Convergence to diffusion waves for nonlinear evolution equations with different end states

Walter Allegretto, Yanping Lin, Zhiyong Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

1 Citation (Scopus)

Abstract

In this paper, we consider the global existence and the asymptotic decay of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects:(E){Mathematical expression} with initial data(I)(ψ, θ) (x, 0) = (ψ0(x), θ0(x)) → (ψ±, θ±) as x → ± ∞, where α and ν are positive constants such that α < 1, s ν < 4 α (1 - α) (s is defined in (1.14)). Under the assumption that | ψ+- ψ-| + | θ+- θ-| 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 similar problem was studied by Tang and Zhao [S.Q. Tang, H.J. Zhao, Nonlinear stability for dissipative nonlinear evolution equations with ellipticity, J. Math. Anal. Appl. 233 (1999) 336-358] for the case of (ψ±, θ±) = (0, 0).
Original languageEnglish
Pages (from-to)244-263
Number of pages20
JournalJournal of Mathematical Analysis and Applications
Volume338
Issue number1
DOIs
Publication statusPublished - 1 Feb 2008
Externally publishedYes

Keywords

  • A priori estimates
  • Decay rate
  • Diffusion waves
  • Energy method
  • Evolution equations

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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