Asymptotic behavior of solution to nonlinear evolution equations with damping

Walter Allegretto, Yanping Lin, Zhiyong Zhang

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4 Citations (Scopus)

Abstract

In this paper, we consider the global existence and the asymptotic behavior of solutions to the Cauchy problem for the following nonlinear evolution equations with ellipticity and dissipative effects(E){(ψt= - (1 - α) ψ - θx+ α ψx x, (t, x) ∈ (0, ∞) × R,; θt= - (1 - α) θ + ν ψx+ 2 ψ θx+ α θx x,) 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, ν < 4 α (1 - α). Under the assumption that | ψ+- ψ-| + | θ+- θ-| is sufficiently small, we show that if the initial data is a small perturbation of the parabolic system defined by (2.4) which are obtained by the convection-diffusion equations (2.1), and solutions to Cauchy problem (E) and (I) tend asymptotically to the convection-diffusion system with exponential rates. Precisely speaking, we derive the asymptotic profile of (E) by Gauss kernel G (t, x) as follows:{Mathematical expression}. The same 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], Nishihara [K. Nishihara, Asymptotic profile of solutions to nonlinear dissipative evolution system with ellipticity, Z. Angew. Math. Phys. 57 (4) (2006) 604-614] for the case of (ψ±, θ±) = (0, 0).
Original languageEnglish
Pages (from-to)344-353
Number of pages10
JournalJournal of Mathematical Analysis and Applications
Volume347
Issue number1
DOIs
Publication statusPublished - 1 Nov 2008
Externally publishedYes

Keywords

  • Asymptotic profile
  • Convection-diffusion system
  • Decay rate
  • Evolution equations

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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