On a shock problem involving a nonlinear viscoelastic bar
We treat an initial boundary value problem for a nonlinear wave equation utt−uxx+K|u|αu+λ|ut|βut=f(x,t) in the domain 0<x<1, 0<t<T. The boundary condition at the boundary point x=0 of the domain for a solution u involves a time convolution term of the boundary value of u at...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2005-11-01
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| Series: | Boundary Value Problems |
| Online Access: | http://dx.doi.org/10.1155/BVP.2005.337 |
| _version_ | 1831547152276389888 |
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| author | Tran Ngoc Diem Alain Pham Ngoc Dinh Nguyen Thanh Long |
| author_facet | Tran Ngoc Diem Alain Pham Ngoc Dinh Nguyen Thanh Long |
| author_sort | Tran Ngoc Diem |
| collection | DOAJ |
| description | We treat an initial boundary value problem for a nonlinear wave equation utt−uxx+K|u|αu+λ|ut|βut=f(x,t) in the domain 0<x<1, 0<t<T. The boundary condition at the boundary point x=0 of the domain for a solution u involves a time convolution term of the boundary value of u at x=0, whereas the boundary condition at the other boundary point is of the form ux(1,t)+K1u(1,t)+λ1ut(1,t)=0 with K1 and λ1 given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of α=β=0, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution (u,P) of this problem up to order N+1 in two small parameters K, λ. |
| first_indexed | 2024-12-17T01:57:58Z |
| format | Article |
| id | doaj.art-d159cf1b5d02473991a53da46acfc18e |
| institution | Directory Open Access Journal |
| issn | 1687-2762 1687-2770 |
| language | English |
| last_indexed | 2024-12-17T01:57:58Z |
| publishDate | 2005-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-d159cf1b5d02473991a53da46acfc18e2022-12-21T22:07:55ZengSpringerOpenBoundary Value Problems1687-27621687-27702005-11-012005333735810.1155/BVP.2005.337On a shock problem involving a nonlinear viscoelastic barTran Ngoc DiemAlain Pham Ngoc DinhNguyen Thanh LongWe treat an initial boundary value problem for a nonlinear wave equation utt−uxx+K|u|αu+λ|ut|βut=f(x,t) in the domain 0<x<1, 0<t<T. The boundary condition at the boundary point x=0 of the domain for a solution u involves a time convolution term of the boundary value of u at x=0, whereas the boundary condition at the other boundary point is of the form ux(1,t)+K1u(1,t)+λ1ut(1,t)=0 with K1 and λ1 given nonnegative constants. We prove existence of a unique solution of such a problem in classical Sobolev spaces. The proof is based on a Galerkin-type approximation, various energy estimates, and compactness arguments. In the case of α=β=0, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution (u,P) of this problem up to order N+1 in two small parameters K, λ.http://dx.doi.org/10.1155/BVP.2005.337 |
| spellingShingle | Tran Ngoc Diem Alain Pham Ngoc Dinh Nguyen Thanh Long On a shock problem involving a nonlinear viscoelastic bar Boundary Value Problems |
| title | On a shock problem involving a nonlinear viscoelastic bar |
| title_full | On a shock problem involving a nonlinear viscoelastic bar |
| title_fullStr | On a shock problem involving a nonlinear viscoelastic bar |
| title_full_unstemmed | On a shock problem involving a nonlinear viscoelastic bar |
| title_short | On a shock problem involving a nonlinear viscoelastic bar |
| title_sort | on a shock problem involving a nonlinear viscoelastic bar |
| url | http://dx.doi.org/10.1155/BVP.2005.337 |
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