Nonlinear evolution equations generated by subdifferentials with nonlocal constraints
Volume 86 / 2009
Banach Center Publications 86 (2009), 175-194
MSC: Primary 35K45; Secondary 35K50.
DOI: 10.4064/bc86-0-11
Abstract
We consider an abstract formulation for a class of parabolic quasi-variational inequalities or quasi-linear PDEs, which are generated by subdifferentials of convex functions with various nonlocal constraints depending on the unknown functions. In this paper we specify a class of convex functions $\{\varphi^t(v;\cdot)\}$ on a real Hilbert space $H$, with parameters $0\le t \le T$ and $v$ in a set of functions from $[-\delta_0,T]$, $0<\delta_0< \infty,$ into $H$, in order to formulate an evolution equation of the form $$ u'(t)+\partial \varphi^t(u;u(t)) \ni f(t),\, 0< t < T,\hbox{ in }H.$$ Our objective is to discuss the existence question for the associated Cauchy problem.
