Papers › Constructive proofs for some semilinear PDEs on H²(e^(|x|²/4),ℝᵈ)
Constructive proofs for some semilinear PDEs on H²(e^(|x|²/4),ℝᵈ)
Maxime Breden, Hugo Chu
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We develop computer-assisted tools to study semilinear equations of the form -Δu -x/2·∇u= f(x,u,∇u) , x∈ℝᵈ. 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 ℒ:= -Δ-x/2·∇ 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.
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