Stability of a problem from viscoelasticity. (English) Zbl 1015.35013
This paper deals with a model hyperbolic equation in \( {\mathbb R}^{2} \) with nonlocal term and a nonlinear term of the type \( f(m) \). The equation mentioned above describes the damped motion of an unbounded one dimensional viscoelastic body on which a kind of attracting force is acting. The author reduces the study of this equation to the investigation of a first order integro-differential system having initial-history data. In Theorem 1 a local existence in time result is shown. In Theorem 2 a global existence result is announced and it is proved the uniform stability in \( x \) of the constant solution for \( t \rightarrow \infty \).
Reviewer: Petar Popivanov (Sofia)
MSC:
35B35 | Stability in context of PDEs |
35B40 | Asymptotic behavior of solutions to PDEs |
35L60 | First-order nonlinear hyperbolic equations |