Travelling Waves in Parabolic PDEs

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Description

Travelling waves are a type of similarity solutions of evolutionary differential equations (partial differential equations (PDEs), delay
differential equations, and integro-differential equations) that are of importance in describing many natural phenomena: combustion waves,
spreading of epidemics, shocks, etc. Study of travelling waves involves proving their existence, their stability in an appropriately defined
sense, and, crucially, estimating the speed of the (physically observed) waves.

In this course we will mainly concentrate of the last problem, in the restricted context of fronts in monostable scalar
reaction-diffusion-advection PDEs in one space dimension; such problems were first studied by Kolmogorov and coauthors in the 1930s.

We will derive variational principles for the minimal speed of the front and see the surprising relationship between the existence of explicit
travelling wave solutions and their physical relevance.

We will also briefly deal with a topological method for proving existence of travelling waves and with stability issues as well with a
recently discovered amazing connection between parabolic PDES and voting models.