Geodesic
Evolution equations governed by Lipschitz continuous non-autonomous forms
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 475-491

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We prove $L^2$-maximal regularity of the linear non-autonomous evolutionary Cauchy \rlap {problem} $$ \dot {u} (t)+A(t)u(t)=f(t) \quad \text {for a.e.\ } t\in [0,T],\quad u(0)=u_0, $$ where the operator $A(t)$ arises from a time depending sesquilinear form $\mathfrak {a}(t,\cdot ,\cdot )$ 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 $H.$
DOI : 10.1007/s10587-015-0188-z
Classification : 35B65, 35K45, 35K90, 47D06
Keywords: sesquilinear form; non-autonomous evolution equation; maximal regularity; convex set 
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     title = {Evolution equations governed by {Lipschitz} continuous non-autonomous forms},
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Sani, Ahmed; Laasri, Hafida. Evolution equations governed by Lipschitz continuous non-autonomous forms. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 475-491. doi: 10.1007/s10587-015-0188-z

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