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Spiralling and other solutions in limiting profiles of competitiondiffusion systems Terracini, Susanna
Description
Reactiondiffusion systems with strong interaction terms appear in many multispecies physical problems as well as in population dynamics. The qualitative properties of the solutions and their limiting profiles in different regimes have been at the center of the community's attention in recent years. A prototypical example is the system of equations \[\left\{\begin{array}{l} \Delta u+a_1u = b_1u^{p+q2}u+cpu^{p2}v^qu,\\ \Delta v+a_2v = b_2v^{p+q2}v+cqu^{p}v^{q2}v \end{array} \right. \] in a domain $\Omega\subset \mathbb{R}^N$ which appears, for example, when looking for solitary wave solutions for BoseEinstein condensates of two different hyperfine states which overlap in space. The sign of $b_i$ reflects the interaction of the particles within each single state. If $b_i$ is positive, the self interaction is attractive (focusing problems). The sign of $c$, on the other hand, reflects the interaction of particles in different states. This interaction is attractive if $c>0$ and repulsive if $c<0$. If the condensates repel, they eventually separate spatially giving rise to a free boundary. Similar phenomena occurs for many species systems. As a model problem, we consider the system of stationary equations: \[ \begin{cases} \Delta u_i=f_i(u_i)\beta u_i\sum_{j\neq i}g_{ij}(u_j)\;\\ u_i>0\;. \end{cases} \] The cases $g_{ij}(s)=\beta_{ij}s$ (LotkaVolterra competitive interactions) and $g_{ij}(s)=\beta_{ij}s^2$ (gradient system for GrossPitaevskii energies) are of particular interest in the applications to population dynamics and theoretical physics respectively. Phase separation and has been described in the recent literature, both physical and mathematical. Relevant connections have been established with optimal partition problems involving spectral functionals. The classification of entire solutions and the geometric aspects of phase separation are of fundamental importance as well. We intend to focus on the most recent developments of the theory in connection with problems featuring anomalous diffusions, nonlocal and non symmetric interactions.
Item Metadata
Title 
Spiralling and other solutions in limiting profiles of competitiondiffusion systems

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20190611T09:04

Description 
Reactiondiffusion systems with strong interaction terms appear in many multispecies physical problems as well as in population dynamics. The qualitative properties of the solutions and their limiting profiles in different regimes have been at the center of the community's attention in recent years. A prototypical example is the system of equations
\[\left\{\begin{array}{l}
\Delta u+a_1u = b_1u^{p+q2}u+cpu^{p2}v^qu,\\
\Delta v+a_2v = b_2v^{p+q2}v+cqu^{p}v^{q2}v
\end{array}
\right.
\]
in a domain $\Omega\subset \mathbb{R}^N$ which appears, for example, when looking for solitary wave solutions for BoseEinstein condensates of two different hyperfine states which overlap in space. The sign of $b_i$ reflects the interaction of the particles within each single state. If $b_i$ is positive, the self interaction is attractive (focusing problems). The sign of $c$, on the other hand, reflects the interaction of particles in different states. This interaction is attractive if $c>0$ and repulsive if $c<0$. If the condensates repel, they eventually separate spatially giving rise to a free boundary. Similar phenomena occurs for many species systems. As a model problem, we consider the system of stationary equations:
\[
\begin{cases}
\Delta u_i=f_i(u_i)\beta u_i\sum_{j\neq i}g_{ij}(u_j)\;\\
u_i>0\;.
\end{cases}
\]
The cases $g_{ij}(s)=\beta_{ij}s$ (LotkaVolterra competitive interactions) and $g_{ij}(s)=\beta_{ij}s^2$ (gradient system for GrossPitaevskii energies) are of particular interest in the applications to population dynamics and theoretical physics respectively.
Phase separation and has been described in the recent literature, both physical and mathematical. Relevant connections have been established with optimal partition problems involving spectral functionals. The classification of entire solutions and the geometric aspects of phase separation are of fundamental importance as well. We intend to focus on the most recent developments of the theory in connection with problems featuring anomalous diffusions, nonlocal and non symmetric interactions.

Extent 
62.0 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: Università di Torino

Series  
Date Available 
20200908

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0394237

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Faculty

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International