Local and Global Existence of Solutions to Scalar Equations on Spatially Flat Universe as a Background with Non-minimal Coupling

TL;DR

This paper establishes the local and global existence of solutions for scalar wave equations on a spatially flat universe background with nonminimal coupling, using energy methods and specific assumptions.

Contribution

It introduces a framework for proving well-posedness of scalar equations with nonminimal coupling on flat universe backgrounds, including global existence under new assumptions.

Findings

Bounded energy norms for finite time solutions

Global solutions exist with decay estimates under specific assumptions

Physical models support the theoretical setup

Abstract

We prove the wellposedness of scalar wave equations on spatially flat universe as a background with nonminimal coupling with the scalar potential turned on by introducing the $k$-order linear energy and the corresponding energy norm. In the local case, we show that both the $k$-order linear energy and the energy norm are bounded for finite time with initial data in $H^{k+1}\times H^{k}$. Whereas in the global case, we have to add three assumptions related to the nonminimal coupling constant, the scale factor of spacetimes, and the form of the scalar that has to be a polynomial with a small positive parameter. Then, we show that the solution does globally exist with a particular decay estimate that depends on the scale factor of the spacetimes. Finally, we provide some physical models that support our general setup.