This is a preview of subscription content, log in via an institution to check access.
References
Antman, S. S., The Theory of Rods, Handbuch der Physik, Vol. VI1/2. Berlin, Heidelberg, New York: Springer 1972.
Bland, D. R., Dilatational waves and shocks in large displacement isentropic dynamic elasticity. J. Mech. Phys. Solids 12, 245–267 (1964).
Caughey, T. K., & R. T. Shield, Instability and the energy criteria for continuous systems. Z. Angew. Math. Phys. 19, 485–492 (1968).
Duhem, P., Recherches sur l'Élasticité. Paris: Gauthier-Villars 1906.
John, F., Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Comm. Pure Appl. Math. 13, 551–585 (1960).
Knops, R. J., & L. E. Payne, Growth estimates for solutions of evolutionary equations in Hilbert space with applications in elastodynamics. Arch. Rational Mech. Anal. 41, 363–398 (1971).
Koiter, W. T., Thermodynamics of Elastic Stability, Third Can. Congr. Appl. MECHS. 29–37 (1971).
Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt =−Au+F(u). Trans. Am. Math. Soc. 192, 1–21 (1974).
Levine, H. A., Some nonexistence and instability theorems for formally parabolic equations of the form Pu t =−Au+F(u). Arch. Rational Mech. Anal. 51, 371–386 (1973).
Levine, H. A., Some uniqueness and growth theorems in the Cauchy problem for Pu tt +Mu t +Nu=0 in Hilbert space. Math. Z. 126, 345–360 (1972).
Levine, H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal. 5, 138–146 (1974).
Levine, H. A., A note on a nonexistence theorem for some nonlinear wave equations. SIAM J. Math. Anal, (in press).
Levine, H. A., & L. E. Payne, Nonexistence theorems for the heat equation with nonlinear boundary conditions and for the porous medium equation backward in time. J. Diff. Eq. (in press).
Movchan, A. A., On the stability of processes relating to the deformation of solid bodies. Arch. Mech. stosow. 15, 659–682 (1963).
Straughan, B., Further nonexistence theorem for abstract nonlinear wave equations. Proc. Am. Math. Soc. (in press).
Author information
Authors and Affiliations
Additional information
Communicated by J. L. Ericksen
Rights and permissions
About this article
Cite this article
Knops, R.J., Levine, H.A. & Payne, L.E. Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics. Arch. Rational Mech. Anal. 55, 52–72 (1974). https://doi.org/10.1007/BF00282433
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00282433