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On parabolic quasi-variational inequalities and state-dependent sweeping processes. (English) Zbl 0923.34018
The authors deal with the evolution problem $-u'(t)\in N_{C(t,u(t))}(u(t)),\quad u(0)= u_0\in C(0,u_0),\tag{$$*$$}$ in the Hilbert space $$H$$, where $$C$$ is a closed convex-valued multifunction and $$N_{C(t,u)}(x)$$ denotes the normal cone to $$C(t,u)$$ at $$x\in C(t,u)$$. Assume that $d(C(t,u), C(s,v))\leq L_1| t-s|+ L_2\| u-v\|,\quad t,s\in [0,T],\quad u,v\in H,$ where $$d$$ stands for the Hausdorff distance.
It is proved that the problem $$(*)$$ has a solution if $$L_2<1$$ and it may have no solution if $$L_2= 1$$. The authors consider also the problem $-v(t)\in N_{\Gamma(v(t))}(v(t))+ f(t),\tag{$$**$$}$ where $$\Gamma$$ is a Lipschitz convex closed-valued multifunction with Lipschitz constant $$L$$. If $$L<1$$ the problem $$(**)$$ admits a solution and if $$L>1$$ the problem $$(**)$$ may have no solutions.

##### MSC:
 34A60 Ordinary differential inclusions 49J40 Variational inequalities 34G20 Nonlinear differential equations in abstract spaces 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations
##### Keywords:
variational inequality; sweeping process; evolution problem
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