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Nonlinear waves in friedman‐robertson‐walker space‐times
Author(s) -
Kovalyov Mikhail
Publication year - 1987
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.3160400504
Subject(s) - mathematics , initial value problem , space (punctuation) , mathematical analysis , wave equation , nonlinear system , hyperbolic partial differential equation , small data , traveling wave , partial differential equation , physics , quantum mechanics , computer science , data mining , operating system
In this paper we consider the initial value problem for the nonlinear wave equation □ u = F ( u , u ′) in Friedman‐Robertson‐Walker space‐time, □ being the D'Alambertian in local coordinates of space‐time. We obtain decay estimates and show that the equation has global solutions for small initial data. We do it by reducing the problem to an initial value problem for the wave equation over hyperbolic space. As byproduct we derive decay and global existence for solutions of the wave equation over the hyperbolic space with small initial data. The same technique with some auxiliary lemmas similar to the ones proved in [6], [7] can be used to generalize the result to the case when F depends also on second derivatives of u in a certain way.