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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Blow up and propagation speed of solutions to the DGH equation

Pages: 657 - 670, Volume 12, Issue 3, October 2009      doi:10.3934/dcdsb.2009.12.657

 
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Yong Zhou - Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China (email)
Zhengguang Guo - Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China (email)

Abstract: A wave-breaking mechanism for solutions with certain initial profiles and propagation speed are discussed in this paper. Firstly, we apply the best constant to give sufficient condition via an appropriate integral form of the initial data, which guarantees finite time singularity formation for the corresponding solution, then we establish blow up criteria via the conserved quantities. Finally, persistence properties of the strong solutions are presented and infinite propagation speed is also investigated in the sense that the corresponding solution $u(x,t)$ does not have compact spatial support for $t>0$ though $u_0 \in C_0^{\infty}(\mathbb{R})$.

Keywords:  Best constant, Integrable equation, Singularity, Propagation speed.
Mathematics Subject Classification:  30C70, 37L05, 35Q58, 58E35.

Received: December 2008;      Revised: May 2009;      Available Online: July 2009.