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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.

##### MSC:
 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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