Homogenization of nonlocal problems
- Reference number
- UKR24-0023
- Project leader
- Rybalko, Volodymyr
- Start and end dates
- 240501-240831
- Amount granted
- 382 000 SEK
- Administrative organization
- Chalmers University of Technology
- Research area
- Computational Sciences and Applied Mathematics
Summary
Many models in mathematical biology and population dynamics take into account nonlocal interactions in the studied systems. In particular, nonlocal models naturally appear in descriptions of dispersal of cells or organisms. In this project we propose to continue the study (started in the project UKR 22- 0004) of spectral problems for nonlocal operators in the context of dispersal models in strongly iinhomogeneous environments. We propose to study localization of eigenfunctions in selfadjoint case. We expect that (similarly to elliptic operators) eigenfunctions corresponding to eigenvalues from the bottom part of the spectrum are localized in neighborhoods of special (hot) points, and asymptotic behavior of these eigenpairs can be established in detail via blow-up analysis. Another important question is to study scaled logarithmic asymptotics of the principal eigenfunction. In the previous work we established that the asymptotic behavior of the principal eigenvalue is described by an additive eigenvalue problem for effective Hamilton-Jacobi equation. We conjecture that scaled logarithmic transformations of the principal eigenfunction converge to a viscosity solution of the effective problem. If time permits we are also planning to address homogenization of random nonlocal spectral problems and generalize the homogenization result obtained for nonlocal spectral problems in the periodic case to random statistically homogeneous environments.
Popular science description
The project continues the study of nonlocal problems started in the framework of the project 22-0004 (Mathematics of living cells). Such problems contain equations with integral terms, as opposed to differential expressions used in traditional models of continuous media. Nonlocal problems appear in mathematical biology and population dynamics, image processing, particle systems, mathematical finance etc. Also they are of great interest because of their mathematical content. Specific goal of this project is to describe qualitative features and important quantitative characteristics of population dynamics in strongly inhomogeneous environments, i.e. those having microstructure. Direct numerical computations for such models are quite time-consuming and we propose to derivation effective (homogenized) models via asymptotic analysis in the limit of small microscopic scale. Questions of this type are studied in the framework of homogenization theory. It is proposed to address question of localization and appearance of new fine scales in symmetric spectral problems. Such phenomena are known and rather well studied for local models described by differential operators. Another important question is to find asymptotics for principal eigenfunction in general case. Finally, if time permits we also planning to address random nonlocal models.