We develop computer-assisted tools to study semilinear equations of the form
−Δu − x2· ∇u = f (x, u, ∇u), x ∈ Rd .
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
Schrödinger equations and to a generalised viscous Burgers equation, and obtain both
radial and non-radial self-similar profiles.