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Long time behaviour of backward difference type methods for parabolic equations with memory in Banach space. (English) Zbl 0913.65139
The behaviour of the solutions of the initial value problem \[ u_t+ Au= \int^t_0 b(t,\tau) Au(\tau)d\tau+ f(t),\quad t>0,\tag{\(*\)} \] with \(u(0)= u_0\), is studied in a Banach space framework. It is shown, under certain conditions, that \[ \| u(t)\|\leq CM(e^{-\sqrt t}\| u_0\|+ \int^t_0 e^{-\sqrt{(t-2)}}\| f(s)\| ds, \] \(t\geq 0\), where \(u(t)\) is a solution of \((*)\). The discrete analogue of \((*)\) is also studied. The results are applied to derive maximum norm stability estimates for piecewise linear finite element approximations in a plane spatial domain. Error estimates are given.

65R20 Numerical methods for integral equations
45N05 Abstract integral equations, integral equations in abstract spaces
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
45K05 Integro-partial differential equations