On properties of solutions to the improved modified Boussinesq equation

Volume 9, Issue 12, pp 6004--6020 http://dx.doi.org/10.22436/jnsa.009.12.08

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Authors

Yuzhu Wang - School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China. Yinxia Wang - School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450011, China.

Abstract

In this paper, we investigate the Cauchy problem for the generalized IBq equation with damping in one dimensional space. When \(\sigma = 1\), the nonlinear approximation of the global solutions is established under small condition on the initial value. Moreover, we show that as time tends to infinity, the solution is asymptotic to the superposition of nonlinear diffusion waves which are given explicitly in terms of the selfsimilar solution of the viscous Burgers equation. When \(\sigma\geq 2\), we prove that our global solution converges to the superposition of diffusion waves which are given explicitly in terms of the solution of linear parabolic equation.

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Keywords

• IMBq equation with damping
• large time behavior
• diffusion waves.

MSC

•  35L30
•  35L75

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