On evolution equations governed by non-autonomous forms. (English) Zbl 1343.35142
The authors consider a linear non-autonomous evolutionary Cauchy problem
\[
u_t (t) + A(t)u(t) = f(t)
\]
where \(A\) is an operator arising from a sesquilinear form on a Hilbert space \(X\) and with a constant domain \(V\). Recently Dier proved a maximal regularity result in \(L^2(0,T; V) \cap H^1 (0,T; H)\). In this paper the authors prove a more general result that provides an alternative proof of the result given by Dier. More precisely, they prove that the solutions of an approximate nonautonomous Cauchy problem in which the sesquilinear form is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form.
Reviewer: Vincenzo Vespri (Firenze)
MSC:
| 35K90 | Abstract parabolic equations |
| 35K45 | Initial value problems for second-order parabolic systems |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 34G10 | Linear differential equations in abstract spaces |
| 35K30 | Initial value problems for higher-order parabolic equations |
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