Weak and strong solvability of parabolic variational inequalities in Banach spaces

Abstract

We consider parabolic variational inequalities having the strong formulation (1) $$ \left\{ {\begin{array}{*{20}c} {\left\langle {u'(t),\,v - \left. {u(t)} \right\rangle + \left\langle {Au(t),} \right.\,v - \left. {u(t)} \right\rangle + \Phi (v) - \Phi (u(t) \geq 0,} \right.} \\ {\forall v \in V^{**} ,\,a.e.\,t \geq 0,} \\ \end{array} } \right. $$ where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacI % cacaaIWaGaaiykaiabg2da9iaadwhadaWgaaWcbaGaaGimaaqabaaa % aa!3BE1! $$u(0) = u_0 $$ for some admissible initial datum, V is a separable Banach space with separable dual % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvamaaCa % aaleqabaGaaiOkaaaakiaacYcacaWGbbGaaiOoaiaadAfadaahaaWc % beqaaiaacQcacaGGQaaaaOGaeyOKH4QaamOvamaaCaaaleqabaGaai % Okaaaaaaa!3FF3! $$V^* ,A:V^{**} \to V^* $$ is an appropriate monotone operator, and % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyKaai % OoaiaadAfadaahaaWcbeqaaiaacQcacaGGQaaaaOGaeyOKH46efv3y % SLgznfgDOjdaryqr1ngBPrginfgDObcv39gaiuaacqWFDeIucqGHQi % cYcaGG7bGaeyOhIuQaaiyFaaaa!4C4A! $$\Phi :V^{**} \to \mathbb{R} \cup \{ \infty \} $$ is a convex, % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4Daiaabw % gacaqGHbGaae4AamaaCaaaleqabaGaaiOkaaaaaaa!3A7D! $${\text{weak}}^* $$ lower semicontinuous functional. Well-posedness of (1) follows from an explicit construction of the related semigroup % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4Eaiaado % facaGGOaGaamiDaiaacMcacaGG6aGaamiDaiabgwMiZkaaicdacaGG % 9bGaaiOlaaaa!4001! $$\{ S(t):t \geq 0\} .$$ Illustrative applications to free boundary problems and to parabolic problems in Orlicz-Sobolev spaces are given.