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...
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Texas State University
20010101

Series:  Electronic Journal of Differential Equations 
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Online Access:  http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html 
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doaj2e79842d2ca74dd185210753f0eb529e20201124T22:39:12ZengTexas State UniversityElectronic Journal of Differential Equations107266912001010120010818Nonsmoothing in a single conservation law with memoryG. GripenbergIt 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 $t>0$. http://ejde.math.txstate.edu/Volumes/2001/08/abstr.htmlconservation lawdiscontinuous solutionmemory. 
collection 
DOAJ 
language 
English 
format 
Article 
sources 
DOAJ 
author 
G. Gripenberg 
spellingShingle 
G. Gripenberg Nonsmoothing in a single conservation law with memory Electronic Journal of Differential Equations conservation law discontinuous solution memory. 
author_facet 
G. Gripenberg 
author_sort 
G. Gripenberg 
title 
Nonsmoothing in a single conservation law with memory 
title_short 
Nonsmoothing in a single conservation law with memory 
title_full 
Nonsmoothing in a single conservation law with memory 
title_fullStr 
Nonsmoothing in a single conservation law with memory 
title_full_unstemmed 
Nonsmoothing in a single conservation law with memory 
title_sort 
nonsmoothing in a single conservation law with memory 
publisher 
Texas State University 
series 
Electronic Journal of Differential Equations 
issn 
10726691 
publishDate 
20010101 
description 
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 $t>0$. 
topic 
conservation law discontinuous solution memory. 
url 
http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html 
work_keys_str_mv 
AT ggripenberg nonsmoothinginasingleconservationlawwithmemory 
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