# The existence of smooth solutions in q-models

## Abstract

The q-models are scenarios that may explain the smallness of the cosmological constant (Klinkhamer and Volovik in Phys Rev D 77:085015, 2008; Phys Rev D 78:063528, 2008; JETP Lett 88:289, 2008; Mod Phys Lett A 31(28):1650160, 2016; JETP Lett 91:259, 2010; Phys Rev D 79:063527, 2009; J Phys Conf Ser 314:012004, 2011). The vacuum in these theories is presented as a self-sustainable medium and include a new degree of freedom, the q-variable, which establishes the equilibrium of the quantum vacuum. In the present work, the Cauchy formulation for these models is studied in detail. It is known that there exist some limits in which these theories are described by an *F*(*R*) gravity model, and these models posses a well posed 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 Ringstrom (The Cauchy problem in general relativity, European Mathematical Society, Warsaw, 2000) play an important role in this discussion.

## Keywords

Cauchy problem Cosmological constant Alternative gravity theories Global hyperbolic spaces## Notes

### Acknowledgements

Both authors are supported by CONICET, Argentina. O.P.S is supported by the Beca Externa Jovenes Investigadores of CONICET. O.P.S warmly acknowledge the Steklov Mathematical Institute of the Russian Academy of Sciences in Moscow, were part of this work has been done, for their hospitality.

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