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Vorlesung

Reaction-diffusion equations: From Semigroup Theory to Pattern Formation

Classical mathematical models of biological or chemical pattern formation have been developed using partial differential equations of reaction-diffusion type. The lecture will be devoted to mathematical analysis of such equations, in particularly different mechanisms of pattern formation.
The topics of the course will include classical semigroup theory for existence, uniqueness and regularity of solutions of reaction-diffusion equations. Starting from linear problems, solutions to semilinear equations will be treated as well and mathematical methods for the analysis of model dynamics, such as theory of invariant rectangles and comparison principles, are developed.
We will focus on two-component reaction-diffusion and reaction-diffusion-ODE systems (an ordinary differential equation coupled to a parabolic equation) which serve as basic models to understand pattern formation mechanisms. The course will cover a classical theory of Turing-type pattern formation (based on diffusion-driven instability of spatially homogenous steady states), and more recent approaches such as models based on the existence of multiple steady states and hysteresis. Analysis of pattern formation involves analysis of two-point boundary value problems, solving spectral problems and applying of singular perturbation methods.

Prerequisites: A basic course in partial differential equations and functional analysis is required.

The lecture will cover the following topics:

Semigroup Theory: Semigroups, Generators, Solutions to abstract linear Parabolic Equations
Stability Theory: Spectral Mapping Theorem for semigroups, Spectral Analysis of the Laplace Operator, Nonlinear Parabolic Equations
Applications: Analysis of model examples - Existence, Uniqueness, Boundedness, Positivity, Dynamics of Solutions
Pattern Formation: Analysis of model examples - Diffusion-driven Instabilities, Steady States and their Stability Analysis

Important information:
The semester will be organized online. Lectures will be streamed live on Microsoft Teams during the regular lecture time. Please send an email to Chris Kowall which includes your mail address for registration at Microsoft Teams, preferably your uni-heidelberg.de account.
Every two weeks, there is discussion session on Wednesday 9.15 o'clock. It is also organized via Microsoft Teams, in the same channel as the lecture. The aim of the discussion session is to answer your questions arising during the lecture and to present further examples.
On Monday, 2.11.2020, an email containing more organizatorial information concerning the lecture will be send to all students enlisted in our Müsli course. So if you haven't registered yet, please do so until Sunday, 1.11.2020.

Lecture

Lecturer: Professor Dr. Anna Marciniak-Czochra

Head Assistant: Chris Kowall

Time: Tuesday 11:15-12:45 (online) und Thursday 11:15-12:45 (online)

Further informations can be found in the following fact sheet.

Literature:

H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations
A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations
K.-J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations
D. Henry: Geometric Theory of Semilinear Parabolic Equations
J. Smoller: Shock Waves and Reaction-Diffusion Equations