On a shock problem involving a nonlinear viscoelastic bar
<p/> <p>We treat an initial boundary value problem for a nonlinear wave equation <inline-formula><graphic file="1687-2770-2005-718156-i1.gif"/></inline-formula> in the domain <inline-formula><graphic file="1687-2770-2005-718156-i2.gif"/>&...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2005-01-01
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| Series: | Boundary Value Problems |
| Online Access: | http://www.boundaryvalueproblems.com/content/2005/718156 |
| Summary: | <p/> <p>We treat an initial boundary value problem for a nonlinear wave equation <inline-formula><graphic file="1687-2770-2005-718156-i1.gif"/></inline-formula> in the domain <inline-formula><graphic file="1687-2770-2005-718156-i2.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2005-718156-i3.gif"/></inline-formula>. The boundary condition at the boundary point <inline-formula><graphic file="1687-2770-2005-718156-i4.gif"/></inline-formula> of the domain for a solution <inline-formula><graphic file="1687-2770-2005-718156-i5.gif"/></inline-formula> involves a time convolution term of the boundary value of <inline-formula><graphic file="1687-2770-2005-718156-i6.gif"/></inline-formula> at <inline-formula><graphic file="1687-2770-2005-718156-i7.gif"/></inline-formula>, whereas the boundary condition at the other boundary point is of the form <inline-formula><graphic file="1687-2770-2005-718156-i8.gif"/></inline-formula> with <inline-formula><graphic file="1687-2770-2005-718156-i9.gif"/></inline-formula> and <inline-formula><graphic file="1687-2770-2005-718156-i10.gif"/></inline-formula> 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 <inline-formula><graphic file="1687-2770-2005-718156-i11.gif"/></inline-formula>, the regularity of solutions is studied also. Finally, we obtain an asymptotic expansion of the solution <inline-formula><graphic file="1687-2770-2005-718156-i12.gif"/></inline-formula> of this problem up to order <inline-formula><graphic file="1687-2770-2005-718156-i13.gif"/></inline-formula> in two small parameters <inline-formula><graphic file="1687-2770-2005-718156-i14.gif"/></inline-formula>, <inline-formula><graphic file="1687-2770-2005-718156-i15.gif"/></inline-formula>.</p> |
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| ISSN: | 1687-2762 1687-2770 |