, thermistor type models, such problems are addressed in the present study for the first time.

More precisely, our system consists of the current flow elliptic double phase equation, involving (p(x), q(x)) Laplacian with modulation coefficient a(x) and conductivity coefficients depend on the temperature as well as the phases of the materials, for the electrostatic potential u, coupled to the heat equation for the temperature T. The right hand side of the heat equation, Joule heat term H, depends on u and T and belongs to L¹ . The system is complemented by Dirichlet and homogeneous Neumann boundary conditions for the electrical potential, as well as Robin boundary condition for the temperature.

We operate with discontinuous (only measurable) variable exponents p(x), q(x) and modulation coefficient a(x), and investigate the full (p(x), q(x)) double phase situation for a(x)>0. Together with the necessity to involve the concept of entropy solutions to the heat equation with right hand side from the space L¹, that are the main challenges of the problem.

Our main result concerns the existence of a weak solution (u,T) to the coupled double phase (p(x), q(x)) - Laplace thermistor model. The solution u, electrostatic potential, is assumed as a weak solution (in a generalized Sobolev space, related to a Musielak-Orlicz space) to the double phase elliptic equation with variable measurable exponents p(x), q(x), modulation a(x)>0 and conductivity coefficients. The solution T is an entropy solution to the heat equation with nonlinear right hand side H from the space L¹ . To tackle the problem, we use a combination of entropy solution techniques and Schauder's fixed-point theorem. We discuss also the electrothermal behavior of heterogeneous organic semiconductor materials with different charge transport mechanisms characterized by different activation energies. Note, that p(x)- Laplace thermistor models were studied very well before [Bulicek, Glitzky, Liero, SIAM J. Math. Anal, 2016; DCDS-S, 2017; Glitzky, Liero, Nonlin. Anal., 2017; DCDS-S, 2021; Liero, Koprucki, Fischer, Scholz, Glitzky, ZAMP, 2015].

Acknowledgment. This work is supported by the Alexander von Humboldt‑Stiftung.

In this talk, we focus on the following chemotaxis system with signal-dependent motility

\begin{align*}

u_{\varepsilon t}&=\Delta(u_\varepsilon e^{-v_\varepsilon})+\varepsilon(\kappa u_\varepsilon-\mu u_\varepsilon^2),\\

0&=\Delta v_\varepsilon -v_\varepsilon+u_\varepsilon

\end{align*}

in a smooth bounded domain $\Omega\in\mathbb{R}^2$ with homogenuous Neumann boundary conditions and parameters $\kappa,\mu>0$ and $\varepsilon\in (0,1)$. It is known that classical solutions are global and bounded. Our aim is to find uniform convergence of solutions $(u_\varepsilon,v_\varepsilon)$ to solutions $(u,v)$ in the limit $\varepsilon\searrow 0$. The main challenge will be to derive $\varepsilon$-independent $L^p$-bounds for all $p\geq 1$, which will be local in time. \\

As a consequence, we can show that for certain initial data $u_0$ the solutions $u_\varepsilon$ exceed any certain value $M>0$ before a time $T(M)>0$ as long as $\varepsilon\in(0,\tilde{\varepsilon}(M))$ for some $\tilde{\varepsilon}(M)>0$. That result is based on the known infinite time blow-up in the $\varepsilon=0$-system that can occur, if $\Omega=B_R(0)$ for some $R>0$ and the initial data $u_0$ of supercritical mass is radially symmetric.

We present a detailed analytical study of a nonlinear PDE model for ant trail formation introduced in [Bertucci, Rakotomalala, Tomašević, 2025]. The model consists of a coupled system of two parabolic equations: one governing the pheromone field produced by the particles, and an active-matter equation describing the particle density driven by this chemical signal. A previous numerical study [Bruna, Schmidtchen, de Wit, 2025] revealed complex behavior in this model. We here provide a rigorous analysis of its dynamical and stationary properties. We establish the existence of a compact global attractor and derive a dimensional lower bound for it. We then prove the existence of families of spot and lane solutions arising along bifurcation branches as the interaction strength increases.

References:

• Existence of spot and lane stationary solutions for an ant active matter PDE model, with Oscar de Wit, arXiv preprint, arXiv:2504.01765, 2026.

• Existence and dimensional lower bound for the global attractor of a PDE model for ant trail formation, with Oscar de Wit, arXiv preprint, arXiv:2504.01765, 2025.