A brief introduction to stability theory for linear PDEs

These are notes related to a 4-lecture minicourse given during June 10-11, 2012, at a workshop just preceeding the SIAM conference on Nonlinear Waves and Coherent Structures in Seattle, WA, USA. The title of the workshop was “The stability of coherent structures and patterns, ” and these four lectures concern stability theory for linear PDEs. The two other parts of the workshop are “Using AUTO for stability problems, ” given by Björn Sandstede and David Lloyd, and “Nonlinear and orbital stability,” given by Walter Strauss. We will focus on one particular method for obtaining linear stability: proving decay of the associated semigroup. Our strategy will be to state the most relevant theorems from semigroup and spectral theory, indicate where in the literature the proofs can be found, and look at some related examples. We will then see how to apply these theorems to reaction-diffusion equations. In particular, we will show that for reaction-diffusion equations, linear stability can be determined simply by computing the spectrum of the associated linearized operator.