Weak and strong solvability of parabolic variational inequalities in Banach spaces
Tóm tắt
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
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$$u(0) = u_0 $$
for some admissible initial datum, V is a separable Banach space with separable dual
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$$V^* ,A:V^{**} \to V^* $$
is an appropriate monotone operator, and
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$$\Phi :V^{**} \to \mathbb{R} \cup \{ \infty \} $$
is a convex,
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$${\text{weak}}^* $$
lower semicontinuous functional. Well-posedness of (1) follows from an explicit construction of the related semigroup
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$$\{ S(t):t \geq 0\} .$$
Illustrative applications to free boundary problems and to parabolic problems in Orlicz-Sobolev spaces are given.