Computation of asymptotic stability for a class of partial differential equations with delay

In this paper the question of asymptotic stability for retarded functional reaction diffusion equations is faced. Due to the infinite dimension of the problem a numerical approach is necessary. Here we propose a technique based on a pseudospectral discretization in time and on a spectral discretization in space of the infinitesimal generator associated to the semigroup of solution operators. Some numerical experiments on a Hutchinson-type equation modeling heat conduction in a rod with spatially variable gain delayed feedback are performed to show the efficiency of the scheme. This work represents a nontrivial extension of previous work of the authors on the computation of asymptotic stability for delay differential equations.