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Existence and convergence of solutions to a singularly perturbed higher order partially differential equation. (English) Zbl 0831.35010

The authors consider the Cauchy problem (1) \(u_t + Df(u) - \delta D^{4n + 2} u = \varepsilon D^2u\), \(u(0,x) = u_0(x)\), where \(\delta, \varepsilon\) are positive parameters, \(D = d/dx\) and \(f\) is a smooth nonlinear map from \(\mathbb{R}\) to \(\mathbb{R}\).
In the first part of the paper, they study the existence of a solution of problem (1) when \(f\) satisfies certain growth condition at infinity. In the second part, they assume that \(\delta\) is sufficiently small compared to \(\varepsilon\), and they state the existence of a subsequence \(\{u^{\varepsilon_k}_{\delta_k} \}\) of solutions of (1), which converges in the sense of distributions [resp. in some \(L^p\)-space] to a solution of the limit equation \(u_t + Df(u) = 0\), as \(\varepsilon\) goes to 0. The proof is based on a modified version of the Tartar compensated compactness method [L. Tartar, Lect. Notes Math. 665, 228-241 (1978; Zbl 0414.35068)]. Related results have been obtained by M. E. Schonbek [resp. B. Cassis] in Commun. Partial Differ. Equations, 7, 959-1000 (1982; Zbl 0496.35058) [resp. ibid. 14, No. 5, 619-634 (1989; Zbl 0703.35043)].
Reviewer: D.Huet (Nancy)

MSC:

35B25 Singular perturbations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
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