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Wave breaking and propagation speed for a class of one-dimensional shallow water equations. (English) Zbl 1230.35101
Summary: We investigate a more general family of one-dimensional shallow water equations. Analogous to the Camassa-Holm equation, these new equations admit blow-up phenomena and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the shallow water equations is proved in the following sense: the corresponding solution \(u(t, x)\) with compactly supported initial data \(u_0(x)\) does not have compact \(x\)-support any longer in its lifespan.

MSC:
35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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