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References
Agmon, S., Unicité et convexité dans les problèmes différentiels, Sem. Math. Sup (1965), University of Montreal Press (1966).
Agmon, S., and Nirenberg, L., Lower Bounds and uniqueness theorems of differential equations in a Hilbert space, Comm. Pure Appl. Math., 20 (1967), pp. 207–229.
Antonov, V.A., Remarks on the problem of stability in Stellar dynamics, Sov. Astr., 4 (1961), pp. 859–867 ≠Ast. Zh., 37 (1960), pp. 918–926.
Brun, L., Sur l'unicité en thermoélasticité dynamique et diverses expressions anologues à la formule de Clapeyron, C.R. Acad. Sci. Paris, 261 (1965), pp. 2584–2587.
Brun, L., Méthodes energétiques dans les systèmes èvolutifs linéaires, Premier partie: Séparation des énergies; Deuxième partie: Théoremès d'unicité, J. de Mechanique, 8 (1969), pp. 125–166, 167–192.
Dafermos, C.M., Wave equations with weak damping, SIAM J. Appl. Math., 18 (1970), pp. 759–767.
Edelstein, W.S., A uniqueness theorem in the linear theory of elasticity with microstructure, Acta. Mech., 8 (1969), pp. 183–184.
Green, A.E., (to appear in Mathematika).
Green, A.E., Knops, R.J., and Laws, N., Large deformations, superposed small deformations and stability of elastic rods, Int. J. Sols. Structs., 4 (1968), pp. 555–577.
Hills, R.N., On the stability of linear dipolar fluids, (to appear).
Hills, R.N., On uniqueness and continuous dependence for a linear micropolar fluid, (to appear).
John, F., A note on "improper" problems in partial differential equations, Comm. Pure Appl. Math., 8 (1955), pp. 494–495.
John, F., Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), pp. 551–585.
Khosrovshahi, G.B., (to appear).
Knops, R.J., and Payne, L.E., Uniqueness in classical elastodynamics, Arch. Rat. Mech. Anal., 27 (1968), pp. 349–355.
Knops, R.J., and Payne, L.E., Stability in linear elasticity, Int. J. Solids Structures, 4 (1968), pp. 1233–1242.
Knops, R.J., and Payne, L.E., On the stability of solutions of the Navier-Stokes equations backward in time, Arch. Rat. Mech. Anal., 29 (1968), pp. 331–335.
Knops, R.J., and Payne, L.E., Continuous data dependence for the equations of classical elastodynamics, Proc. Camb. Phil. Soc., 66 (1969), pp. 481–491.
Knops, R.J., and Payne, L.E., Hölder stability and logarithmic convexity, IUTAM Symposium on Instability of Continuous Systems, Herrenalb (1969).
Knops, R.J., and Payne, L.E., On uniqueness and continuous data dependence in dynamical problems of linear thermoelasticity, Int. J. Structures Solid, 6 (1970), pp. 1173–1184.
Knops, R.J., and Payne, L.E., Growth estimates for solutions of evolutionary equations in Hilbert space with applications in elastodynamics, Arch. Rat. Mech. Anal., 41 (1971), pp. 363–398.
Knops, R.J., and Steel, T.R., Uniqueness in the linear theory of a mixture of two elastic solids, Int. J. Eng. Sci., 7 (1969), pp. 571–577.
Knops, R.J., and Steel, T.R., On the stability of a mixture of two elastic solids, J. Comp. Mats., 3 (1969), pp. 652–663.
Laval, G., Mercier, C., and Pellat, R., Necessity of the energy principles for magnetostatic stability, Nuclear Fusion, 5 (1965), pp. 156–158.
Levine, H.A., Logarithmic convexity and the Cauchy problem for abstract second order differential inequalities, J. Diff. Eqns., 8 (1969), pp. 34–55.
Levine, H.A., On a theorem of Knops and Payne in dynamical linear thermoelasticity, Arch. Rat. Mech. Anal., 38 (1970), pp. 290–307.
Levine, H.A., Instability and nonexistence of global solutions to non-linear wave equations of the form Putt=− Au+F (u). Lecture Notes, University of Dundee (1972).
Levine, H.A., Logarithmic convexity and the Cauchy problem for P(t)utt+M(t)ut+N(t)u=0 in Hilbert space, (to appear).
Levine, H.A., and Payne, L.E., Instability and nonexistence of global solutions to non-linear evolutionary equations of the form Putt=A(F (Au)) and some examples, (to appear).
Levine, H.A., and Payne, L.E., Instability, growth and nonexistence of global solutions to non-linear evolutionary equations of the form Put=A(F (Au)) and some examples, (to appear).
Levine, H.A., Paper in this volume.
Levine, H.A., Knops, R.J., and Payne, L.E., (to appear).
Murray, A., and Protter, M., Asymptotic behaviour of solutions of second order systems of partial differential equations, (to appear).
Murray, A., Uniqueness and continuous dependence for the equations of elastodynamics without strain energy function, (to appear).
Naghdi, P.M., and Trapp, J.A., A uniqueness theorem in the theory of Cosserat surface, J. Elast., 2 (1972), pp. 9–20.
Ogawa, H., Lower bounds for solutions of hyperbolic inequalities, Proc. Amer. Math. Soc., 16 (1965), pp. 853–857.
Ogawa, H., Lower bounds for solutions of differential inequalities in Hilbert space, Proc. Amer. Math. Soc., 16 (1965), pp. 853–857.
Ogawa, H., Lower bounds for the solutions of parabolic differential inequalities, Can. J. Math., 19 (1967), pp. 667–672.
Payne, L.E., On some non-well-posed problems for partial differential equations, Numerical Solutions of Non-Linear Differential Equations, Wiley (1964), pp. 239–263.
Pucci, C., Sui problemi di Cauchy non "ben posti", Rend. Acad. Naz. Lincei, 18 (1955), pp. 473–477.
Steel, T.R., Applications of a theory of interacting continua, Quart. J. Mech. Appl. Math., 20 (1967), pp. 57–72.
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Knops, R.J. (1973). Logarithmic convexity and other techniques applied to problems in continuum mechanics. In: Knops, R.J. (eds) Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Lecture Notes in Mathematics, vol 316. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069622
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