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Initial value problems and first boundary problems for a class of quasilinear wave equations. (English) Zbl 0822.35094
Summary: The initial value problems and the first boundary problems for the quasilinear wave equation \[ u_{tt}- [a_ 0+ na_ 1 (u_ x)^{n-1} ]u_{xx}- a_ 2 u_{xxtt} =0 \] are considered, where \(a_ 0, a_ 2>0\) are constants, \(a_ 1\) is an arbitrary real number, \(n\) is a natural number. The existence and uniqueness of classical solutions are proved by the Galerkin method.

35L70 Second-order nonlinear hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Galerkin method
Full Text: DOI
[1] Zhang Shanyuan, Zhuang Wei, Strain Solitary Waves in the Nonlinear Elastic Rods,Acta Mechanica Sinica,20 (1988), 58–66 (in Chinese).
[2] Zhou Yulin and Fu Hongyuan, Nonlinear Hyperbolic Systems of Higher Order of Generalized Sine-Gorden Type,Acta Mathematicae Sinica,26 (1983), 234–249 (in Chinese). · Zbl 0538.35055
[3] Gagliardo, E., Ulterlori Proprieta di alcune classl di funzioni inpiu variablli.Ric. Mat.,8 (1959), 24–51. · Zbl 0199.44701
[4] Nirenberg, L., On Eillptic Partial Differential Equations (Lecture II),Ann. Sc. Norm. Super. Pisa, S.3:13 (1952), 115–162.
[5] Yoslda, K., Functional Analysis, Springer-Verlag, Berlin, 1978
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