We develop computer-assisted tools to study semilinear equations of the form \(\begin{aligned} -\Delta u -\frac{x}{2}\cdot \nabla {u}= f(x,u,\nabla u),\quad x\in \mathbb {R}^d. \end{aligned}\) 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 \(\mathcal {L}:= -\Delta -\frac{x}{2}\cdot \nabla \) 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.
