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Solution semigroup and invariant manifolds for functional equations with infinite delay. (English) Zbl 0798.35153
Fully nonlinear diffusion-type equations with infinite delay of the form \[ \dot u(t)= Au(t)+ Lu_ t+ g(u(t), u_ t)\tag{1} \] are considered and it is shown that a resolvent operator \(R\) satisfying \(\dot R(t)= AR(t)+ LR_ t\), \(R(0)= I\), \(R_ 0= 0\), exists by considering interpolation spaces between \(D(A)\) and \(X\). Then the solution of (1) is given by \[ u(t)= R(t) x+\int^ t_ 0 R(t- s)(L\phi_ s+ h(s))ds, \] where \(h(s)= g(u(s),u_ s)\). However, \(R\) is not a semigroup. In fact, it is shown that \[ S(t)\left({x\atop \phi}\right)= \left({R(t)x+ \textstyle{\int^ t_ 0}R(t- s)L\phi_ sds\atop \phi_ t+ R_ tx+\textstyle{\int^ t_ 0} R_{t-s} L\phi_ s ds}\right) \] is a semigroup with the usual properties. This enables the existence of stable and unstable manifolds to be proved.

35R10 Functional partial differential equations
45K05 Integro-partial differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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