Weak formulation of stochastic parabolic systems with subdifferential boundary conditions

Abstract

AbstractThis paper discusses the mathematical formulation of the stochastic parabolic system, which is described by the multivalued operator called subdifferential operator, that plays an important role in the modeling of the free boundary problem such as viscoelastic and obstacle problems. For various deterministic distributed systems containing a subdifferential operator, the existence of the strong or weak solution is already shown.With respect to the stochastic distributed system, on the other hand, the existence of a strong solution has not been shown except for the system containing a subdifferential operator of a special convex function. As to the existence of the weak solution, there is a report of the case where the subdifferential operator appears in the interior of the system, but there is no report on the general stochastic system containing a subdifferential operator in the boundary condition. Another point is that a strong constraint is imposed on the initial value and the input function in the proof for the existence of the strong solution.From those viewpoints, this paper aims at ameliorating such a constraint and formulates the system by the weak solution. First, the existence condition for the weak solution is shown. Then it is shown that there exists the maximal solution in the set of weak solutions and an estimate for the maximal solution is derived which has a stronger significance than in the past formulation.