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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.

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