Stability, Instability and Concentration Phenomena in Reaction-Diffusion Equations: A Geometric Approach.

• Grant number: 18/10033-7
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: September 01, 2018
• End date: July 31, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Arnaldo Simal Do Nascimento
• Grantee: Carolinne Stefane de Souza
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil

Abstract

Many different mathematical models in reaction-diffusion evolution processes can be described through partial differential equations of parabolic type.In the study of qualitative theory of such equations the question of existence of stationary solutions has a major role inasmuch as quite often one has a gradient flow and in the majority of these cases the flux global dynamic will, to a great extension, be determined by the stable and unstable stationary solutions.The importance of stable stationary solutions lie in the fact that the in the associated physical models, they correspond to the only realizable and time-persistent states.Roughly speaking, the main aim of the current project is the better understanding of the basic mechanisms acting in these processes that induce the existence of stable and unstable stationary solutions in environments with quite different geometries: Euclidian domains, surfaces and thin corrugated domains along a space curve.Usually this is accomplished via identification of the meaningful parameters in the different environments and subsequently the search for novel informations through a rigorous mathematical analysis. These phenomena have been studied for quite some time, and as such better understood, in the case of Euclidian domains. Yet in surfaces or in thin corrugated domains along a spatial curve, the geometry of the domains come into play and it expresses itself via different curvature definitions.