Evolution equations governed by Lipschitz continuous non-autonomous forms
Tóm tắt
We prove L
2-maximal regularity of the linear non-autonomous evolutionary Cauchy problem
$$\dot u(t) + A(t)u(t) = f(t){\text{ for a}}{\text{.e}}{\text{. }}t \in \left[ {0,T} \right],{\text{ }}u(0) = {u_0}$$
, where the operator A(t) arises from a time depending sesquilinear form a(t, ·, ·) on a Hilbert space H with constant domain V. We prove the maximal regularity in H when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of H.
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