Abstract
The first basic question is to identify the possible configurations of the systems (3.1), namely the set X of initial data for which the time evolution is well defined and which is mapped onto itself by time evolution. In the mathematical language, one has to find the functional space X for which the Cauchy problem is well posed. In order to see this, one has to give conditions on U′(ϕ) and to specify the class of initial data or, equivalently, the class of solutions one is interested in. Here one faces an apparently technical mathematical problem, which has also deep physical connections.
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References
K. Jőrgens, Mat. Zeit. 77, 291 (1961).
I. Segal, Ann. Math. 78, 339 (1963).
See e.g. V. Arnold, Ordinary Differential Equations, Springer 1992, Chap. 4; G. Sansone and R. Conti, Non-linear Differential Equations, Pergamon Press 1964.
To our knowledge the proof of global existence of solutions of (4.6) for initial data in \( H_{loc}^2 \oplus L_{loc}^2 \) first appeared in Ref. I, although the validity of such a result was conjectured by W. Strauss, Anais Acad. Brasil. Ciencias 42, 645 (1970), p. 649, Remark: “The support restrictions on u o(x), u 1(x), F(x, t, 0) could probably be removed by exploiting the hyperbolic character of the differential equation …”.
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Strocchi, F. (2008). General Properties of Solutions of Classical Field Equations. In: Symmetry Breaking. Lecture Notes in Physics, vol 732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73593-9_4
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