Self-similar Solutions of the Cubic Wave Equation

Abstract

We prove that the focusing cubic wave equation in three spatial dimensions has a countable family of self-similar solutions which are smooth inside the past light cone of the singularity. These solutions are labeled by an integer index $n$ which counts the number of oscillations of the solution. The linearized operator around the $n$-th solution is shown to have $n+1$ negative eigenvalues (one of which corresponds to the gauge mode) which implies that all $n>0$ solutions are unstable. It is also shown that all $n>0$ solutions have a singularity outside the past light cone which casts doubt on whether these solutions may participate in the Cauchy evolution, even for non-generic initial data.

14 pages, 1 figure

References in corpus (3)

• On blowup for semilinear wave equations with a focusing nonlinearity
• Self-similar solutions of semilinear wave equations with a focusing nonlinearity
• Universality of global dynamics for the cubic wave equation

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• All solutions of the n = 5 Lane-Emden equation
• Scattering for defocusing energy subcritical nonlinear wave equations
• On the profile of energy concentration at blow-up points for sub-conformal focusing nonlinear waves
• Co-dimension one stable blowup for the supercritical cubic wave equation