The maximum principle and sign changing solutions of the hyperbolic equation with the Higgs potential

TL;DR

This paper investigates the maximum principle for linear hyperbolic equations and explores sign-changing solutions in semilinear equations with Higgs potential, supported by numerical simulations of bubble formation in de Sitter spacetime.

Contribution

It introduces new results on the maximum principle and sign-changing solutions for hyperbolic equations with Higgs potential, including numerical evidence of bubble creation.

Findings

Numerical simulations show bubble formation in de Sitter spacetime.

Sign-changing solutions exist for the semilinear Klein-Gordon equation.

The maximum principle applies to certain linear hyperbolic equations.

Abstract

In this article we discuss the maximum principle for the linear equation and the sign changing solutions of the semilinear equation with the Higgs potential. Numerical simulations indicate that the bubbles for the semilinear Klein-Gordon equation in the de Sitter spacetime are created and apparently exist for all times.