International Journal of Mathematics and Mathematical Sciences
Volume 3 (1980), Issue 1, Pages 79-102
doi:10.1155/S0161171280000063
Existence and decay of solutions of some nonlinear parabolic variational inequalities
Mitsuhiro Nakao1
and Takashi Narazaki2
1Department of Mathematics, College of General Education, Kyushu University, Fukuoka, Japan
2Department of Mathematical Sciences, Tokai University, Kanagawa, Japan
Abstract
This paper discusses the existence and decay of solutions u(t) of the variational inequality of parabolic type: <u′(t)+Au(t)+Bu(t)−f(t), v(t)−u(t)>≧0for ∀v∈Lp([0,∞);V)(p≧2) with v(t)∈K a.e. in [0,∞), where K is a closed convex set of a separable uniformly convex Banach space V, A is a nonlinear monotone operator from V to V* and B is a nonlinear operator from Banach space W to W*. V and W are related as V⊂W⊂H for a Hilbert space H. No monotonicity assumption is made on B.