# 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(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...

Full description

Main Author: Article English 2001-01-01 Electronic Journal of Differential Equations http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html
id doaj-2e79842d2ca74dd185210753f0eb529e Article doaj-2e79842d2ca74dd185210753f0eb529e2020-11-24T22:39:12ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912001-01-0120010818Nonsmoothing 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(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\$. http://ejde.math.txstate.edu/Volumes/2001/08/abstr.htmlconservation lawdiscontinuous solutionmemory. DOAJ English Article DOAJ G. Gripenberg G. Gripenberg Nonsmoothing in a single conservation law with memory Electronic Journal of Differential Equations conservation law discontinuous solution memory. G. Gripenberg G. Gripenberg Nonsmoothing in a single conservation law with memory Nonsmoothing in a single conservation law with memory Nonsmoothing in a single conservation law with memory Nonsmoothing in a single conservation law with memory Nonsmoothing in a single conservation law with memory nonsmoothing in a single conservation law with memory Texas State University Electronic Journal of Differential Equations 1072-6691 2001-01-01 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\$. conservation law discontinuous solution memory. http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html AT ggripenberg nonsmoothinginasingleconservationlawwithmemory 1725710252695879680