Constructive proofs for some semilinear PDEs on $H^2(e^|x|^2/4,\mathbbR^d)$
<|reference_start|>Constructive proofs for some semilinear PDEs on $H^2(e^|x|^2/4,\mathbbR^d)$: 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.<|reference_end|>
@article{breden2024constructive,
    title={Constructive proofs for some semilinear PDEs on
  $H^2(e^{|x|^2/4},\mathbb{R}^d)$},
    author={Maxime Breden and Hugo Chu},
    journal={arXiv preprint arXiv:2404.04054},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.04054},
    primaryClass={math.AP cs.NA math.NA}
}