Title of project:
Qualitative Theory of Partial Differential Equations

Code:
ND2021B

Supervisor: Prof. N. Dirr

Project description:

Certain partial differential equations of the form

∂tu(x,t)=∆u(x,t)+f(x,u(x,t))
can be transformed in an integral equation and solved by a Peano-Iteration in a similar way as it is done for ODEs. The difference is however, that we have to work in infinite dimensional vector spaces instead of Rn. Combining ideas from ODEs and Functional Analysis, a lot can be said about the qualitative behaviour of such equations (stability, long-time behaviour etc.) similar to the ODE case.
These equations, called reaction-diffusion equations, have applications in chemistry, biology and physics.
A project should always contain some proofs, but, depending on the interest of the student, could then focus on numerics or on analysis.
Background Reading:
D. Henry, Geometric Theory of Semilineat Parabolic Differential Equations Springer Lecture Notes in Mathematics 840
L.C. Evans, Partial Differential Equations, AMS Grad. Studies in Math. 19

Project offered as double module, single module, or both:
Double

Prerequisite 3rd year modules:
Measure Theory, Ordinary Differential