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...
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2005-11-01
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Series: | Boundary Value Problems |
Online Access: | http://dx.doi.org/10.1155/BVP.2005.337 |
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doaj-d159cf1b5d02473991a53da46acfc18e2020-11-24T22:01:28ZengSpringerOpenBoundary 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 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Tran Ngoc Diem Alain Pham Ngoc Dinh Nguyen Thanh Long |
spellingShingle |
Tran Ngoc Diem Alain Pham Ngoc Dinh Nguyen Thanh Long On a shock problem involving a nonlinear viscoelastic bar Boundary Value Problems |
author_facet |
Tran Ngoc Diem Alain Pham Ngoc Dinh Nguyen Thanh Long |
author_sort |
Tran Ngoc Diem |
title |
On a shock problem involving a nonlinear viscoelastic bar |
title_short |
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_sort |
on a shock problem involving a nonlinear viscoelastic bar |
publisher |
SpringerOpen |
series |
Boundary Value Problems |
issn |
1687-2762 1687-2770 |
publishDate |
2005-11-01 |
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. |
url |
http://dx.doi.org/10.1155/BVP.2005.337 |
work_keys_str_mv |
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