zbMATH — the first resource for mathematics

On the existence of solutions to the equation \(u_{tt}=u_{xxt}+\sigma (u_x)_x\). (English) Zbl 0415.35018

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
Full Text: DOI
[1] Adams, R.A, Sobolev spaces, (1975), 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, () · 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 Englewood Cliffs, N. J · Zbl 0144.34903
[8] Greenberg, J.M, On the existence, uniqueness, and stability of the equation ϱ0xtt = E(xx)xxx + λxxxt, 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(ux)uxx + λuxtx = θutt, J. math. anal. appl., 31, 406-417, (1970)
[10] Greenberg, J.M; McCamy, R.C; Mizel, V.J, On the existence, uniqueness and stability of solutions of the equation σ′(ux)uxx + λuxxt = ϱ0utt, 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 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, ()
[14] Tsutsumi, M, Some nonlinear evolution equations of second order, (), 950-955 · Zbl 0258.35017
[15] Yosida, K, Functional analysis, (1971), 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.