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Reliable solution of parabolic obstacle problems with respect to uncertain data. (English) Zbl 1099.35054
A parabolic initial-value variational inequality is considered in an abstract setting. To prove that this problem has a unique solution \(u\), a penalized equation is introduced and its penalization parameter is let to pass to zero. In the next part of the paper, the input data of the variational inequality are assumed to be uncertain, i.e., the convex set of test and trial functions, the initial state, the operator on the left-hand side, and the right hand-side of the inequality can depend on a parameter \(\omega \) from a compact admissible set \(A\). As a consequence, \(u\) is \(\omega \)-dependent. Moreover, a criterion functional \(\Phi \) dependent on both \(u\) and \(\omega \) is considered. The goal is to maximize \(\Phi (\omega ,u(\omega ))\) over \(A\). Still in an abstract setting, it is shown that the maximization problem has at least one solution. In the final part of the paper, the general abstract scheme is applied to a particular obstacle problem.

35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
49J40 Variational inequalities
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