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* - u -x2 u= f (x, u, u), xᵈ. 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: = - -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 Schr\"odinger equations and to a generalised viscous Burgers equation, and obtain both radial and non-radial self-similar profiles.