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Breakdown in Burgers-type equations with saturating dissipation fluxes. (English) Zbl 0946.35039
Convection-diffusion equations of the form \(u_t+f(u)_x =Q(u_x)_x\) show an interesting feature that is different from the conventional Burgers equation: While solutions of the Cauchy problem with small initial data remain smooth, solutions with large initial data may develop discontinuities in finite time. The authors restrict themselves to the symmetric situation where \(f\) and \(Q\) are odd functions of their arguments and the initial condition is an odd function, too. Under these assumptions two situations are considered: If \(Q\) is either monotone and bounded or of the special form \(Q(s)= \nu{s \over 1+s^2}\), then a large class of initial data leads to a blow up of gradients within finite time. The same holds for the boundary value problem with inhomogeneous boundary conditions. Numerical experiments illustrate the results.

MSC:
35K55 Nonlinear parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
35L67 Shocks and singularities for hyperbolic equations
35K15 Initial value problems for second-order parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
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