  We formulate a class of nonlinear {evolution} partial differential equations
(PDEs) as linear optimization problems on moments of positive measures
supported on infinite-dimensional vector spaces. Using sums of squares (SOS)
representations of polynomials in these spaces, we can prove convergence of a
hierarchy of finite-dimensional semidefinite relaxations solving approximately
these infinite-dimensional optimization problems. As an illustration, we report
on numerical experiments for solving the heat equation subject to a nonlinear
perturbation.
