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Nonsmoothing in a single conservation law with memory. (English) Zbl 0963.35118
Author’s abstract: It is shown that, provided the nonlinearity \(\sigma\) is strictly convex, a discontinuity in the initial value \(u_0(x)\) of the solution of the equation \[ {\partial \over \partial t} ( u(t,x) + \int_0^t k(t-s) (u(s,x) - u_0(x)) ds) + \sigma(u)_x (t,x) = 0, \] where \(t > 0\) and \(x \in \mathbb{R}\), is not immediately smoothed out even if the memory kernel \(k\) is such that the solution of the problem where \(\sigma\) is a linear function is continuous for \(t>0\).
Reviewer: E.Feireisl (Praha)

MSC:
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
45K05 Integro-partial differential equations
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