Asymptotics of dissipative nonlinear evolution equations with ellipticity: different end states

作者：Duan RJ; Zhu CJ^*

来源：Journal of Mathematical Analysis and Applications, 2005, 303(1): 15-35.

DOI：10.1016/j.jmaa.2004.06.007

• evolution equations
• diffusion waves
• decay rate
• energy method
• a priori estimates

In this paper, we consider the global existence and asymptotic behaviors of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:
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with initial data
(psi, theta)(x, 0) = (psi(0)(x), theta(0)(x)) --> (psi(+/-), theta(+/-)) as x --> +/-infinity, (I)
where alpha and nu are positive constants such that alpha < 1, nu < 4alpha(1, alpha). Under the assumption that \psi(+) - psi(-)\ + \nutheta(+) - theta(-)\ is sufficiently small, we show that if the initial data is a small perturbation of the diffusion waves defined by (2.5) which are obtained by the diffusion equations (2.1), solutions to Cauchy problem (E) and (I) tend asymptotically to those diffusion waves with exponential rates. The analysis is based on the energy method. The same problem was studied by Tang and Zhao [J. Math. Anal. Appl. 233 (1999) 336-358] for the case of (psi(+/-), theta(+/-)) = (0, 0).

• 出版日期2005-3-1
• 单位华中师范大学