Special Session 127: Nonlocal models arising in biology and ecology

Elliptic relaxation of a degenerate Cahn-Hilliard tumour growth model

Matteo Fornoni Università degli Studi di Milano Italy Co-Author(s):    Cecilia Cavaterra, Maurizio Grasselli, Benoit Perthame

Abstract: We consider a phase-field system modelling solid tumour growth. This system consists of a Cahn-Hilliard equation coupled with a nutrient equation. The former is characterised by a degenerate mobility and a singular potential of a single-well type, promoting short-range attraction and long-range repulsion for better biological consistency. Both equations are subject to suitable reaction terms which model proliferation and nutrient consumption. Chemotactic effects are also taken into account. Adding an elliptic regularisation, depending on a suitable relaxation parameter, in the equation for the chemical potential, we prove the existence of a weak solution to an initial and boundary value problem for the relaxed system. Then, we let the relaxation parameter go to zero, and we recover the existence of a weak solution to the original system.

Asymptotic models for tumor growth

Rafael Granero Belinchon Universidad de Cantabria Spain Co-Author(s):    Martina Magliocca

Abstract: In this talk we will present two new models of tumor growth. These models take the form of nonlinear and nonlocal evolutionary PDE and they allow for arbitrary multilayer geometries. Furthermore, we will present the well-posedness theory associated to these equations.

Nonlocal-to-Local Convergence for a Diffuse Interface Model for Two Phase Flow with Matched Densities

Christoph Hurm University of Regensburg Germany Co-Author(s):    Patrik Knopf, Andrea Poiatti

Abstract: We prove convergence of a sequence of strong solutions to a diffuse interface model for the two-phase flow of incompressible fluids with matched densities with a nonlocal Cahn-Hilliard equation to the strong solution to the corresponding system with a standard local Cahn-Hilliard equation. The analysis is done in the case of constant viscosity. The proof is based on an energy method and convergence of the linearized operators in the Cahn-Hilliard equations.

Traveling wave-like solutions to a time-fractional Fisher-KPP type equation

Hiroshi Ishii Hokkaido University Japan Co-Author(s):

Abstract: In this talk, we study a time-fractional Fisher-KPP type equation in which the time derivative is given by the Caputo derivative. The main focus is on the long-time behavior of front solutions. After briefly reviewing the background of the equation, we present numerical results and discuss the qualitative properties of the solutions. It is known that this equation does not admit classical traveling wave solutions. To describe the long-time behavior, we assume that the solution asymptotically behaves like a traveling wave solution and analyze possible wave profiles to which the solution may converge. Based on this analysis, we construct traveling wave-like solutions that describe the asymptotic behavior of the front solution.

Traveling waves in a virus infection model of cell-to-cell transition

Yoshihisa Morita Ryukoku University Japan Co-Author(s):

Abstract: Some {\it in vitro} experiments have observed the propagation of viral infection through cell-to-cell contact. Motivated by this phenomenon, we consider a cell-to-cell infection model consisting of variables representing the concentrations of target cells, eclipse-phase cells, infectious cells, and dead cells. The model describes the spread of infection through spatially discrete coupling between neighboring infectious and target cells. Numerical simulations demonstrate the emergence of traveling waves representing the propagation of the infectious state. We establish the existence of such traveling waves by reformulating the system as a nonlinear integral equation and applying a fixed-point argument. We also show that an integro-differential formulation of the model admits traveling wave solutions.This talk is based on the work of Asai-Iwami-Morita (2026).

Well-posedness for damped hyperbolic equation with critical Hartree type nonlinearity

Chao Yang AGH University of Krakow Poland Co-Author(s):

Abstract: In this talk, we are concerned with the well-posedness of the initial boundary value problem for a hyperbolic equation with strong damping, weak damping, and Hartree type nonlinearity with Hardy-Littlewood-Sobolev critical exponent. Local existence of the solution is established via the Galerkin method and the Banach fixed point theorem. To address difficulties arising from the damping structure, the nonlocal characteristic of the Hartree type nonlinearity, and the lack of compactness caused by the critical exponent, we develop a suitable potential well framework and prove the invariance of associated manifolds. This framework allows us to obtain global existence and finite time blowup results at different initial energy levels. We further analyze the dynamical behavior of solutions, including asymptotic properties and blowup time estimates. Finally, we examine the continuous dependence of global solutions on initial data and coefficients of dampings.