Constructive proofs for some semilinear PDEs on $$H^2(e^{|x|^2/4},\mathbb {R}^d)$$

Abstract We develop computer-assisted tools to study semilinear equations of the form $$\begin{aligned} -\Delta u -\frac{x}{2}\cdot \nabla {u}= f(x,u,\nabla u),\quad x\in \mathbb {R}^d. \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>·</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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 $$\mathcal {L}:= -\Delta -\frac{x}{2}\cdot \nabla $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>·</mml:mo> <mml:mi>∇</mml:mi> </mml:mrow> </mml:math> 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.