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Nonlinear initial value problems with a singular perturbation of hyperbolic type

Published online by Cambridge University Press:  14 November 2011

R. Geel
R. Geel
Affiliation:
Ubbo Emmius Institute, Department of Mathematics, Postbus 2056, 9704 CB Groningen, The Netherlands
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Synopsis

This paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation consisting of a nonlinear first order part and a linear second order part. The second order part of the differential operator contains a small factor ε and can therefore be considered as a perturbation of the nonlinear first order part of the operator.

The existence of a solution u together with pointwise a priori estimates for this solution are established by applying a fixed point theorem for nonlinear operators in a Banach space.

It is shown that the difference between the solution u and the solution w of the unperturbed nonlinear initial value problem (which follows from the original problem by putting ε = 0) is of order ε, uniformly in compact subsets of ℝ2 where w is sufficiently smooth.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 3-4 , 1981 , pp. 333 - 345
Copyright
Copyright © Royal Society of Edinburgh 1981

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References

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