  We develop computer-assisted tools to study semilinear equations of the form
\begin{equation*} -\Delta u -\frac{x}{2}\cdot \nabla{u}= f(x,u,\nabla u) ,\quad
x\in\mathbb{R}^d. \end{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 $\mathcal{L}:= -\Delta
-\frac{x}{2}\cdot \nabla$ 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 Schr\"odinger
equations and to a generalised viscous Burgers equation, and obtain both radial
and non-radial self-similar profiles.
