Differential and Integral Equations

Solvability of nonautonomous parabolic variational inequalities in Banach spaces

Matthew Rudd

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We consider nonautonomous parabolic variational inequalities having the strong formulation \begin{align*} \langle { u'(t) }{ v - u(t) } \rangle + \langle { A(t) u(t) }{ v - u(t) }\rangle + \Phi(t, v ) - \Phi(t,u(t)) \geq 0, \\ \forall \, v \in V^{**}, \ a.e. \ t \geq s, \end{align*} where $u(s) = u_{s}$ for some admissible initial datum, $V$ is a separable Banach space with separable dual $V^{*}$, $A(t) : V^{**} \rightarrow V^{*}$ is an appropriate family of monotone operators, and $\Phi(t,\cdot) : V^{**} \rightarrow \mathbb R \cup \{ \infty \}$ is a family of convex, weak* lower-semicontinuous functionals. Well-posedness follows from an explicit construction of the related evolution family $\{ U(t,s) : t \geq s \}$. Illustrative applications are given.

Article information

Differential Integral Equations, Volume 17, Number 9-10 (2004), 1093-1122.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49J40: Variational methods including variational inequalities [See also 47J20]
Secondary: 34G25: Evolution inclusions 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40]


Rudd, Matthew. Solvability of nonautonomous parabolic variational inequalities in Banach spaces. Differential Integral Equations 17 (2004), no. 9-10, 1093--1122. https://projecteuclid.org/euclid.die/1356060315

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