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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
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