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 {equation*} -Δ u -x/2· ∇{u}= f(x,u,∇ u) , x^d. {equation*} 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 -x/2· ∇ 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 Schr\"odinger equations and to a generalised viscous Burgers equation, and obtain both radial and non-radial self-similar profiles.