Andrews, Graham On the existence of solutions to the equation \(u_{tt}=u_{xxt}+\sigma (u_x)_x\). (English) Zbl 0415.35018 J. Differ. Equations 35, 200-231 (1980). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 44 Documents MSC: 35G20 Nonlinear higher-order PDEs 35A07 Local existence and uniqueness theorems (PDE) (MSC2000) 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B45 A priori estimates in context of PDEs 35B50 Maximum principles in context of PDEs Keywords:existence of solutions; nonlinear third-order equation Citations:Zbl 0397.35011 PDFBibTeX XMLCite \textit{G. Andrews}, J. Differ. Equations 35, 200--231 (1980; Zbl 0415.35018) Full Text: DOI References: [1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101 [2] Antman, S. S., Ordinary differential equations of nonlinear elasticity. I. Foundations of the theories of nonlinearly elastic rods and shells, Arch. Rational Mech. Anal., 61, 307-351 (1976) · Zbl 0354.73046 [3] Chu, S. C.; Diaz, J. B., On “in the large” applications of the contraction principle, (Hale, J. K.; LaSalle, J. P., Differential Equations and Dynamical Systems (1967), Academic Press: Academic Press New York) · Zbl 0231.34006 [4] Chueh, K. N.; Conley, C. C.; Smoller, J. A., Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., 26, 373-392 (1977) · Zbl 0368.35040 [5] Clements, J., Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26, 745-752 (1974) · Zbl 0252.35044 [6] Dafermos, C. M., The mixed initial boundary value problem for the equations of nonlinear one dimensional viscoelasticity, J. Differential Equations, 6, 71-86 (1969) · Zbl 0218.73054 [7] Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall: Prentice-Hall Englewood Cliffs, N. J · Zbl 0144.34903 [8] Greenberg, J. M., On the existence, uniqueness, and stability of the equation \(ϱ_0X_{ tt } = E(X_x)X_{ xx } + λX_{ xxt } \), J. Math. Anal. Appl., 25, 575-591 (1969) · Zbl 0192.44803 [9] Greenberg, J. M.; MacCamy, R. C., On the exponential stability of solutions of \(E(u_x)u_{ xx } + λu_{ xtx } = θu_{ tt } \), J. Math. Anal. Appl., 31, 406-417 (1970) · Zbl 0219.35010 [10] Greenberg, J. M.; McCamy, R. C.; Mizel, V. J., On the existence, uniqueness and stability of solutions of the equation \(σ\)′\((u_x)u_{ xx } + λu_{ xxt } = ϱ_0u_{ tt } \), J. Math. Mech., 17, 707-728 (1968) · Zbl 0157.41003 [11] Krasnosel’skii, M. A., Topological Method in the Theory of Nonlinear Integral Equations (1964), Pergamon: Pergamon Elmsford, N. Y · Zbl 0111.30303 [12] MacCamy, R. C.; Mizel, V. J., Existence and nonexistence in the large of solutions of quasilinear wave equations, Arch. Rational Mech. Anal., 25, 299-320 (1967) · Zbl 0146.33801 [13] Reed, M., Abstract Nonlinear Wave Equations, (Lecture Notes in Mathematics No. 507 (1976), Springer-Verlag: Springer-Verlag Berlin) [14] Tsutsumi, M., Some nonlinear evolution equations of second order, (Proc. Japan Acad., 47 (1971)), 950-955 · Zbl 0258.35017 [15] Yosida, K., Functional Analysis (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0217.16001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.