Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\mathbb{R}^d)$
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917208183013376 |
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| author | Breden, Maxime Chu, Hugo |
| author_facet | Breden, Maxime Chu, Hugo |
| contents | We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -Δu -\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 $\mathcal{L}:= -Δ-\frac{x}{2}\cdot \nabla$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\mathbb{R}^d)$ Breden, Maxime Chu, Hugo Analysis of PDEs Numerical Analysis 35C06, 35J61, 65G20, 65N15 We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -Δu -\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 $\mathcal{L}:= -Δ-\frac{x}{2}\cdot \nabla$ 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. |
| title | Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\mathbb{R}^d)$ |
| topic | Analysis of PDEs Numerical Analysis 35C06, 35J61, 65G20, 65N15 |
| url | https://arxiv.org/abs/2404.04054 |