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Maximal regularity for nonautonomous evolution equations. (English) Zbl 1072.35103
This paper concerns perturbation results for maximal $$L_p$$ regularity of operators $A\in L_1(J, {\mathcal L}(E_1,E_0))\cap {\mathcal L}(W^1_p(J,E_0)\cap L_p(J,E_1), L_p(J,E_0)),$ i.e. for the property that $$\partial_t+A$$ is an isomorphism from $$\{u\in W^1_p(J,E_0)\cap L_p(J,E_1)):u(0)=0\}$$ onto $$L_p(J,E_0)$$. Here $$J:=[0,T]$$, $$T>0$$, $$1<p<\infty$$, $$E_0$$ and $$E_1$$ are Banach spaces, and $$E_1$$ is continuously embedded into $$E_0$$ and dense in $$E_0$$. In particular, it is shown that $$A$$ possesses the maximal $$L_p$$ regularity property if $$\tau\in J\mapsto A(\tau)\in{\mathcal L}(E_1,E_0))$$ is continuous and if all the constant operators $$A(\tau)$$ posses the maximal $$L_p$$ regularity property. The main idea is to use stability results for bounded invertibility, based on Neumann series.

##### MSC:
 35K90 Abstract parabolic equations 47D06 One-parameter semigroups and linear evolution equations
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