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...
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| Format: | Article |
| Language: | English |
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Texas State University
2001-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html |
| _version_ | 1818955902076583936 |
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| author | G. Gripenberg |
| author_facet | G. Gripenberg |
| author_sort | G. Gripenberg |
| collection | DOAJ |
| 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 $$ {partial over partial t} Big( u(t,x) + int_0^t k(t-s) (u(s,x)-u_0(x)),ds Big) + sigma(u)_x(t,x) = 0, $$ 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 $t>0$. |
| first_indexed | 2024-12-20T10:45:26Z |
| format | Article |
| id | doaj.art-2e79842d2ca74dd185210753f0eb529e |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-20T10:45:26Z |
| publishDate | 2001-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-2e79842d2ca74dd185210753f0eb529e2022-12-21T19:43:24ZengTexas 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. |
| spellingShingle | G. Gripenberg Nonsmoothing in a single conservation law with memory Electronic Journal of Differential Equations conservation law discontinuous solution memory. |
| title | 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_short | Nonsmoothing in a single conservation law with memory |
| title_sort | nonsmoothing in a single conservation law with memory |
| topic | conservation law discontinuous solution memory. |
| url | http://ejde.math.txstate.edu/Volumes/2001/08/abstr.html |
| work_keys_str_mv | AT ggripenberg nonsmoothinginasingleconservationlawwithmemory |