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.

Show me the PDF file (133K), TEX file, and other files for this article.

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

Return to the EJDE web page