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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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