{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/constructive-proofs-for-some-semilinear-pdes","title":"Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\\mathbb{R}^d)$","arxiv_id":"2404.04054","date":"2024-04-05","proceeding":null,"authors":["Maxime Breden","Hugo Chu"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2404.04054v1","url_pdf":"https://arxiv.org/pdf/2404.04054v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"constructive-proofs-for-some-semilinear-pdes","repo_url":"https://github.com/huggzz/hermite-laguerre_proofs","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}