Kenmochi, Nobuyuki

EPrint Series of Department of Mathematics, Hokkaido University

Recent Development of Parabolic Quasi-Variational Inequalities
Nobuyuki Kenmochi
(kenmochi@faculty.chiba-u.jp)
Department of Mathematics, Faculty of Education
Chiba University, Chiba, 263-8522 Japan
In this talk we discuss a class of parabolic quasi-variational inequalities of the form
u′(t) + ∂ϕt(u; u(t))  f(t), 0 < t < T, in X∗,
u(t) = u0(t) for − δ0 ≤ t ≤ 0, inX
under the following setting: X is a real reﬂexive Banach space compactly embedded in a
real Hilbert space H , so X ⊂ H ⊂ X∗, where X∗ is the dual of X and δ0 and T are ﬁxed
positive numbers; ϕs(v; z), −δ0 ≤ s ≤ t, is a proper l.s.c. convex function in z on X for
v belonging to a suitable class of functions from [−δ0, t] into X in which ϕs(v; ·) depends
continuously on v in a non-local way; u0 is a prescribed initial function given on [−δ0, 0].
As a simple example shows, the existence question for the above type of quasi-
variational inequalities are quite delicate. We shall give suﬃcient conditions for the
existence of a local in time solution of our problem. Moreover, in order to illustrate
our conditions we apply our abstract result to a concrete obstacle problem having the
obstacle depending on the unknown function.


Recent development of parabolic quasi-variational inequalities

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EPrint Series of Department of Mathematics, Hokkaido University