The existence of smooth solutions in q-theories

TL;DR

This paper investigates the Cauchy problem in q-theories, demonstrating the existence of smooth solutions under certain initial conditions and extending the understanding of their mathematical well-posedness beyond known limits.

Contribution

It proves the well-posedness of the Cauchy problem in q-theories without relying on the F(R) limit, showing solutions exist for finite times under specific initial conditions.

Findings

The Cauchy problem is well posed in q-theories.

Smooth solutions exist for finite time with specific initial conditions.

Mathematical theorems about nonlinear systems support these results.

Abstract

The q-models are scenarios that may explain the smallness of the cosmological constant [1]-[7]. The vacuum in these theories is presented as a self-sustainable medium and include a new degree of freedom, the q-variable, which stablish the equilibrium of the quantum vacuum. In the present work, the Cauchy formulation for these models is studied. It has been already noted that there exist some limits where these theories are described by an F(R) model, which posses a well formulated Cauchy problem. This paper shows that the Cauchy problem is well posed even not reaching this limit. By use of some mathematical theorems about second order non linear systems, it is shown that these scenarios admit a smooth solution for at least a finite time when some specific type of initial conditions are imposed. Some technical conditions of [11] play an important role in this discussion.