Hai-Xia Meng, Yu-Xia Wang
Abstract:
In this article, we give a variational formulation for traveling wave
solutions that decay exponentially at one end of the cylinder for
parabolic equations. The variational formulation allows us to obtain the
monotone dependence of the velocity on the domain and the nonlinearity,
since the velocity is related to the infimum. In particular, we apply this
method to Ginzburg-Landau-type problems and a scalar
reaction-diffusion-advection equation in infinite cylinders.
For the former, we not only obtain the existence, non-existence,
boundedness and regularity of the solutions,
but also obtain the monotone dependence of the velocity on
the nonlinearity and the domain. For the later, we obtain the monotone
dependence of the velocity on the nonlinearity and the domain besides the
existence, uniqueness, monotonicity and asymptotic behavior at infinity
of the solutions. Moreover, we deduce that the influence of the advection
on the traveling waves is different from a flow along the cylinder axis
considered in many articles.
Submitted February 28, 2014. Published June 20, 2014.
Math Subject Classifications: 35K57.
Key Words: Variational formulation; traveling waves; wave velocity.
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Hai-Xia Meng School of Mathematics and Physics, Lanzhou Jiaotong University Lanzhou, Gansu 730070, China email: menghx08@lzu.edu.cn |
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| Yu-Xia Wang School of Mathematical Sciences University of Electronic Science and Technology of China Chengdu, Sichuan 611731, China email: yxwang_10@lzu.edu.cn |
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