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Effects of a saturating dissipation in Burgers-type equations. (English) Zbl 0888.35097
The author considers a model equation \[ u_t+ f(u)_x= \nu Q(u_x)_x,\quad \nu>0, \] assuming that \(f\) is a smooth function and that the flux function \(Q\) satisfies the conditions \[ |Q(s)|\leq 1,\;Q'(s)>0\quad\text{for all }s,\quad Q'(s)\to 0\quad\text{as }|s|\to\infty. \] A purpose of the paper is to study an interaction between nonlinear convection and nonlinear diffusion with a saturating dissipation flux. It is proved that if the downstream state is below a critical threshold, the upstream-downstream transition is smooth; above this threshold a part of this transition must be accomplished via a discontinuous jump. It is proved also that the solution to a corresponding Cauchy problem with sufficiently small compact or periodic data preserves its initial regularity.

35Q53 KdV equations (Korteweg-de Vries equations)
76R99 Diffusion and convection
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