American Institute of Mathematical Sciences

Advanced
October  2009, 12(3): 657-670. doi: 10.3934/dcdsb.2009.12.657

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, China, China

Received  December 2008 Revised  May 2009 Published  July 2009

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, Propagation speed., Integrable equation, Singularity.
Mathematics Subject Classification: 30C70, 37L05, 35Q58, 58E3.
Citation: Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657
[1]

S. Bonafede, G. R. Cirmi, A.F. Tedeev. Finite speed of propagation for the porous media equation with lower order terms. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 305-314. doi: 10.3934/dcds.2000.6.305
[2]

David Henry. Infinite propagation speed for a two component Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 597-606. doi: 10.3934/dcdsb.2009.12.597
[3]

Ivan C. Christov. On a C-integrable equation for second sound propagation in heated dielectrics. Evolution Equations & Control Theory, 2019, 8 (1) : 57-72. doi: 10.3934/eect.2019004
[4]

Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
[5]

Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543
[6]

Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4031-4053. doi: 10.3934/dcdsb.2019049
[7]

Jianqing Chen. Best constant of 3D Anisotropic Sobolev inequality and its applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 655-666. doi: 10.3934/cpaa.2010.9.655
[8]

Julián Fernández Bonder, Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Communications on Pure & Applied Analysis, 2002, 1 (3) : 359-378. doi: 10.3934/cpaa.2002.1.359
[9]

Antoine Benoit. Finite speed of propagation for mixed problems in the $WR$ class. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2351-2358. doi: 10.3934/cpaa.2014.13.2351
[10]

Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112
[11]

Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867
[12]

Patrick W. Dondl, Michael Scheutzow. Positive speed of propagation in a semilinear parabolic interface model with unbounded random coefficients. Networks & Heterogeneous Media, 2012, 7 (1) : 137-150. doi: 10.3934/nhm.2012.7.137
[13]

Elena Trofimchuk, Manuel Pinto, Sergei Trofimchuk. On the minimal speed of front propagation in a model of the Belousov-Zhabotinsky reaction. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1769-1781. doi: 10.3934/dcdsb.2014.19.1769
[14]

Zhibo Cheng, Jingli Ren. Periodic and subharmonic solutions for duffing equation with a singularity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1557-1574. doi: 10.3934/dcds.2012.32.1557
[15]

Jerry Bona, H. Kalisch. Singularity formation in the generalized Benjamin-Ono equation. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 27-45. doi: 10.3934/dcds.2004.11.27
[16]

Sungrim Seirin Lee. Dependence of propagation speed on invader species: The effect of the predatory commensalism in two-prey, one-predator system with diffusion. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 797-825. doi: 10.3934/dcdsb.2009.12.797
[17]

I. Baldomá, Tere M. Seara. The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 323-347. doi: 10.3934/dcdsb.2008.10.323
[18]

Shota Sato, Eiji Yanagida. Forward self-similar solution with a moving singularity for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 313-331. doi: 10.3934/dcds.2010.26.313
[19]

Jerry L. Bona, Stéphane Vento, Fred B. Weissler. Singularity formation and blowup of complex-valued solutions of the modified KdV equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4811-4840. doi: 10.3934/dcds.2013.33.4811
[20]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (25)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences