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


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

Citations:

Zbl 0397.35011
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References:

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[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)
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