Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 08, pp. 1-8.
Title: Nonsmoothing in a single conservation law with memory
Author: G. Gripenberg (Helsinki Univ. of Technology, Finland)
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 $t>0$ 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 $t>0$.
Submitted September 11, 2000. Published January 11, 2001.
Math Subject Classifications: 35L65, 35L67, 45K05.
Key Words: conservation law; discontinuous solution; memory.