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.