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Convergence de la solution de l’équation de Burgers visqueuse non homogène vers une solution stationnaire
Hisao Fujita YASHIMA 1), Nardjes TROUDI 2)
, and Naima KHELOUFI3)
École Normale Supérieure de Constantine, Algérie
Mathematics Subject Classification: (MSC 2010) 35K59, 35B4; 42A75
Key words : Nonhomogeneous viscous Burgers equation, asymptotic
behavior, convergence to the stationary solution.
Abstract:
We consider the nonhomogeneous viscous Burgers equation ∂tu + u∂xu = ε∂x2u + ε2μsinx on ℝ and prove that there exists a number μ > 0 such that, if |μ| < μ, then the solution of this equation converges for t →∞ to the stationary solution u(x) = εw(x) with w(x) which does not depend on ε. To prove it, we transform the equation into a linear parabolic equation and prove the convergence of the solution of the latter equation for t →∞.