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The large-time solution of Burgers’ equation with time-dependent coefficients. II. Algebraic coefficients. (English) Zbl 1356.35209

Summary: In this paper, we consider an initial-value problem for Burgers’ equation with variable coefficients \[ u_t+\Phi(t)uu_x=\Psi(t)u_{xx},\quad -\infty<x<\infty, \quad t>0, \] where \(x\) and \(t\) represent dimensionless distance and time, respectively, while \(\Psi(t),\Phi(t)\) are given continuous functions of \(t(>0)\). In particular, we consider the case when the initial data has algebraic decay as \(|x|\rightarrow \infty\), with \(u(x,t)\rightarrow u_+\) as \(x \rightarrow \infty\) and \(u(x,t) \rightarrow u_{-}\) as \(x \rightarrow - \infty\). The constant states \(u_+\) and \(u_{-}(\neq u_+)\) are problem parameters. We focus attention on the case when \(\Phi(t)=t^{\delta}\) (with \(\delta > -1\)) and \(\Psi(t)=1\). The method of matched asymptotic coordinate expansions is used to obtain the large-\(t\) asymptotic structure of the solution to the initial-value problem over all parameter values.
For Part I see [the second author, Stud. Appl. Math. 136, No. 2, 163–188 (2016; Zbl 1332.35329)].

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35C20 Asymptotic expansions of solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 1332.35329
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References:

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