Nonsmoothing in a single conservation law with memory

• Main Author: G. Gripenberg
• Format: Article
• Language: English
• Published: Texas State University 2001-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: conservation law; discontinuous solution; memory.
• Online Access: http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html

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 $xin mathbb{R}$, 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 $t>0$.