On solutions to the Holm-Staley \(b\)-family of equations. (English) Zbl 1189.37083

The paper is concerned with the family of one-dimensional shallow water equations (the Holm-Staley \(b\)-family of equations) \[ y_t+uy_x+bu_xy=0,(1) \] derived recently by D. D. Holm and M. F. Staley [Phys. Lett., A 308, No. 5–6, 437–444 (2003; Zbl 1010.35066)]. Here \(u\) denotes the velocity field, \(y(x,t)=(1-\partial_x^2)u(x,t)\) and \(b\not=0,2,3\). The author establishes the existence of local solutions of equation \((1)\) with the initial date \(u_0\in H^s(\mathbb{R})\) and \(s>3/2\), the non-existence of global solutions (for \(1<b\leq 3\), \(u_0\in H^2(\mathbb{R})\) is odd and \(u_{0x}(0)\leq 0\)), the existence of global solutions (if \(u_0\in H^3(\mathbb{R})\) and \(y_0=(1-\partial_x^2)u_0\) is one sign) and the blow-up in finite time of solutions for (1) under some assumptions on \(u_0\). A detailed description of the corresponding strong solution in its lifespan with \(u_0\) being compactly supported is also studied. Namely, he shows that for any fixed time \(t>0\), the corresponding solution \(u(x,t)\) behaves as \(u(x,t)=L(t)e^{-x}\) for \(x\gg 1\) and \(u(x,t)=l(t)e^x\) for \(x\ll -1\) with a strictly increasing function \(L(t)>0\) and a strictly decreasing function \(l(t)<0\), respectively.


37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35L05 Wave equation
26A12 Rate of growth of functions, orders of infinity, slowly varying functions


Zbl 1010.35066
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