Optimal decay rates of solutions to dissipative nonlinear evolution equations with ellipticity

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Abstract

We derive the optimal decay rates of solution to the Cauchy problem for a set of nonlinear evolution equations with ellipticity and dissipative effects {ψt = -(1 - α)ψ - θx + αψxx, θt = -(1 - β)θ + vψx + 2ψθx + αθxx, with initial data (ψ, θ)(x, 0) = (ψ0(x), θ0(x)) → (ψ±, θ ±) as x → ±∞, where α and v are positive constants such that α < 1, v < α(1 - α), which is a special case of (1.1). We show that the solution to the system decays with the same rate to that of its associated homogenous linearized system. The main results are obtained by the use of Fourier analysis and interpolation inequality under some suitable restrictions on coefficients α and v. Moreover, we discuss the asymptotic behavior of the solution to general system (1.1) at the end.
Original languageEnglish
Pages (from-to)399-418
Number of pages20
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume57
Issue number3
DOIs
Publication statusPublished - 1 May 2006
Externally publishedYes

Keywords

  • Evolution equation
  • Fourier transform
  • Interpolation inequality
  • Optimal decay rate

ASJC Scopus subject areas

  • Mathematics(all)
  • Physics and Astronomy(all)
  • Applied Mathematics

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