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Nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation. (English) Zbl 1207.35056

The authors investigate the nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation. They do not require the strength of the rarefaction waves to be small, and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in \(H^1(\mathbb R)\), while for a general smooth nonlinear flux function, we need to ask for the \(L^2\)-norm of the initial perturbation to be small, but the \(L^2\)-norm of the first derivative of the initial perturbation can be large and, consequently, the \(H^1(\mathbb R)\)-norm of the initial perturbation can also be large. This improves the result of L. Ruan, W. Gao and J. Chen [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 68, No. 2, 402–411 (2008; Zbl 1387.35536)], where rarefaction waves for the generalized KdV-Burgers-Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small.

MSC:

35B35 Stability in context of PDEs
35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations
35Q53 KdV equations (Korteweg-de Vries equations)

Citations:

Zbl 1387.35536
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References:

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