Abstract
This paper deals with existence of solutions u(x 1, x 2, x 3) = u(x, t) of a nonlinear wave equation of the form
for large times t. Here u′ stands for the gradient vector
We assume that the a ik (U) are in C ∞ in a closed ball |U| in ≦ δ in ℝ 4, and that
, so that (1a) goes over into the classical linear wave equation
for “infinitesimal” u. The solution u of (1a) is to be found from initial conditions for t = 0.
This is a continuation of the author’s paper Delayed Singularity Formation of Solutions of Nonlinear Wave Equations in Higher Dimensions, Comm. Pure Appl. 29, 1976, pp. 649–682, referred to as (*) in the sequel.
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Bibliography
Klainerman, S., Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33, 1980, pp. 43–101.
Shatah, J., Global existence of small solutions to nonlinear evolution equations, preprint.
Klainerman, S., and Ponce, G., Global, small amplitude solutions to nonlinear evolution equations, preprint.
John, F., Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math. 34, 1981, pp. 29–51.
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John, F. (1983). Lower Bounds for the Life Span of Solutions of Nonlinear Wave Equations in Three Dimensions. In: Moser, J. (eds) Fritz John. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-5406-5_40
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DOI: https://doi.org/10.1007/978-1-4612-5406-5_40
Publisher Name: Birkhäuser, Boston, MA
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