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19.—Concavity and the Evolutionary Properties of a Class of General Materials*

Published online by Cambridge University Press:  14 February 2012

R. N. Hills  and
R. J. Knops
R. N. Hills
Affiliation:
Heriot-Watt University, Edinburgh.
R. J. Knops
Affiliation:
Heriot-Watt University, Edinburgh.
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Extract

Concavity arguments have been used by Knops, Levine and Payne [1] to discuss evolutionary properties of weak solutions to an abstract non-linear differential equation in a Hilbert space. These authors demonstrate that provided the non-linearity is suitably restricted and for specified initial data, the norm of the solution becomes unbounded in a finite time. In other words, the solution possesses a finite escape time.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 72 , Issue 3 , 1975 , pp. 239 - 243
Copyright
Copyright © Royal Society of Edinburgh 1975

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References

References to Literature

[1]Knops, R. J., Levine, H. A. and Payne, L. E. 1974. Non-existence, instability and growth theorems for solutions of a class of abstract non-linear equations with applications to nonlinear elastodynamics. Archs Ration. Mech. Analysis, 55, 52.CrossRefGoogle Scholar
[2]Gurtin, M. E., 1973 Thermodynamics and the energy criterion for stability. Archs Ration. Mech. Analysis, 52, 93.CrossRefGoogle Scholar
[3]Hills, R. N. and Knops, R. J. Qualitative results for some general classes of material behaviour To appear.Google Scholar
[4]Levine, H. A., Uniqueness and growth of weak solutions to certain linear differential equations in Hilbert space. J. Diff. Eqns. (In press).Google Scholar
[5]Knops, R. J. and Wilkes, E. W., 1973.Theory of elastic stability, Handbuch der Physik, VIa/3 (Ed. C., Truesdell). Berlin:Springer-Verlag.Google Scholar
[6]Green, A. E. and Laws, N., 1972. On a global entropy production inequality. Quart. Jl Mech. Appl. Math., 15, 1.CrossRefGoogle Scholar
[7]Green, A. E. and Laws, N., 1972. On the entropy production inequality. Archs Ration. Mech. Analysis, 45, 47.CrossRefGoogle Scholar