Abstract
We consider a nonlinear version of the Wheeler–DeWitt equation which was introduced by Cooper, Susskind, and Thorlacius in the context of two-dimensional quantum cosmology. We establish the existence of global solutions to the Cauchy problem and Goursat problems which, both, arise naturally in physics. Our method of proof is based on a nonlinear transformation of the Wheeler–DeWitt equation and on techniques introduced by Baez and collaborators and by Tsutsumi for nonlinear wave equations.
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Baez, J.C., Zhou, Z.: The global Goursat problem on \({\mathbb{R}}\times S^{1}\). J. Funct. Anal. 83, 364–382 (1989)
Baez, J.C., Segal, I.E., Zhou, Z.: The global Goursat problem and scattering for nonlinear wave equations. J. Funct. Anal. 93, 239–269 (1990)
Cooper, A., Susskind, L., Thorlacius, L.: Two-dimensional quantum cosmology. Nucl. Phys. B 263, 132–162 (1991)
Dias, J.P., Figueira, M.: The simplified Wheeler–DeWitt equation: the Cauchy problem and some spectral properties. Ann. Inst. Henri Poincaré: Phys. Théorique 54, 17–26 (1991)
Dias, J.P., Figueira, M.: The Cauchy problem for a nonlinear Wheeler–DeWitt equation. Ann. Inst. Henri Poincaré: Anal. Non-Linear 10, 99–107 (1993)
Dias, J.P., Figueira, M.: On a class of solutions for the simplified Wheeler–DeWitt equation with a massless single scalar field. Ricerche di Mat. 44, 145–155 (1995)
Dias, J.P., Figueira, M.: On a singular limit for a class of solutions of the simplified Wheeler–DeWitt equation with a massless single scalar field. Rendiconti di Mat. Serie VII 13, 529–542 (1993)
Gibbons, G.W., Grischchuk, L.P.: What is a typical wave function for the universe? Nucl. Phys. B 313, 736–748 (1989)
Hartle, J.B., Hawking, S.W.: Wave function of the Universe. Phys. Rev. D 28, 2960–2975 (1983)
Hawking, S.W.: The quantum state of the universe. Nucl. Phys. B 239, 257–276 (1984)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non-Linéaires. Dunod, Paris (1969)
Nguyen, L.H., Parwani, R.R.: Nonlinear quantum cosmology. Gen. Relativ. Gravit. 41, 2543–2560 (2009)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness. Academic Press, London (1975)
Tsutsumi, M.: Nonrelativistic approximation of nonlinear Klein–Gordon equations in two space dimensions. Nonlinear Ann. T.M.A. 8, 637–643 (1984)
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Dias, JP., LeFloch, P.G. A new approach to the Cauchy and Goursat problems for the nonlinear Wheeler–DeWitt equation. Nonlinear Differ. Equ. Appl. 25, 10 (2018). https://doi.org/10.1007/s00030-018-0503-0
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DOI: https://doi.org/10.1007/s00030-018-0503-0