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Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations. (English) Zbl 0765.45009

The authors consider the Volterra equation in a Banach space \(X\): \[ u(t)=f(t)+\int^ t_ 0a(t-s)Au(s)ds,\;t\geq 0 \] where \(f:\mathbb{R}_ +\to X\), \(a\in L^ 1_{loc}(\mathbb{R}_ +)\) and \(A:D(A)\subset X\to X\) is a closed linear operator with dense domain under the assumption of the existence of its resolvent \(S(t)\); they investigate the convergence of \(\lim_{t\to\infty}S(t)x\) for all \(x\in X\) and a general result for the convergence in the strong sense is proved by means of vector valued Laplace transform methods.
A detailed application is given to a model of a linear incompressible viscoelastic fluid governed by the equations \(u_ t(t,x)=\int^ t_ 0\Delta u(t-s,x)da(s)-\nabla p(t,x)+g(t,x)\), \((\nabla\circ u)(t,x)=0\), \(x\in\Omega\), \(t>0\); \(u(t,x)=0\), \(x\in\partial\Omega\), \(t>0\); \(u(0,x)=u_ 0(x)\), \(x\in\Omega\), where \(a,g\) and \(u_ 0\) are given.
Reviewer: G.Di Blasio (Roma)

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45M05 Asymptotics of solutions to integral equations
45K05 Integro-partial differential equations
47G10 Integral operators
47D03 Groups and semigroups of linear operators
76A10 Viscoelastic fluids
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