scholarly journals On the Existence of Global Weak Solutions for a Weakly Dissipative Hyperelastic Rod Wave Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Haibo Yan ◽  
Ls Yong ◽  
Yu Yang ◽  
Yang Wang
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Global Weak Solutions ◽  
Initial Value

Assuming that the initial valuev0(x)belongs to the spaceH1(R), we prove the existence of global weak solutions for a weakly dissipative hyperelastic rod wave equation in the spaceC([0,∞)×R)⋂‍L∞([0,∞);H1(R)). The limit of the viscous approximation for the equation is used to establish the existence.

scholarly journals TheH1(R)Space Global Weak Solutions to the Weakly Dissipative Camassa-Holm Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Zhaowei Sheng ◽  
Shaoyong Lai ◽  
Yuan Ma ◽  
Xuanjun Luo
Keyword(s):  
Cauchy Problem ◽  
Weak Solutions ◽  
Space Variable ◽  
Global Weak Solutions ◽  
Initial Value ◽  
Dissipative Term ◽  
First Order ◽  
Holm Equation ◽  
The Cauchy Problem ◽  
Derivatives Of

The existence of global weak solutions to the Cauchy problem for a generalized Camassa-Holm equation with a dissipative term is investigated in the spaceC([0,∞)×R)∩L∞([0,∞);H1(R))provided that its initial valueu0(x)belongs to the spaceH1(R). A one-sided super bound estimate and a space-time higher-norm estimate on the first-order derivatives of the solution with respect to the space variable are derived.

The Global Existence and Semiclassical Limit of Weak Solutions to Multidimensional Quantum Drift-diffusion Model

2007 ◽  
Vol 7 (4) ◽  
Author(s):  
Xiuqing Chen
Keyword(s):  
Diffusion Model ◽  
Weak Solutions ◽  
Semiclassical Limit ◽  
Periodic Boundary Conditions ◽  
Drift Diffusion Model ◽  
Global Weak Solutions ◽  
Drift Diffusion ◽  
Initial Value ◽  
Fourth Order Parabolic System ◽  
Entropy Estimate

AbstractWe establish the global weak solutions to quantum drift-diffusion model, a fourth order parabolic system, in two or there space dimensions with large initial value and periodic boundary conditions and furthermore obtain the semiclassical limit by entropy estimate and compactness argument.

scholarly journals Global Weak Solutions to a Generalized Hyperelastic-rod Wave Equation

2005 ◽  
Vol 37 (4) ◽  
pp. 1044-1069 ◽  
Author(s):  
G. M. Coclite ◽  
H. Holden ◽  
K. H. Karlsen
Keyword(s):  
Wave Equation ◽  
Weak Solutions ◽  
Global Weak Solutions

scholarly journals The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Meng Wu
Keyword(s):  
Sobolev Space ◽  
Nonlinear Equation ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Global Weak Solutions ◽  
Initial Value ◽  
Local Strong Solutions

The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev spaceC([0,T);Hs(R)) ∩C1([0,T);Hs-1(R))provided that the initial value lies inHs(R)withs>3/2. Meanwhile, we prove the existence of global weak solutions inL∞([0,∞);L2(R))for the equation.

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

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