Electron. J. Diff. Eqns., Vol. 2001(2001), No. 08, pp. 1-8.

Nonsmoothing in a single conservation law with memory

G. Gripenberg

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
$$ {\partial \over \partial t} \Big( u(t,x) + \int_0^t k(t-s) (u(s,x)-u_0(x))\,ds \Big) + \sigma(u)_x(t,x) = 0, $$
where $t greater than 0$ 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 $t greater than 0$.

Submitted September 11, 2000. Published January 11, 2001.
Math Subject Classifications: 35L65, 35L67, 45K05.
Key Words: conservation law, discontinuous solution, memory.

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Gustaf Gripenberg
Institute of Mathematics
Helsinki University of Technology
P.O. Box 1100, FIN-02015 HUT, Finland
e-mail: gustaf.gripenberg@hut.fi
http://www.math.hut.fi/~ggripenb

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