Blowup issues for a class of nonlinear dispersive wave equations
Blowup issues for a class of nonlinear dispersive wave equations
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DOI：
10.1016/j.jde.2014.03.008
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发表时间：
2014-03
影响因子：
2.4
通讯作者：
L. Brandolese;M. F. Cortez
中科院分区：
文献类型：
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作者：
L. Brandolese;M. F. Cortez
在本文中，我们考虑实线上的非线性色散波动方程 u t− u t x x+[f (u)] x−[f (u)] x x x+[g (u)+ f ″(u) 2 u x 2] x= 0，为了适当选择函数 f 和 g，包括众所周知的模型，例如用于研究弹性杆内部振动的 Dai 方程或模拟浅层水波传播的 Camassa-Holm 方程水。我们建立了局部空间爆炸准则（即，仅涉及单点邻域中数据u 0 的属性的准则），简化并扩展了该方程的早期爆炸准则。我们的论点适用于有限和无限能量情况，产生强解的有限时间爆炸，其行为可能不同，如 x→+∞ 和 x→−∞。
In this paper we consider the nonlinear dispersive wave equation on the real line, u t− u t x x+[f (u)] x−[f (u)] x x x+[g (u)+ f ″(u) 2 u x 2] x= 0, that for appropriate choices of the functions f and g includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa–Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (ie, a criterion involving only the properties of the data u 0 in a neighborhood of a single point) simplifying and extending earlier blowup criteria for this equation. Our arguments apply both to the finite and infinite energy cases, yielding the finite time blowup of strong solutions with possibly different behavior as x→+∞ and x→−∞.