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On a shock problem involving a nonlinear viscoelastic bar
Boundary Value Problems volume 2005, Article number: 718156 (2005)
Abstract
We treat an initial boundary value problem for a nonlinear wave equation in the domain
,
. The boundary condition at the boundary point
of the domain for a solution
involves a time convolution term of the boundary value of
at
, whereas the boundary condition at the other boundary point is of the form
with
and
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
, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution
of this problem up to order
in two small parameters
,
.
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Long, N.T., Dinh, A.P.N. & Diem, T.N. On a shock problem involving a nonlinear viscoelastic bar. Bound Value Probl 2005, 718156 (2005). https://doi.org/10.1155/BVP.2005.337
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DOI: https://doi.org/10.1155/BVP.2005.337