Constructive proofs for some semilinear PDEs on H2(e|x|^2/4,Rd)

We develop computer-assisted tools to study semilinear equations of the form (Formula presented.) Such equations appear naturally in several contexts, and in particular when looking for self-similar solutions of parabolic PDEs. We develop a general methodology, allowing us not only to prove the existence of solutions, but also to describe them very precisely. We introduce a spectral approach based on an eigenbasis of L:=-Δ-x2·∇ in spherical coordinates, together with a quadrature rule allowing to deal with nonlinearities, in order to get accurate approximate solutions. We then use a Newton–Kantorovich argument, in an appropriate weighted Sobolev space, to prove the existence of a nearby exact solution. We apply our approach to nonlinear heat equations, to nonlinear Schrödinger equations and to a generalised viscous Burgers equation, and obtain both radial and non-radial self-similar profiles.