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On the formation of scalar viscous shocks problem. (English) Zbl 1161.35444

Consider a viscous scalar conservation law with smooth, possibly non-convex flux. Assume that the (arbitrarily large) initial data remains in a small neighborhood of given states \(u', u''\) as \(x \rightarrow \pm \infty\) with \(u'\), \(u''\) connected by a stable shockprofile. We then show that the solution eventually forms a viscous shock. The time needed for the shock to appear is the main focus of the present analysis.

MSC:

35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
35B40 Asymptotic behavior of solutions to PDEs
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