Nonsmoothing in a single conservation law with memory
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 $$ {partial over partial t} Big( u(t,x) + int_0^t k(ts) (u(s,x)u_0(x)),ds Big) + sigma(u)_x(t,x) = 0, $$ where $t>0$ and $xin mathbb{R}$, is not...
Main Author:  

Format:  Article 
Language:  English 
Published: 
Texas State University
20010101

Series:  Electronic Journal of Differential Equations 
Subjects:  
Online Access:  http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html 
Summary:  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 $$ {partial over partial t} Big( u(t,x) + int_0^t k(ts) (u(s,x)u_0(x)),ds Big) + sigma(u)_x(t,x) = 0, $$ where $t>0$ and $xin mathbb{R}$, 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 $t>0$. 

ISSN:  10726691 