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Zbl 1224.35263
Quintero, José Raúl
A remark on the Cauchy problem for the generalized Benney-Luke equation. (English)
[J] Differ. Integral Equ. 21, No. 9-10, 859-890 (2008). ISSN 0893-4983

Summary: In this article we address the well posedness of the Cauchy problem associated with the generalized Benney-Luke equation $$\Phi _{tt}-\Delta \Phi +a\Delta ^2\Phi -b\Delta \Phi _{tt}+\theta \left (\Phi _{t}\left [\partial _{x}[(\partial _{x}\Phi )^{p}]+\partial _{y}[(\partial _{y}\Phi )^{p}]\right ]+ 2\left [(\partial _{x}\Phi )^{p}\Phi _{xt}+(\partial _{y}\Phi )^{p}\Phi _{yt}\right ]\right )+\beta \nabla \cdot (\vert \nabla \Phi \vert ^{m}\nabla \Phi )=0$$ in $\Bbb {R}^{1+2}$ under a reasonable ``physical" initial condition, which is imposed from the formal derivation of the Benney-Luke water wave model.

MSC 2000:: *35L30 Higher order hyperbolic equations, initial value problems
35L76
35Q35 Other equations arising in fluid mechanics
76B15 Wave motions (fluid mechanics)

Keywords: Cauchy problem; generalized Benney-Luke equation; well posedness

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