Blow up and propagation speed of solutions to the DGH equation

Yong Zhou; Zhengguang Guo

doi:10.3934/dcdsb.2009.12.657

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2009, Volume 12, Issue 3: 657-670. Doi: 10.3934/dcdsb.2009.12.657

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Blow up and propagation speed of solutions to the DGH equation

Yong Zhou^1, and
Zhengguang Guo^1,

1.
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004

Received: December 2008

Revised: May 2009

Published: May 2009

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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 $u_0 \in C_0^{\infty}(\mathbb{R})$.

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Mathematics Subject Classification: 30C70, 37L05, 35Q58, 58E35.

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