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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\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {L}:= -\\Delta -\\frac{x}{2}\\cdot \\nabla $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mi>\u0394<\/mml:mi>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mfrac>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                            <mml:mo>\u00b7<\/mml:mo>\n                            <mml:mi>\u2207<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    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\u2013Kantorovich 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\u00f6dinger equations and to a generalised viscous Burgers equation, and obtain both radial and non-radial self-similar profiles.\n                  <\/jats:p>","DOI":"10.1007\/s00211-025-01504-4","type":"journal-article","created":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T07:01:20Z","timestamp":1761289280000},"page":"2097-2143","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Constructive proofs for some semilinear PDEs on $$H^2(e^{|x|^2\/4},\\mathbb {R}^d)$$"],"prefix":"10.1007","volume":"157","author":[{"given":"Maxime","family":"Breden","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hugo","family":"Chu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,24]]},"reference":[{"key":"1504_CR1","unstructured":"Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables Applied Mathematics Series (1970)"},{"issue":"2","key":"1504_CR2","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1080\/03605309908820681","volume":"15","author":"J Aguirre","year":"1990","unstructured":"Aguirre, J., Escobedo, M., Zuazua, E.: Self-similar solutions of a convection diffusion equation and related semilinear elliptic problems. 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